FIGURING IT OUT

NUNO CRATO

FIGURING IT OUT
ENTERTAINING ENCOUNTERS WITH
EVERYDAY MATH

123
Nuno Crato
Universidade Técnica de Lisboa
Inst. Superior de Economia e Gestão
Dept. Matemática
Rua Miguel Lupi 20
1200 Lisboa
Portugal
ncrato@iseg.utl.pt

Original Portuguese edition published by Gradiva Publicações lda., Lisboa, Portugal,

2008 original title: A Matemática das Coisas

ISBN 978-3-642-04832-6 e-ISBN 978-3-642-04833-3

DOI 10.1007/978-3-642-04833-3
Springer Heidelberg Dordrecht London New York
Library of Congress Control Number: 2010930692

© Springer-Verlag Berlin Heidelberg 2010

This work is subject to copyright. All rights are reserved, whether the whole or part of the
material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data
banks. Duplication of this publication or parts thereof is permitted only under the provisions of
the German Copyright Law of September 9, 1965, in its current version, and permission for use
must always be obtained from Springer. Violations are liable to prosecution under the German
Copyright Law.
The use of general descriptive names, registered names, trademarks, etc. in this publication does
not imply, even in the absence of a specific statement, that such names are exempt from the relevant
protective laws and regulations and therefore free for general use.

Cover design: eStudio Calamar S.L.

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)

 PREFACE

“When I tell people that I am a mathematician, they jokingly ask if

I could help them balance their bank account. Then, when I tell them
I make lots of counting mistakes, they think I must be a pretty mediocre
mathematician.”
That is what one mathematician friend of mine once told me, but it
could as easily have come from just about any mathematician, as almost
everyone in this field complains of how misunderstood the profession
is. There really are a lot of people who have no idea what it is that
mathematicians do.
Math, of course, is an integral part of our daily lives. The 20th cen-
tury could not have been the most revolutionary one hundred years
in the history of science, as indeed it was, without the extraordinary
advances that took place in the field of mathematics. Computers could
not have been created without binary logic, group theory and the
mathematical concept of information. Telephones would not work if
mathematicians hadn’t developed the statistical study of signals and
the algorithms to digitalize and compress data. Automated traffic lights
would no doubt effect chaos, rather than order, if advances in a field of
mathematics called Operations Research had not occurred.
But despite its crucial importance, mathematics is frequently viewed
as an insular, even irrelevant field into which few interesting people
venture, and which has little to contribute to our daily lives. Even the
well-educated often demonstrate a surprising ignorance of the history
of mathematics and its advancements.
I would venture that if you asked an intellectual to name two
or three renowned 20th century philosophers, there would not be
many who could not respond without hesitation. I would also say that
most reasonably educated people could easily name two or three great

v
vi PREFACE

contemporary composers. Many of them would also have little diffi-

culty in identifying half a dozen modern schools of art, from cubism to
minimalism. But mathematicians and fields of mathematics? Few peo-
ple know who David Hilbert was or what the formalist school was, or
the important part that Andrey Kolmogorov and John von Neumann
played with respect to probability studies.
This book is full of stories about math, with few equations, lots of
examples and many applications. Math is a fascinating science, of funda-
mental importance for our history and always present in our daily lives.
Many things would not be possible without math: Picasso’s art, online
bank transactions, house numbers and A4 paper sizes, modern maps
and the defeat of Hitler. Math applications appear where you would least
expect them. The history of math is the history of winners.

Lisboa, Portugal Nuno Crato

 CONTENTS

Preface v

Everyday Matters

The Dinner Table Algorithm 3

Cutting the Christmas Cake 7

Oranges and Computers 11

When Two and Two Don’t Make Four 15

Getting More Intelligent Every Day 19

The Other Lane Always Goes Faster 23

Shoelaces and Neckties 27

Number Puzzles 31
Tossing a Coin 35

The Switch 39
Eubulides, The Heap and The Euro 43

The Earth is Round

How GPS Works 49

Gear Wheels 53

February 29 57
The Nonius Scale 61

Pedro Nunes’ Map 65

vii
viii CONTENTS

Lighthouse Geometry 71

Asteroids and Least Squares 75

The Useful Man and the Genius 79

Secret Affairs

Alice and Bob 85

Inviolate Cybersecrets 89

Quantum Cryptography 93

The FBI Wavelet 97

The Enigma Machine 101

Art and Geometry

The Vitruvian Man 107

The Golden Number 111

The Geometry of A4 Paper Sizes 115
The Strange Worlds of Escher 119

Escher and the Möbius Strip 123

Picasso, Einstein and the Fourth Dimension 127

Pollock’s Fractals 131

Voronoi Diagrams 137

The Platonic Solids 141

Pythagorean Mosquitoes 145

The Most Beautiful of All 147

Mathematical Objects

The Power of Math 153

Doubts in the Realm of Certainty 157
 CONTENTS ix

When Chance Enhances Reliability 161

The Difficulty of Chance 167

Conjectures and Proofs 171

Mr. Benford 173

Financial Fractals 179

Turing’s Test 183

DNA Computers 187

Magical Multiplication 191

π Day 193
The Best Job in the World 197

Out of this World

Electoral Paradoxes 201

The Melon Paradox 205

The Cupcake Paradox 207

Infinity 209
Unfair Games 211

Monsieur Bertrand 213

Boy or Girl? 215

A Puzzle for Christmas 217

Crisis Time for Easter Eggs 221

Index 223
EVERYDAY MATTERS
 THE DINNER TABLE ALGORITHM

If you want to invite some friends to a dinner party, but your dining
table will only accommodate four people, then you might be faced with a
dilemma: how do you choose three compatible dinner companions from
among your five closest friends? Your buddy Art has recently broken up
with his girlfriend Betty, who is now dating Charlie. Charlie and Art
have managed to remain friends, but Charlie is not speaking to Dan,
who won’t go anywhere without Eva, who can’t stand Art. So how can
you choose your three dinner companions to have a pleasant, hassle-
free evening? The best way, believe it or not, would be to make use of an
algorithm, which is a set of rules that enable you to search systematically
for an answer.
Algorithms are much loved by mathematicians as well as computer
scientists. Even though some algorithms are very complex, the simplest
can sometimes be the most effective. In our case we can follow a system-
atic process of trial and error, which may be quite an efficient algorithm,
despite its apparent simplicity.
So let us start by choosing your friend A. Under the circumstances,
we immediately see that you cannot possibly also invite your friend B.
You could invite C, but he wouldn’t come unless B was also invited.
And so it goes on. It seems there is no scenario under which A could
be included, which means we need to start again, this time with B, and
keep going until we have found three companionable friends for the din-
ner party. Will that be possible in this case? Or will we have to give up,
forced to admit that human relationships are more complicated than
algorithms?

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_1, 3


C Springer-Verlag Berlin Heidelberg 2010
4 FIGURING IT OUT

This type of problem is known as a satisfiability problem.

Mathematicians call them SAT problems, which keeps things simpler.
The dinner party mentioned above is an example of a “2-SAT” problem,
as each restriction contains two variables (“A or B”, “A and C”, etc.). The
problem would become more complicated if Art, Charlie and Dan were
inseparable, i.e., if we had to take three variables in each restriction into
account (“A and B and C” or “A and C or D”).
Such problems are known as “3-SAT” problems. And it is also pos-
sible to imagine restrictions of a more general type, which give rise to
“k-SAT” problems.
Although this example may seem trivial, a similar approach can be
applied to many basic tasks, such as drawing up timetables in large
schools, organizing conferences, or planning flight schedules for air-
lines. It is the basis of a new branch of mathematics called computational
complexity, which aims to study and classify problems in terms of their
inherent difficulty. When such SAT problems could only be solved
manually, one after the other, it was difficult to study many of their
characteristics. However, from the time it became possible to employ
computers to solve them, attempts have been made to study the com-
plexity of the processes used to solve them, i.e., the algorithms, and to
evaluate the time that it takes a computer to solve them.
In 1959, Richard Karp was still a 24-year-old mathematician who
had just earned a Ph.D. from Harvard and begun to work at the IBM
research laboratory at Yorktown Heights, NY. At the time, computers
were in their infancy, but the invention of transistors made it possible
to incorporate more and more elaborately designed circuits. Karp’s task
at IBM was to find an automatic process for designing circuits with as
few transistors as possible. Written as a computer program, the algo-
rithm he wrote was limited to checking out all the possible circuits
and calculating their costs. Later, in 1985, when he was presented with
the prestigious Turing Award given by the Association for Computing
Machinery, Karp recalled that although this approach seemed simple, it
contained a basic problem: “The number of circuits that the program
had to comb through grew at a furious rate as the number of input vari-
ables increased, and, as a consequence, we could never progress beyond
 THE DINNER TABLE ALGORITHM 5

the solution of toy problems.”1 Karp, who spent 10 years more at IBM
before becoming a professor at Caltech, had identified a phenomenon
that came to occupy the attention of hundreds of researchers and to
generate thousands of studies: the problems might well be simple and
the technique might be easy to apply, but they could rapidly grow to
become impossible to solve, even when using the most powerful com-
puters. Mathematicians, logicians and computer scientists spent many
years subsequently trying to devise more efficient algorithms, but always
arrived at the same result: there are problems that can be resolved sim-
ply and that have a complexity that increases in a controlled fashion,
and there are problems that quickly become impossible to solve because
their complexity increases exponentially with the number of variables
and restrictions.
At present, a distinction is made between the “type P” problems,
in which the complexity increases in polynomial time with the rise
in the number of variables, and the “non-P” type problems, in which
this does not happen. In particular, there is a class of non-P problems
that are all reducible to each other and whose solution can be checked
in polynomial time. These are the so-called “NP-complete” problems
(nondeterministic polynomial). Even though solutions for these prob-
lems can be checked efficiently, to find such solutions there are known
algorithms that increase dramatically in computing time (more than
polynomially) as their dimension grows. These problems thus become
impractical when the number of variables increases. It is still not known
if type NP-complete problems are amenable to a type P approach. This
question was also posed by Karp in 1985 during his Turing Award
speech, but even today remains a major unsolved issue in computer
science. Specialists assume that these are two different and irreducible
types of problems, but they have not been able to prove this yet.
Our dinner table dilemma, which is a 2-SAT problem, is of type P.
Even if we had to select thirty persons from a group of 50 instead of hav-
ing to choose three of our five friends, a computer program could find

1
From Karp’s 1985 Turing Award lecture “Combinatorics, complexity, and randomness”
(in http://awards.acm.org)
6 FIGURING IT OUT

a solution rapidly or indicate that there is no possible solution, which

would be equally important to know.
And if we were, say, holding an event at the UN and had to select
300 persons from a list of 500 possible guests, this would indeed keep
the computer busy for a little longer, but we would still have an answer
in a reasonable amount of time.
Strangely enough, though, we enter another world entirely when we
move on to a 3-SAT problem by inserting restrictions such as “either
not including Art and Betty or Charlie”. We then cross the line dividing
type P problems, for which we will eventually find a solution, from NP-
complete problems, when having a few dozen friends is enough to make
it impossible for any computer in the world to organize our dinner table
in time.
 CUTTING THE CHRISTMAS CAKE

When a small cake has to be cut in two pieces to be shared by two people,
and the person who cuts the cake is also the person who chooses which
half to take, then there is no guarantee that one of the two people will
not be disadvantaged. The best way to avoid any complaints about the
division of the cake is for one person to cut the cake and the other to
choose which half to take. This way, it is in the first person’s interest to
divide the cake as fairly as possible, as otherwise he or she might very
well end up with the smaller piece. It is a wise solution, requiring that
two persons, basically motivated by egotism, cooperate with one another
in such a way that neither is deprived of a fair share.
This well-known anecdote is applicable to many situations in our
day-to-day lives, and not only ones involving cakes. However, the prob-
lem becomes more difficult when the cake has to be divided among more
than two persons. How would you divide a cake among three people, for
instance? Two cut and one chooses? Couldn’t two of them conspire to
deprive the third of a fair share? And what if many more people wanted
a slice of the action? What if a cake had to be shared by twenty equally
sweet-toothed persons?
That is not a trivial problem, and mathematicians are beginning to
develop algorithms for equal shares. These algorithms can be applied in
very diverse areas, ranging from personal matters like the sharing of an
inheritance to affairs of state such as establishing international borders.
The “one cuts, the other chooses” algorithm can be directly applied
to some situations in which more than two people are involved. If four
people want a slice of the cake, for example, the algorithm is applied in

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_2, 7


C Springer-Verlag Berlin Heidelberg 2010
8 FIGURING IT OUT

two steps. We start by assigning the four people to two groups, each con-
taining two members. One of the groups then cuts the cake in two, and
the other group chooses its half. In the second step each group divides
its half of the cake, using the established procedure of one person cutting
and the other choosing.
It is easy to see that this repetitive method can also work well with
eight participants or indeed with any number that is a power of two. It
is not so simple to find a solution when three persons want to share the
cake. But if you think carefully about it, you will find a solution to this
problem. Can you suggest a way?
However, mathematicians don’t like methods that work only in spe-
cial cases; they prefer to devise algorithms that can be more widely
applied. The ideal would be to find methods that could be applied
to any number of persons. One of these methods, first proposed by
the Polish mathematicians Stefan Banach (1892–1945) and Bronislaw
Knaster (1893–1980), resolves the problem utilizing what has been
dubbed the “moving knife procedure”. It is easier to explain if we take a
loaf cake as an example.
The persons who want a slice of the cake gather round it while one
of them begins to slide the knife along the cake. The knife keeps moving
until one of the participants says “Stop!”. At this precise moment the
knife stops moving and a slice is cut from the cake and handed to the
person who said “Stop”. This person then has a slice that he or she con-
siders to be at least a fair share of the cake – if he or she had thought that
the knife had not yet traveled far enough to provide a fair share, then he
or she would have remained silent. Now the others also had the chance
to say “Stop”, but they did not do so. So presumably they did not con-
sider that the slice of cake offered was larger than a fair share – otherwise
they would have claimed this slice.
After being given a slice, the first participant leaves the game while
the knife continues to travel along the cake until one of the remaining
participants says “Stop!” and is given the corresponding slice. This pro-
cess is repeated until only two participants remain in the game. At this
stage, the first person to speak receives the slice that is cut and the other
receives the remainder of the cake.
 CUTTING THE CHRISTMAS CAKE 9

Stop! Stop!

The moving knife method can be used to divide a homogeneous cake into equal
portions for an arbitrary number of persons. One person moves a knife along the
cake until one of the participants says “Stop!” and claims the slice of cake that is cut
at that point. The procedure is continued until another participant claims a slice, and
so on until the cake has been divided into slices for each person

The interesting thing about this method is that, even considering the
fallibility of each of the participants in assessing the right moment to say
“Stop”, none of them can claim that he or she has been disadvantaged.
If any person has not in fact received their fair share, then it is their own
fault, as he or she did not speak up at the right time.
This method seems to be perfect, but it fails to take some inter-
esting aspects into consideration. It works well with a homogeneous
cake, but would it work with a cake that has various ingredients that
are distributed irregularly, like a Christmas cake? Would it be possible
to devise an algorithm that guarantees that each person ends up with
an equal quantity of glacé cherries, almonds, sultanas and dough? An
answer to this question is provided by a theorem the Polish mathemati-
cian Hugo Steinhaus (1887–1972) proved in the 1940’s and that came to
be known by the curious name of the “ham sandwich theorem”. Let us
take a three-dimensional object with three components such as a sand-
wich consisting of bread, butter and ham – it does not matter if these
components are distributed equally or not, are concentrated in different
areas or are spread uniformly. What this theorem proves is that there is
always a plane that divides the object in two halves in such a way that
each half contains an equal quantity of the three components. In other
words, even if the ham or the butter are distributed unequally, there is
always a way to cut the sandwich into two completely equal halves.
10 FIGURING IT OUT

In the case of a two-dimensional object an equal division works

only with two components. Let us suppose that salt and pepper are
spread on a table, for example. Steinhaus’s theorem proves that there
is always a straight line that divides the surface of the table into two sec-
tions containing equal quantities of salt and pepper. If there were three
ingredients, let us say salt, pepper and sugar, it is easy to imagine a con-
centration of the substances in three different places so that it would
be impossible to draw a straight line that would divide them equally.
Generally the theorem states that for n dimensions there is always a
hyperplane that simultaneously divides n components equally. As it
seems that we live in a three-dimensional world, and as the Christmas
cake has many more ingredients than just three, we have just learned
that no knife exists that can cut slices of Christmas cake containing equal
quantities of all the ingredients.
 ORANGES AND COMPUTERS

For more than 2000 years mathematics has been making progress by
means of rigorous proofs, based on explicit assumptions and logical
arguments. The arguments should be faultless. But how can their valid-
ity be checked? This has always been the subject of debate and has never
been completely resolved. The issue was rekindled at the end of the 20th
century, when some prestigious mathematical journals accepted proofs
completed with the help of computers. Should these proofs be accepted
as legitimate? Should they even be considered mathematical proofs?
One of these disputes involved a well-known and easily understood
problem: what is the best way to stack spheres? Is it the way that super-
markets sometimes stack oranges, in little pyramids structured in layers,
with each orange sitting in the space between those on the layer below?
This system seems more efficient than piling one orange exactly on top
of another, for instance. But aren’t there other more efficient ways to
stack them?
Legend has it that this particular mathematical problem originated
in a question that the English explorer Sir Walter Raleigh (1552–1618)
posed to the scientist Thomas Harriot (1560–1621). Raleigh was inter-
ested in finding a procedure for rapidly estimating the quantity of his
munitions. For this purpose, he wanted to be able to calculate the num-
ber of cannonballs in each pile simply by inspecting it, without having
to count them. Harriot was able to provide him with a correct and sim-
ple answer for square pyramidal piles: if each side of the bottom layer of
the pile has k cannonballs, then the stack consists of k(1 + k)(1 + 2 k)/6
cannonballs. So, for instance, if the bottom layer of a square pyramidal

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_3, 11


C Springer-Verlag Berlin Heidelberg 2010
12 FIGURING IT OUT

pile has four balls on each side, then the pile has a total of 30 balls. You
could check this yourself by stacking thirty oranges of your own.
Harriot studied various ways of stacking balls. Years later, he
brought up the problem in a discussion with the German astronomer
Johannes Kepler (1571–1630), who posed an even more interesting
question: what is the most efficient way of packing spheres?
Kepler conjectured that the best way would be to put balls in parallel
layers, with each layer disposed along a hexagonal grid. Balls on layers
below and above should be inserted on the spaces formed by the balls
on the other layers. Kepler concluded that there was no better solution
than this one but he was unable to prove it mathematically. Centuries
passed, and the problem became known as the sphere-packing problem.
The astronomer’s supposition became known as the Kepler conjecture.
It was always admitted that the supposition was true, but nobody ever
succeeded in proving it with absolute certainty.
Then in 1998, Thomas C. Hales, a mathematics professor at the
University of Michigan, surprised the scientific community by provid-
ing a proof. After this, Kepler’s conjecture seemed to have ceased being
a simple hypothesis and to have become a perfectly proven theorem.
However, there was a problem with all this. Just one minor problem. . .
the proof had been derived with the help of a computer.
Hales had explicitly resolved many of the steps that were required
to prove the hypothesis, but he had left others to be tested automati-
cally using software specially written for this purpose. He claimed that
combining the results from the computer with his own work would
unquestionably prove the theorem. This was not the first time that
a proof had been made with the assistance of a computer. In 1976,
Wolfgang Haken and Kenneth Appel, from the University of Illinois,
had also used a computer to attain another of the great goals of math-
ematics – the proof of the four colors theorem, which posits that four
colors are sufficient to color a flat map in such a way that no two adja-
cent regions have the same color. And in 1996 Larry Wos and William
McCune, of the Argonne Laboratory in the USA, used logical software to
provide proof of another famous supposition, the “Robbins conjecture”,
a deep statement in mathematical logic.
 ORANGES AND COMPUTERS 13

As soon as Hales announced his achievement, the Annals of

Mathematics, a prestigious scientific journal, offered to publish his work,
but as is usual in academic circles, only after it had been peer-reviewed,
that is reviewed by fellow experts. It then took years of work before a
panel of 12 experts declared that they had been defeated by the enor-
mity of the task. They confirmed that they were 99% certain that the
proof was valid, but they could not succeed in independently verify-
ing all the steps the computer had performed. The editor of the journal
regretfully wrote to Hales that while the experts had approached their
task with unprecedented vigor, they had become completely exhausted
before being able to complete the verification.
The editors of Annals of Mathematics did eventually decide to accept
the work performed by Hales, though they would only publish those
parts that had been verified via explicit logical reasoning, as is nor-
mal in the field of mathematics. The computational parts of Hales’
proof were published in another, more specialized journal, Discrete and
Computational Geometry. The provision of computer-generated proof
has thus been implicitly admitted into the realm of pure mathematics,
but it continues to be regarded with suspicion. Will this ever change?
 WHEN TWO AND TWO DON’T
MAKE FOUR

Two and two always makes four. But the four can result from the sum
“two plus two” or from the sum “one plus three”. It would seem impos-
sible to differentiate between the two fours. However, this problem has
a tremendous practical importance for statistics.
In 1919, two American political scientists, William Ogburn and Inez
Goltra, published a study on the voting behavior of Oregon women who
had recently registered to vote for the first time. The two investiga-
tors only knew the total number of votes cast in the election, but had
no information on voting patterns according to gender. “Even though
the method of voting makes it impossible to count women’s votes” they
wrote, “one wonders if there is not some indirect method of solving the
problem”.1 They decided to estimate the correlation between the num-
ber of votes cast in each district with the number of women who had
voted in that district. In this way, in the districts with more women,
they could attribute the departures from the mean to the higher num-
ber of women voters. Still, as the investigators themselves conceded,
their method was fallible, as there could have been another explanation:
men could have changed their voting habits in those districts that had a
greater number of women.
The problem of reconstructing individual behavior from aggregate
data came to be known as the ecological inference problem (as ecology
is the science that is concerned with the relationships between the

1
W. F. Ogburn and I. Goltra, Political Science Quarterly 34, 413–433, 1919.

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_4, 15


C Springer-Verlag Berlin Heidelberg 2010
16 FIGURING IT OUT

elements and their environment), but very few basic steps were taken
to solve it.
Thirty years later an American sociologist named William Robinson
published a study that decisively influenced the future methodology
of the social sciences. Essentially, Robinson showed that the existing
methods at that time did not permit the reconstruction of partial data
from aggregate data, and he afterwards coined the expression “ecolog-
ical fallacy” to describe the faulty inferences that could be drawn as a
result. Robinson’s study cast doubt on several strands of sociological
investigation. Geopolitical studies, which were flourishing in France,
Germany and the USA, practically ground to a halt when the validity
of the methods then used was questioned.
However, the ecological inference problem is still a pressing ques-
tion in applied statistics. The questions posed by the studies are too
important for scientists to simply accept that no solution exists. The
prime example that is usually cited is the attempt to understand the
political and electoral success of the Nazi party in the early 1930s. In
this case it is necessary to differentiate between the groups and classes
that supported Hitler’s rise to power. The sociologists have based such
studies on the data for each electoral district, for which only aggregate
data is available. They have no other option.
Another prime example of the importance of ecological inference is
taken from epidemiology. The total number of persons affected during
an outbreak of disease is often known, but the specific areas of the pop-
ulation that are most affected are often much less evident. The data are
aggregated in the hospitals, but in less developed countries it is always
very difficult to process them so that the zones where the epidemic is
spreading most rapidly can be pinpointed quickly. An efficient method
for comparing aggregate data with the existing parceled information
(for instance, in some better-organized health centers) could be used
to detect the origin of the epidemic and to help save many human lives.
Yet another example comes from marketing. The success or failure
of an advertising campaign in attracting new customers can usually be
measured, as can the age and income distribution of the target popu-
lation. Nevertheless, it may be too costly to carry out the research that
 WHEN TWO AND TWO DON’ T MAKE FOUR 17

could pinpoint the age groups and social groups that showed the best
responses to the campaign, so this effort is often not made, even though
the resulting knowledge could provide essential data.
The methods used to date for ecologic inference have not been very
successful, and at times they have even been disastrous. People usually
cite ridiculous examples, such as a study carried out by a group of Israeli
sociologists to forecast the number of voters who would remain loyal to
the Labor Party, which resulted in a negative number of voters! Or the
example of a US opinion-polling company that concluded that 120% of
blacks in Louisiana would vote for the Democrats!
Gary King, a statistician and political scientist at Harvard, has suc-
ceeded in finding new solutions to the ecological inference problem.
True percentage of black voters

Estimated proportion of black voters

Comparing his estimates with results obtained at a later date, Gary King found a
remarkable fit with the actual outcome. In the diagram the 3262 electoral districts in
Louisiana are represented by a number of dots proportional to the number of voters
in each district. Almost all the elements are located along the diagonal line, indicat-
ing that the estimated fraction and the actual fraction of black voters are practically
identical
18 FIGURING IT OUT

His method is much more complex than the normal multiple-variable

procedures, as it is non-linear. The algorithm starts by analyzing the
smallest units that can possibly be obtained.
This data is used to calculate the logical limits for each subgroup.
For example, if a thousand voters voted for a certain candidate, the
number of women who supported this candidate cannot be less than
zero or greater than one thousand. These limits, which may seem trivial,
introduce non-linearities into the statistical tools. The next step in the
algorithm is to estimate a more probable value that maximizes the cor-
relation of the estimated figures for each subgroup with the fragmentary
parceled data that exists on some of the subgroups. Finally, these esti-
mated figures are compared with the known values for some subgroups
and then corrected.
The method is obviously quite complex, and a whole book was
needed to explain it in detail.2 The important fact is that Gary King
tested the algorithm in more than 16,000 cases, and his estimates were
shown to be remarkably close to the actual figures. The American
Political Science Association (APSA) awarded him the Gosnell Prize for
the “best methodological study” of the year, and the US National Science
Foundation (NSF) was equally laudatory. Its director Frank Scioli said
“I expect Gary King’s solution will contribute to the production of more
accurate, insightful data analysis in a variety of research studies, lead-
ing to more informed policy-making and better understanding of our
economy and society, ”3

2
Gary King published his study in A Solution to the Ecological Inference Problem, Princeton
University Press, 1997. The author has also made computer programs that permit the applica-
tion of his method available on the internet. These programs, which run in the Windows envi-
ronment and in the GAUSS language, are available free of charge at http://gking.harvard.edu.
3
Quotation from http://www.nsf.gov/news/news_summ.jsp?cntn_id=102784
 GETTING MORE INTELLIGENT
EVERY DAY

Ever since intelligence tests were first invented, almost one hundred
years ago, there has been a spectacular upsurge in their average results.
The increase has been most surprising in the less specific tests, such as
the intelligence quotient (IQ) tests, which are supposed to assess various
types of intelligence. Could this be true, or are there major errors in the
test concepts? This is a difficult question, and psychologists, statisticians
and psychometricians do not agree on how to interpret the test results.
One thing seems to be certain: the increase in IQ test results is nei-
ther an isolated phenomenon nor a statistical artifact. The tests have
been calibrated over the course of many years and have been taken by
millions and millions of people all over the world, and everywhere the
same phenomenon is observed: average persons who are tested today
using old tests achieve results that would have been classified as highly
intelligent just a few decades ago.
The arguments about what intelligence tests measure are as old as
the tests themselves. Nevertheless, there is not a single scientist who
today asserts that IQ tests are meaningless, just as there is no specialist
who believes that they are infallible.
Intelligence tests, which establish a coefficient internationally
known as IQ, attempt to measure various components such as memory,
reading skills, spatial visualization, and arithmetical and logical capabili-
ties. For a long time, in accordance with the statistical studies performed
by Charles Spearman, it was accepted that these various components
were closely correlated, and that there was a “general g factor” that
underlay all the measurements. The IQ measurement was worked out

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_5, 19


C Springer-Verlag Berlin Heidelberg 2010
20 FIGURING IT OUT

on the basis of a variety of tests. It was thought that the inherent errors
in the different types of tests would even out the results, and the mean
would indicate the g factor that was quantified in the IQ.
The theory of a single intelligence and of a “general g factor” has
come to be viewed with increasing skepticism, however, particularly
after the publication of the studies by Howard Gardner. The alterna-
tive view that he put forward holds that there are different types of
intelligences, what he called multiple intelligences. A person with great
mathematical and logical capabilities, for example, could still be defi-
cient in the area of intuitive interpretation, just as a person with great
powers of spatial visualization might have difficulty understanding ele-
mentary algebra. However, Gardner s followers have also been criticized
for ignoring the correlation between the different aspects of intelligence,
and the view that is prevalent today is more balanced.
Whether a g factor exists or not, the reality is that standardized tests
combine a battery of partial tests that are weighted to produce an overall
measurement. This measurement is then compared with the measured
results attained by a great number of individuals within the same age
group, and is then standardized so that the mean is 100 and the stan-
dard deviation is 15 points. In this way the mean intelligence for each
age group at each period of time is always set at 100. The standardiza-
tion, based on a Gauss curve, ensures that 95% of individuals have an
IQ between 70 and 130. A value below 70 is considered to show mental
deficiency, and a value above 130 to show exceptional intelligence.
The gain in IQ values is apparent when individuals take tests from
bygone days. For example, it has been calculated that American children
today have a mean IQ of 120 when assessed by the criteria used in 1932,
which means that about 25% of them would be considered exceptionally
intelligent, compared with only 2.25% in 1932.
How can this development be explained? Scientists offer different
opinions. One of the most common explanations is that test-taking
strategies have been learned and perfected over time. It seems obvi-
ous that today’s young people and adults are much more familiar
with standardized tests than those who took them at the beginning
of the 20th century. Both scholastic standards and familiarity with
 GETTING MORE INTELLIGENT EVERY DAY 21

9 32 9
1 19 7

70 100 130
Intelligence Coefficient-IQ

When a sample of American children were subjected to the Stanford-Binet intelli-

gence test in 1932, the results showed a good fit with a Gauss curve with a mean
of 100 and a standard deviation of 15. Only 2.25% of the children scored over 130.
When children of the same age took the same test in recent years and the results
were calibrated with the weighting factors used in 1932, the mean was 120, with 25%
of the children scoring more than 130, so they would be considered exceptionally
intelligent. Is that possible?

multiple-choice tests have increased tremendously, which is why this

explanation appears very plausible.
While these factors do have some influence, the problem is that
they only explain a small part of the phenomenon. On the one hand,
an increase in scholastic standards would imply that the tests that have
shown the greatest gains would be those most closely related to learning
themes. But that is not the case. Rather, it is the more basic tests, such
as those based on Raven’s Progressive Matrices, that have resulted in
the highest increases. On the other hand, comparative studies of young
people exposed to different levels of scholastic standards show that
scholastic standards can only explain a small part of the phenomenon,
which means that this factor is not solely responsible for the huge gains
that have been recorded over the past 100 years.
Nor does the explanation involving increased familiarity with stan-
dardized testing seem to hold water. Only a slight difference can be
discerned when the results obtained by intensively coached children
are compared with those of their less well-prepared counterparts. This
22 FIGURING IT OUT

difference in the mean results is limited to 5 or 6 points. Significant dif-

ferences are not achieved even when young people and adults take the
same test multiple times. This fact provides evidence of the consistency
of this type of assessment, and reassures researchers that whatever it is
that IQ tests are measuring, they are measuring it well.
The psychometrician Richard Lynn, from the University of Ulster
in Northern Ireland, has documented that it is improvements in nutri-
tion and hygiene that are largely responsible for vast improvements in
our physical and mental health. Lynn goes on to argue that this has in
turn produced an increase in mean stature, brain size and intellectual
capacity.
Although this explanation seems credible, it too has been the object
of counter-attacks by a great many psychologists and statisticians, who
argue that such an explanation would imply a considerable increase in
the intellectual capacities of one generation compared to the previous
one, and that the facts do not support this. For example, nobody asserts
that the average person today is a genius compared to the average indi-
vidual at the beginning of the last century. The improvement seems to
be limited to a certain type of abstract intelligence shown by testing.
The American psychologist and psychometrician Ulric Neisser,
from Cornell University, has offered a simple and convincing explana-
tion for this phenomenon. Neisser studied the types of tests in which
the gains were most remarkable, and noted that the tests using Raven’s
Progressive Matrices were primarily responsible for the IQ gains. Well,
these tests measure the capacity for abstract reasoning as well as for
interpreting diagrams. Neisser argues that the 20th century saw a real
explosion in the field of audiovisual media, from street advertising to
films, cartoons and computer games. Neisser concluded that there are
different forms of intelligence that develop better according to different
types of experience. In fact, he said, we are much more expert than our
grandparents with regard to visual analysis, but not with regard to other
forms of intelligence.
THE OTHER L ANE ALWAYS GOES FASTER

Jack and Anna leave their respective homes at 8 and have to drive over
a bridge to get to the office where they both work. The traffic begins
to back up long before they reach the bridge, but each of them handles
the situation differently. Whereas Anna remains calm and stays in the
right-hand lane, Jack, whose car is behind hers, soon switches over to
the left-hand lane and overtakes her. Up ahead, his lane comes to a halt,
and Jack is forced to sit there and watch as the cars in the right-hand
lane now pass him. Then, taking advantage of a gap, he abruptly decides
to rejoin the right-hand lane. A bad decision, as right then his new lane
stops again. He waits a short time, frustrated and unable to do anything,
until once more a gap opens up in the other lane. He makes use of it to
change lanes again in an even riskier maneuver than last time. Now, he
feels as if he is gaining ground, until the traffic grinds to a halt yet again.
This pattern is repeated over and over. Anna, on the other hand, simply
stays put in the right-hand lane. Despite all of his risky lane switches,
Jack survives the perils of the road, and eventually gets to work, even
arriving on time. He thinks his driving maneuvers have paid off, until
he sees that Anna has already parked her car and is walking into the
office building.
One way or another, all drivers have experienced this. It is paradox-
ical, but the other lane always seems to go faster. Nevertheless, as soon
as we join this “fast” lane, it turns out that in fact the lane we have just
left is now the faster-moving one. Drivers who constantly change lanes
jeopardize their own safety and that of others, but on average, ironically,
they do not actually end up going any faster.

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_6, 23


C Springer-Verlag Berlin Heidelberg 2010
24 FIGURING IT OUT

This is an intriguing problem. So much so that two statisticians

decided to investigate it using a mathematical model and computer sim-
ulations. Donald A. Redelmeier, from the University of Toronto, and
Robert J. Tibshirani, from Stanford University, jointly published a short
article on the subject in Nature. Later they published a more compre-
hensive study in Chance. Their conclusions are surprising. The illusion
that the other lane is moving faster is based on various objective factors.
Subjective opinions are only part of the problem.
The behavior of drivers in a busy lane is never completely constant.
There are always some whose speed fluctuates, some who accelerate
more fiercely and some more gently, some drivers who are just slightly
anxious, while others lose their cool entirely. To complicate matters, the
distance between vehicles is a (non-linear) function of their speed. The
greater the speed, the greater the distance between the cars has to be.
The result is that, from a given volume of forward-moving traffic, a line
of vehicles moves erratically, stopping and starting, even if there are no
traffic lights or junctions to impede the flow of traffic. When the line
starts going again, the cars do not all advance at the same time. Each
vehicle only moves forward when the vehicle in front moves. When the
line stops, the same thing happens, with the cars at the front stopping
first and those behind only stopping later. That is why a busy line of cars
has an oscillating longitudinal movement.
When two or more adjacent lanes are moving forward, with each of
them oscillating in stop-and-go movements, there are moments when
each car is overtaken by those in another lane and also moments when
the opposite happens, i.e. when each car passes those in the other lanes.
The objective of the statistical study carried out by Redelmeier and
Tibshirani was to compare these moments. For this purpose they used a
computer to simulate two lanes with similar movements to those expe-
rienced by real lanes full of vehicles, but they did not permit drivers to
change lanes.
This simulation is not simple, and it is necessary to include the
inherent randomness of each vehicle. In many traffic studies, such
individual parameters are not taken into account, as only the overall
 THE OTHER LANE ALWAYS GOES FASTER 25

The “Stop-and-go” Phenomenon

Two adjacent lanes of traffic can be moving at the same average speed, but even so each vehicle
spends more time being passed than in passing the others. This is shown by this example with two
adjacent lanes moving erratically in “stop-and-go” mode. In this example no vehicles change lanes.

1st Image: the vehicles in the upper lane are passing those in the lower lane. Vehicle 1 , which had stopped, is starting
to move and vehicle 2 begins to be passed by other vehicles

2nd Image: vehicle 1 stops again, but vehicle 2 continues to be passed by other vehicles

3rd Image: vehicle 2 is no longer being passed

4th Image: the vehicle in the lower lane beside vehicle 1 starts to move, so vehicle 1 starts to be passed by other vehicles

5th Image: vehicle 2 only starts to move now, vehicle 1 continues to be passed by other vehicles

6th Image: vehicle 2 has arrived beside vehicle 1 , which only now stops being passed by other vehicles

At the start the two vehicles were side by side, and they are still side by side at the end, but each of them was moving
for less time than it was being passed

traffic flow is of interest. But it is the unpredictable behavior of individ-

ual drivers that accounts for delays and for slowing down the overall
flow. That is why the study carried out by these two statisticians is so
interesting.
When setting up the simulation to run on the computer, Redelmeier
and Tibshirani concentrated on randomly selected individual vehicles
as reference points and compared two times: the mean time taken by a
vehicle to pass another, and the mean time taken to be passed. That is
when the surprises started.
26 FIGURING IT OUT

As long as the traffic density is low and the traffic flows smoothly,
the model predicts equilibrium between the movement of the two lanes.
The time during which each vehicle is passed, is, on average, more or less
the same as the time that it takes to pass the vehicles in the other lane, so
there is long-term equilibrium. However, when traffic density increases,
the times taken to pass or be passed start to differ, as each vehicle spends
more time being passed than passing others. Despite this, all the vehicles
take the same average time to travel a given distance.
This is a surprising fact. How can we spend more time being passed
by the vehicles in the other lane than in passing them, even though we
all take the same time to travel the same distance?
Although the explanation is simple, it is still difficult to visualize.
When we pass the cars in the other lane, this happens because they have
stopped or are traveling more slowly. At this point their lane is more
bunched up, with less distance between the vehicles, so that we can pass
many vehicles quickly. Let us suppose that we pass 50 in a minute. When
it is our turn to be passed, on the other hand, it is the vehicles in the other
lane passing us that are traveling faster, and so are more widely spaced
out. More time, maybe 2 minutes, is required for an equal number (50)
of vehicles to pass us. For this reason each vehicle spends more time
being passed than in passing the others, but in the end we all arrive at the
same time. What’s the moral of this story? The risks entailed in frequent
lane changes are not usually worth the trouble!
 SHOELACES AND NECKTIES

Mathematicians just love problems taken from real life that are easy to
formulate. Often they turn out to be the most difficult, and therefore
frequently the most interesting. This creates great enthusiasm among
them for such apparently trivial questions as finding the best way to lace
your shoes!
Shoe-lacing patterns have been studied by the mathematician John
Halton, who considered them to be particular cases of the famous travel-
ing salesman problem. This is a well-known and difficult mathematical
problem, inspired by a real-life situation: a salesman wants to pass
through a specific number of towns, visiting each one only once, but his
starting and ending points are fixed. The pathway of a shoelace is equiv-
alent to the salesman’s route, with the eyes (the holes through which
the lace passes) representing the towns. The shortest pathway for the
shoelaces is equivalent to determining the shortest route between all the
towns.
This problem was approached anew by the Australian mathe-
matician Burkard Polster in a study published in Nature, one of the
most prestigious scientific journals in the world. Polster systematically
studied the various ways of lacing shoes.
On the surface, it would seem that there are only one or two
accepted methods of lacing our shoes. However, people in different cul-
tures tend to lace their shoes in many different ways. To take only two
examples, consider the different methods normally used in the USA
and in Europe. In the U.S., shoelaces are usually threaded in opposing
zigzags, and when seen from above they seem to be crossed, while in

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_7, 27


C Springer-Verlag Berlin Heidelberg 2010
28 FIGURING IT OUT

Europe they are typically threaded in alternating zigzags in such a way

that the eyes of the shoes seem joined horizontally by the shoelaces when
viewed from above. There is also the shoe-shop method in which the
shoelace makes a continual zigzag from top to bottom and then returns
in a diagonal line. Which of these do you think is the most efficient
method?
The first curious fact is that there are actually an astronomical num-
ber of options when it comes to lacing shoes. For shoes with two rows
of five eyes, for example, Polster verified that there are 51,840 different
ways to thread the laces. This number rises to the millions when the
number of eyes increases.
Polster restricted himself to ways of threading the laces that neces-
sarily use all the eyes and allow them to be pulled together by applying
pressure to the laces: for example, laces may not pass through three suc-
cessive eyes on the same side, as this would not exert any individual
pressure on the eye in the middle. Then he defined the efficiency crite-
ria. He stated that the security of the binding should be maximized and
the compression of the laces should be minimized.
Comparing the three above-mentioned systems, Polster verified
that the most economical method is the American one, with the second-
best system depending on the number of eyes. If there are four or more
pairs of eyes, the European method is superior to the shoe-shop method.
In the case of three pairs of eyes they are equal. And in the case of only
one or two pairs of eyes, the problem is trivial, as all three methods are
equally good. If you try to verify this, you will see that it is not difficult.
However, Polster did not restrict himself to studying just these three
methods. Taking only the above-mentioned restrictions into account, he
analyzed the problem and discovered that the most economical system
is not, in fact, any of the three commonly used methods. Instead, he
found a less well-known way to lace shoes, called the “bow-tie” method,
which appears to be the most efficient of all.
As far as the criterion of maximum security of the binding was con-
cerned, he did not find any esoteric method, which is comforting. After
all, the American and shoe-shop methods are the best. When the rows
 SHOELACES AND NECKTIES 29

American method European method Shoe-shop method “Bow-tie” method

Four different ways to lace your shoes

of eyes are farther apart, the shoe-shop method is the strongest. When
the rows are close together, the American system is preferable.
Polster was probably inspired to write his mathematical work on
shoelaces by an equally curious study that the computational physicists
Thomas Fink and Yong Mao had published several years previously. It
considered the various ways to tie a tie, a subject that gave rise to a book
the pair published in 1999, called The 85 Ways to Tie a Tie: The Science
and Aesthetics of Tie Knots.
Their study begins with a brief history of neckties and then explores
the mathematical theory of knots. The two physicists try to identify
every possible type of tie knot, but limiting themselves to those that can
be tied in less than 10 moves. Even so, they find 85 different ways of tying
a tie. The simplest of these requires only three moves. You start by plac-
ing the tie with the outward side facing your shirt, and the odd number
of turns ensures that it ends up with the outward side facing outwards,
as usual. This is called the “oriental knot” and is seldom used in west-
ern dress. Next is a four-move knot, which is more widely used. Things
become a bit more complicated when the number of moves required
30 FIGURING IT OUT

to complete the knot increases. One of the more impressive eight-move

knots is the Windsor, which the eponymous Duke actually never used,
but is similar to the bulky knots he used to wear. It comes in handy when
a larger-volume knot is desired. Many other knots are also described.
However, no matter which knot is in or out of fashion, you can be sure
mathematics will always be able to describe it.
 NUMBER PUZZLES

In this case there’s only

LEAD one cell left to complete
the column... a piece of
cake!

LEAD
What’s The grid has a total
the game’s of 81cells, subdivided
in 9 cell squares,
basis? called “regions” The same number, 3,
is on the two rows
The goal is to fill each cell above and in the left
with a number from 1 to 9, column. In this region,
without repeating numbers there’s only one cell
in each region, each column, left to place it.
and each row.

How
to ADVICE
play
Sometimes...
it’s harder to
continue, because
there are several
possibilities. Here, for
example, if we want
In some regions the to introduce number
The whole grid is a Latin
decisions are simple. 1, we’d have four
square: numbers can’t be
In this particular options. Some people
repeated neither in a row
case, numbers 3, choose one cell
nor in a column
6, and 7 are LEAD tentatively, and then
missing. We insert scratch it, if necessary
Another 7 (which has no (in computation we
example other possible position) call this a
and carry on to the 6... “backtracking”).

The puzzle I am about to describe has an intriguing name, and one

you’ve surely heard about. You can write it as “sudoku” or “su doku”,
as you prefer. It comes from the Japanese: “su” means “number”
or “counting”, and “doku” means “single” or “unique”. In order to

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_8, 31


C Springer-Verlag Berlin Heidelberg 2010
32 FIGURING IT OUT

complete this puzzle, you have to insert numbers into empty boxes. And
there is only one solution.
But, contrary to what you might assume, the sudoku did not actually
originate in Japan. Rather, it first appeared in the 1970s in the New York
magazine Dell Pencil Puzzles and Logic Problems. At that time it was
called “Number Place”, not sudoku. In 1984, it appeared in a Japanese
newspaper under the name “Suuji wa dokushin ni kagiru”, which was
subsequently abbreviated to “su doku”, and it soon became a popular
Japanese pastime. In 1997, a retired judge from New Zealand by the
name of Wayne Gould became so enthusiastic about the puzzle that he
began to write a computer program to study it. It took him 6 years to fin-
ish the program, but by then he had become so experienced at sudokus
that he could solve them in record time. Gould then persuaded the
London Times to make use of his expertise and passion for the puzzles,
and when they agreed, sudoku fever hit Europe.
On November 12, 2004, the venerable old Times kicked off the
sudoku phenomenon in Europe by publishing the puzzles in its daily
editions. A few days later, the London Daily Mail hired another puzzle
provider and also started to print sudokus. Many British newspapers,
such as the Sun, Daily Telegraph, Observer, and Guardian, jumped on
the bandwagon, and the result is that today practically every British
newspaper treats its readers to a daily sudoku. In Portugal, where I am
from, various daily papers quickly followed the British example, and the
same happened all across Europe. The U.S. soon joined in the sudoku
craze. The first puzzle was published there in 2004 by the Conway Daily
Sun of New Hampshire. In 2005, the New York Post started publishing
sudokus regularly.
Now, sudokus appear everywhere. Puzzles can even be downloaded
to cellphones. From New Zealand to Serbia, from Israel to South Africa,
millions of people spend leisure time entering numbers in sudokus every
day of the week.
The sudoku is a puzzle that is typical of the 21st century. It consists
of numbers, not words, it can travel the globe extremely rapidly, across
language barriers. You can do sudokus on several websites (for instance
 NUMBER PUZZLES 33

on www.sudoku.com.au) and you can even compete in real time with

people from all over the world (www.sudokufun.com).
Sudokus can range from the easy to the fiendishly difficult. The eas-
iest of them can be solved by anyone in just a few minutes, while the
more complex ones can take even an experienced enthusiast hours to
solve. But only rarely are they so taxing that they make a fan give up in
frustration.
Today, everybody is familiar with the sudoku structure. The puzzle
consists of a large square divided into nine smaller squares on each side,
so it has a total of 81 boxes that have to be completely filled using digits
from 1 to 9 (colors or other symbols are also used in some cases). Each
digit may appear only once in each row and in each column. Technically
sudokus are said to be “Latin squares”.
The history of Latin squares is as old as it is interesting. Apparently
Latin squares were first conceived by the Swiss mathematical genius
Leonhard Euler (1707–1783), in the context of problems affecting
resources. Euler (pronounced “oiler”) posed a problem concerning six
ranks of officers and six types of regiments. He then tried to combine 36
officers within the 6 regiments in such a way that each regiment would
be assigned one officer from each rank. As is normal in the case of math-
ematical problems, Euler formulated various conjectures with regard to
these “magic squares”. One of them, concerning what became known as
“orthogonal squares”, continued to perplex mathematicians throughout
the modern age, until finally S.C, Bose, S.S. Shrikhande, and E.T. Parker
solved the mystery completely in 1960.
Latin squares have also been used in statistics for the design of
experiments. Ronald A. Fisher (1890–1962), for example, who is usu-
ally considered the father of modern statistics, made use of them in
experiments in which three different factors were combined completely.
A classic example is a study of four brands of tires fitted on four differ-
ent vehicles. To prevent either the type of vehicle or the position of the
tire (back or front, right or left) from interfering with the conclusions
drawn from the experiment, one tire from each manufacturer is fitted
to each vehicle, with each brand of tire being in a different position in
34 FIGURING IT OUT

each case. For instance, tire brand A is placed on the left front wheel of
vehicle 1, tire brand B on the right front wheel of the same vehicle, and
so on, always ensuring that two tires of the same brand are never fitted
to the same vehicle and no tire brand is fitted twice to a wheel in the
same position on any of the cars. Fisher then constructed a Latin square
by creating a table with the rows occupied by the tires on each vehicle
and the columns occupied by the four possible positions (front or back,
right or left). It can be seen that the absence of repetitions favors statisti-
cal analysis, as the resistance of each tire brand is evaluated on different
vehicles and in different positions. Thus the wear and tear on the tires
cannot be attributed to differences between the vehicles or differences in
the wheel stress, and the effects of inevitable errors are reduced.
A sudoku is a Latin square that is already partially filled with digits –
the challenge for the player is to enter all the numbers to create a com-
plete Latin square. Even before sudokus were invented, this problem
had been the subject of many scientific studies. Computational scien-
tists showed that it was a difficult problem of the “NP-complete” type.
Curiously, the difficulty in solving each problem of this type depends on
the number of boxes that have already been filled in. As can be easily
understood, if only a few boxes are filled in at the start, the problem is
easy to solve as there are many possible solutions. On the other hand,
if many boxes have been filled in, there are few hypotheses left and the
problem is just as easy to solve. The greatest difficulty arises in inter-
mediate cases, a situation that is known as “phase transition”. In simple
Latin squares the phase transition occurs at or about the magical propor-
tion when 42% of the boxes have been filled in. This is not exactly the
case with the sudoku, as there are additional restrictions, and sudokus
that have been properly designed only have one possible solution.
The great innovation of sudokus is the creation of regions inside
each Latin square. The large square of 81 boxes is divided into nine
smaller squares, each containing nine boxes. You might think that this
would make the problem more complicated, but that is not true. It
becomes easier and also more interesting. Why don’t you try a sudoku?
 TOSSING A COIN

If we toss a coin and it comes up heads, we don’t find this unusual. It is

just as likely that it will come up heads as tails. But if it comes up heads
five times in a row, we would say that that it is unusual. And what if the
order was different? Let’s call heads 1 and tails 0 to make things simpler
from now on. For example, if the sequence was 01001, would that also
be unusual?
This problem confuses many people. Why don’t you try it out? Ask
a friend which of the following sequences is most likely: 11111, 10001
or 10110. You can be quite sure that your friend will say that the first
sequence is the most unusual, that the second is a little more likely, and
that the third is quite normal.
Well, your friend is wrong. Any one of these three events is equally
likely. If you take any sequence of five tosses of a coin, the probability of
it happening is 1/2 × 1/2 × 1/2 × 1/2 × 1/2, which is about 3%. You just
have to remember that it is equally likely that the coin will fall on either
side each time to realize that any sequence is as probable as any other.
What happens is that we see a simple pattern in the sequence with
five consecutive 1 s, and we know that this pattern is unusual. But we
don’t discern any pattern in the 10110 sequence, so we don’t differenti-
ate it from 10010, for instance, or from any other sequence without an
apparent pattern.
If you are still not convinced, why not try this experiment. Write
down a sequence of five zeros and ones. Then toss a coin five times in a
row many times. You will see that it is very unlikely that your previously
defined sequence will appear.

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_9, 35


C Springer-Verlag Berlin Heidelberg 2010
36 FIGURING IT OUT

Mathematicians really like examples featuring coins, as they provide

a simple model for discussing many complex phenomena.
We live in a world of probabilities. Very few things are absolutely
certain, even things we regard as quite sure, such as court judgements.
When the first forensic evidence based on comparison with DNA was
introduced, some lawyers said that it did not constitute absolute proof,
as scientists confirmed that there was a possibility that two persons
could have certain DNA segments that coincided, though they would,
of course, de facto, be different persons. This probability might only
be about one in a hundred million, which is equivalent, when faced
with confirmation by DNA analysis, to almost absolute certainty. But
lawyers and judges are not used to quantifying the probability of errors
in their statements and decisions, and therefore these numbers may not
appear satisfactory to them. Nevertheless, one in a hundred million is
very unusual indeed, so it provides a more certain proof than practically
any error in a human decision.
The example using the coins can be applied to many real-life events.
If you were in a European country where license plate numbers are
sequential, and you saw car number AAA 111 you would be astounded.
However, license plate AAA 111 (if it exists) is just as unlikely as GXF
472 or any other combination of letters and digits. Once again, it is a
question of our tendency to separate what we see as a strange pattern
from things that do not seem to contain any pattern whatsoever.
In our daily lives it is often difficult to consider probability from a
rational standpoint. For example, we might be easily convinced that it is
more likely that a traffic light will turn red as we approach it than that
it will remain green. But we only think that way about it because it is
annoying when a light turns red and forces us to stop, though we hardly
notice the light at all when it is green and we can keep going without
impediment. If we wanted to keep a careful score, we would have to
define the period of time during which the event “the light turned red”
could happen. In fact, if we are driving at a normal speed, the time from
seeing a traffic light to reaching it can be longer than the cycle of the
traffic light. In that case the light will always be red at some time as we
approach it.
 TOSSING A COIN 37

No, there is certainly no god of chance who randomly or consciously

decides to persecute us or favour us. We only see the hand of gods when
our power of reasoning plays tricks on us.
 THE SWITCH

In a marvelous book written several years ago, Witold Rybczynski won-

dered which invention would turn out to be the tool of the millennium.
After checking out various possible contenders, he chose the screw and
the screwdriver. Appropriately enough, his book is called One Good
Turn: A Natural History of the Screwdriver and the Screw.
Rybczynski’s choice is arguable, just like any other. Imagine if you
were asked to make such a choice. What would you select? The tele-
phone? The airplane? The radio? Well, personally I would choose a very
humble instrument indeed – the switch.
To be exact, the switch isn’t really an instrument; it’s more of a
theoretical instrument with many varied practical applications. As an
example, let us take the everyday electrical switch used to turn lights on
and off. Another much more dramatic example of a switch are the points
used in railroad or streetcar tracks. Older readers may well recall these.
In the old days you would have seen the driver or guard get out of a car-
riage carrying a type of rod, walk over to the rails, and insert the rod.
Then, turning it like an old-time automobile starting lever, he would
manually move the points over to the desired track. This was a frequent
scene in cities with streetcars. Today the railroad tracks still have points,
but now they are activated by means of an remotely operated electrome-
chanical system, although I suppose there may still be some manually
operated ones left on little-used tracks.
What the train driver uses to activate these points is simply a com-
mutator, a kind of double switch that selects the track the train is meant
to take. Points are a great invention. Right at the start of the railroad

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_10, 39


C Springer-Verlag Berlin Heidelberg 2010
40 FIGURING IT OUT

era, before they were introduced, railroad tracks could not be inter-
connected. Locomotives that ran on one track could not move to any
other track. The invention of points enormously advanced the range of
applications of the railroads, which in turn would be unthinkable today
without these switches.
In the 19th century, as railroad lines were being developed, the
switch became the basis for a means of communication that advanced in
line with the railroads: the telegraph. This electrical instrument is sim-
ply the long-distance communication of signals controlled by a switch.
At one end of an electric wire an operator moves a “transmitter” (a sort
of crank) that connects and disconnects the electric current and thus
creates impulses. Some impulses are long, some short, representing
the dots and dashes of the Morse code. At the other end of the wire
another operator receives these signals, which arrive in the form of audi-
ble movements of a lever. A switch at one end operates a lever at the
other.
The telegraph was succeeded by the telephone. At the beginning
it was only an object of curiosity, but then it became a communi-
cations system for private lines. Its inventor, Alexander Graham Bell
(1847–1922), started with several telephones on his desk, each con-
nected to a line that went to a different place. This system soon reached
its limits: imagine having to have as many phones on our desks as we
have friends and colleagues we wish to call! It became necessary to find
a system that would be able to route the calls along the lines. In this way
each person would need only one phone and one line. It was essential
to create one or more telephone exchanges that could route the calls by
operating the necessary switches. The first telephone exchange was inau-
gurated in January 1878 in New Haven, Connecticut, and was operated
manually. It actually took a long time before telephone networks were
automated, first using electromechanical and then electronic systems.
Today’s telephone system consists of a gigantic network of switches.
The symbols of our time, the computer and the internet, have
resulted in the greatest concentration of switches of all, involving an
extremely intricate series of connections controlled by super-rapid
switches called transistors.
 THE SWITCH 41

What makes this all so curious is that the future of computer

communications depends on our ability to design ever faster switches.
Today, fiber optics, which provide faster communications, are replacing
electric wires. In recent years, the advances made in fiber optic technol-
ogy have been vastly greater than those in computer chips. However,
an emerging significant obstacle to faster communications is the rel-
atively slow electronic control of the much faster optical signals. The
future will belong to switches that work using light only, that are con-
trolled by light, and eliminate any loss of time during the transformation
of the light signal into an electric signal via electric commutation. The
progress of our communications once again depends on improvements
being made by that wonderful, earlier invention, the switch.
 EUBULIDES, THE HEAP AND THE EURO

The euro coins have been in circulation for a few years now, so people
in the eurozone should all be able to identify them. Why, then, are there
still many people who get confused by them? Some people find it diffi-
cult to distinguish a two-cent coin from 5 cents, while others get the 10
and 20 cent coins mixed up, or confuse the 20 cents with the 50 cents.
Is that our fault, or is it the design of the coins that is to blame for our
confusion?
The designers of the euro coins decided coin sizes should increase
slightly with their value. So, for example, the 20 cent coins are a little
bigger than the 10 cent coins, and 50 cent pieces are also a bit bigger
than the 20 cent coins. All three coins are made from the same alloy. A
similar thing happens with the one and two euro coins, which have a
nickel alloy, and with the one, two, and five cents, which are made from
a copper alloy. It seems to be a rational system, but it doesn’t yield the
best results.
In my country, Portugal, when the escudo was still used, nobody
got the 50 escudo coin mixed up with the 100 escudo coin, even
though the 100 was smaller in diameter than the 50 escudos. In this
case the logical solution of size increasing with value was completely
ignored. Furthermore, these two coins were made from different alloys
and in different styles, which prevented any mix-ups. This principle
is used in other monetary systems too, including in US coins, for
which there is no link between size and value, and different alloys
are used for coins of similar values, which helps to differentiate them.
The fact that in the case of the euro design the same alloys are used

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_11, 43


C Springer-Verlag Berlin Heidelberg 2010
44 FIGURING IT OUT

for adjacent sizes and values of coins does not make them easy to
identify.
This reminds me of the paradox of the heap, formulated more than
2300 years ago by the Greek philosopher Eubulides. Eubulides, a con-
temporary of Aristotle, was born in Miletus and later lived in Megara,
where he was a leading member of the so-called Megarian school of
philosophy. Among other celebrated paradoxes, he created the “sorites”
paradox (a Greek word meaning “heap”, derived from “soros”, which
means a hill). Eubulides started by positing that “A heap does not con-
sist of one grain of sand, or of two, or even of one hundred grains, and
we cannot create a heap by just adding one more grain of sand to a pile
of grains of sand”.
Everybody agreed. Then Eubulides argued “If we have a pile and we
add a grain to it, we still have a pile. It seems that we will never have a
heap, no matter how many times we add a grain to the original pile. . .”
Now, we know this is not true: after a great number of grains is added to
the pile, we get a heap. The number of grains required can be very great
indeed, but we will eventually have a heap.
Eubulides then concluded “This means that at a certain point you
had a pile that became a heap by adding to it one grain of sand, but this
is impossible as we have agreed before.”
Now, what would you reply?! That he is playing with words?! But
are words not intended to express thoughts? Where is the fault in
this line of thinking? In the 20th century, this paradox was considered
by various philosophers, logicians and mathematicians, from Gottlob
Frege (1848–1925) and Bertrand Russell (1872–1970) to some contem-
porary logicians. Interest in the paradox was rekindled recently by a new
analysis published by the Swedish-born Oxford philosopher Timothy
Williamson (Vagueness, Routledge, 1994).
Various solutions have been proposed for this paradox. One ratio-
nal approach confirms that undefined attributes such as “heap of
sand” do exist, with a zone between what is undoubtedly a heap
and what is not, or not yet, a heap. But in this case it would be
necessary to eliminate undefined attributes from all rigorous logi-
cal reasoning. Frege concluded that everyday language would become
 EUBULIDES , THE HEAP AND THE EURO 45

irretrievably paradoxical if we were to use it with its greatest intended

rigor.
Another solution is to negate the validity of inductive reasoning, for
example, by adopting randomized logic and taking intermediate degrees
of truth in the propositions into consideration. Yet another solution
would be to consider the problem as just a question of perception. Just
as happened with similar heaps of sand, we get the 10 cent coins mixed
up with the 20 cent coins and also 20 cents with 50 cents. And, just as
happens when we compare a single grain of sand with a heap, we find it
easy to tell a 10 cent and a 50 cent coin apart.
This problem is just as important in theoretical economics. In
order to calculate “indifference curves” (for instance, in combinations
of goods that consumers take to be equivalent), we encounter prob-
lems that are difficult to resolve in practice. If you want to buy a car,
do you see any real difference in price between one that costs 15,000
dollars and one that costs 15,001 dollars? That is unlikely. But buyers
beware, as the marketing professionals seem to have studied the philos-
ophy of Eubulides. As we don’t see much difference between 15,000 and
15,001, or between 15,001 and 15,002. . . without knowing quite how it
happened, we typically end up leaving an automobile dealership driv-
ing a car with many more features and accessories than we need, having
inevitably spent more money than we originally intended.
THE EARTH IS ROUND
 HOW GPS WORKS

For thousands of years man navigated by the stars. But since the
invention of GPS, we have replaced the Pole Star, the Southern Cross
and the Sun with artificial satellites as our primary navigational guides.
This may seem far less romantic, but you have to admit that the inner
workings of GPS are intriguing. How does it work? Are there satellites
watching us from the skies and following our every move, our every

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_12, 49


C Springer-Verlag Berlin Heidelberg 2010
50 FIGURING IT OUT

position? The short answer is, not really, though the reality, while less
frightening, is much more fascinating.
GPS stands for “Global Positioning System”, a system first created
by the U.S. Department of Defense in the 1970s and 1980s, but which
is accessible to the public. The system consists of three elements: a net-
work of satellites, terrestrial control stations, and users operating GPS
receivers.
The network of satellites comprises 24 devices positioned across six
different orbital planes in such a way that, no matter where you are in
the world, it is always possible to receive signals from at least four of
the twenty-four satellites at any given time. The terrestrial control sta-
tions monitor the satellites, always tracking their exact position in space.
These stations then control the satellites so that they can, in turn, trans-
mit their precise position. The GPS receivers are operated by users who
receive signals from the satellites and calculate their position based on
those signals. It is an extremely complex system.
To understand how it all works, imagine that you are on a hiking
vacation in a remote area. I like to picture myself in the Alentejo region
in southern Portugal, where I used to spend my summer holidays. Now,
imagine you are there and have lost your way. However, you do know
that you are somewhere between the villages of Ourique and Castro
Verde. Then, imagine that there is a church in each of these villages and
that the bells of each church can be heard for dozens of miles all around,
and that both sets of church bells are rung every hour precisely, and that
each has a distinctive sound, so it is easy to tell them apart.
Suddenly, you hear the church bells of Ourique. The precise time
is 17.3 s after midday. This means that the sound of these bells, rung
at exactly 12 o’clock, took 17.3 s to reach you. As you know that the
speed of sound is about 1125 ft/s, you are in a position to calculate your
distance from Ourique. Multiplying the speed by the time elapsed, you
know that you are about 19,500 ft, or 6500 yards away. Seconds later, you
hear the bells of Castro Verde and look at your watch: it is exactly 26.6 s
after noon. Calculating the distance in the same way, you work out that
you are about 29,925 ft or almost 10,000 yards from Castro Verde. So
just where are you?
 HOW GPS WORKS 51

Now draw two circles on the map. Center the first one at Ourique
with a radius of 6500 yards: this connects all the points at which you
would hear the church bells of Ourique at exactly 17.3 s after noon,
which means you are located somewhere on this circle. Following the
same course, trace another circle centered at Castro Verde with a radius
of 10,000 yards. You are also somewhere on this circle. So you have to
be at a point where the two circles intersect. They intersect at two places.
You are at one of those points.
You are in the middle of a field, but you notice that one of the two
points of intersection marked on the map is near the freeway. Clearly,
you are not there, or you would be dodging cars, and not in the middle
of a vast field. Therefore, you must be at the other point. You check the
map and see that you are very close to a very small village called Cabeça
da Serra.
GPS works in a very similar way. The satellites emit signals in
the form of electromagnetic waves (not sound waves). The GPS device
receives these signals and is able to measure its distance from each of
the satellites by means of wave interference. As it knows the precise
positions of the satellites (recorded in the so-called almanac, which is
continually updated), the device also knows the coordinates of your
position. Your GPS device then indicates your position by pinpointing
these coordinates in the map integrated into its navigation software.
If a device receives signals from two satellites, it can measure two
position coordinates (latitude and longitude) after eliminating the geo-
graphical ambiguity, as once more there are two possible locations. The
GPS receivers can eliminate the false hypothesis after an initialization
period, during which they receive multiple signals from various satel-
lites, just as you eliminated the location beside the freeway with the help
of some additional information.
But a GPS device requires signals from four satellites. With the
third signal, the receiver can measure the altitude with respect to sea
level. With the fourth signal, it synchronizes its internal clock. The four
signals that it receives simultaneously enable it to calculate these four
coordinates. Time is like a fourth dimension.
52 FIGURING IT OUT

The signals emitted by the satellites travel at the speed of light, so

the measurements must be extremely precise. Atmospheric conditions
have to be considered, as they modify the velocity of the waves and cause
noise interference in the signals. The orbital velocity of the satellites also
has to be taken into account, as this affects the frequency of the signals
that are received. To increase the precision, the effects of the terrestrial
gravitational field and of relative movement also have to be considered.
GPS is an incredibly sophisticated and successful marriage of science
and modern technology. However, its underlying principle is as simple
and beautiful as two church bells pealing in the countryside.
 GEAR WHEELS

Reproduction of an Antikythera Mechanism model, recently constructed by John

Gleave

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_13, 53


C Springer-Verlag Berlin Heidelberg 2010
54 FIGURING IT OUT

A person examining the interior of a mechanical clock cannot fail to

be amazed at the number of gear wheels it contains. These gear wheels
ensure that the clock’s hands revolve at a certain speed by converting the
oscillations of the internal energy source, usually at one-second inter-
vals, into a one-hour cycle for the minute hand and a twelve-hour cycle
for the hour hand.
To make this conversion it would be possible to construct a gear
wheel with only one tooth and to connect it to the energy source, which
would make such a wheel turn once per second. Then it would be pos-
sible to mesh this gear wheel with another wheel with 3,600 teeth. Each
time the first wheel made a complete rotation, the second wheel would
advance by one of its 3,600 teeth. When the first wheel had rotated 3,600
times, that is after 1 h, the second wheel would have completed a single
rotation. So this second wheel could be connected to the minute hand.
However, a gear wheel with only one tooth would be highly unstable
and fragile, just as a wheel with 3,600 teeth would simply be too mas-
sive. So in their wisdom, clock designers chose to utilize a series of gear
wheels that transmit the initial movement from one to another, succes-
sively reducing their cycle of rotation. Two gear wheels are mounted on
each shaft in such a way that a shaft is rotated by one wheel and simul-
taneously moves the other wheel, which in turn meshes with a third
wheel on another shaft, and so on. The final rate of conversion of the
original movement is the product of the gear ratios at each individual
step. To obtain the desired rate of 3,600 : 1 there could be a series of
steps, for instance, we could write the following product of fractions:
36/10 × 50/10 × 20/5 × 10/5 × 25/10 × 100/10, with the first number
in each fraction representing the number of teeth on the first gear wheel
and the second the number of teeth on the wheel with which it meshes.
Obviously there are many possible combinations of ratios and so there
are many possible combinations of wheels. The clockmaker’s skill lies
in designing an optimum series of gear wheels, neither too big nor too
small, that produce the desired ratio.
In the 18th century scientists developed mathematical algorithms to
solve this problem iteratively. Essentially, these algorithms factorize the
 GEAR WHEELS 55

numerator and the denominator of the desired ratio, i.e., they write each
of the two numbers as a product of prime numbers. For example, if we
want to arrive at a ratio of 28/45, we can write (2 × 2 × 7)/(3 × 3 × 5)
and look for various combinations. In this case, (2 × 7)/(3 × 3) × 2/5,
that is 14/9 × 2/5, is one of the possible solutions if you want to use four
gear wheels.
The problem becomes more complicated when it is impossible or
impractical to arrive at exactly the desired ratio, and only an approx-
imation can be achieved. For instance, if the desired ratio is 997 :
1999, both numbers are prime numbers, so the simplest exact solu-
tion would be to make one gear wheel with 997 teeth and another
with 1999, which does not appear to be practical as the gear wheels
would be enormous. But a 1 to 2 or 10 to 20 approximation is rea-
sonable enough in this case, as 997 divided by 1999 is 0.498..., very
close to one half. Better approximations would be possible as it is
known that, no matter which ratio is the final goal, it is always possi-
ble to arrive at a sufficiently precise approximation by using systems
of simpler gear wheels. But solving the problem can be very labori-
ous. After all, it was only in the 19th century that sufficiently efficient
algorithms were developed to obtain systems that delivered satisfactory
approximations.
Fascinatingly, up until a few decades ago it was thought that all these
techniques had been mastered only recently. But in 1901, the chance
discovery of fragments of a metal mechanism on the seabed close to
the Greek island of Antikythera refuted this idea. When it was found,
the mechanism was in very poor condition, and it was difficult to tell
what it was, and therefore what its possible significance could be. There
was a great deal of restorative work to do before that could be known.
Finally, in 1974, Derek J. De Solla Price (1922–1983), a science histo-
rian at Yale University, solved the mystery. He concluded that it was, in
fact, a truly significant discovery: it was a mechanism designed to repro-
duce the apparent movements of the sun and moon, including the phase
changes of our satellite. It was what we today call a planetary clock or an
orrery.
56 FIGURING IT OUT

Planetary clocks are very rare devices, as they are costly to construct,
and very complicated. Their operating system consists of gear wheels
that control the movements of separate markers, usually spheres, with
each marker representing a planet.
The most surprising thing about the Antikythera mechanism, how-
ever, is the precision of the ratio it uses for the lunar and solar periods.
The ratio was achieved using six gear wheels, 64/38 × 48/24 × 127/32,
resulting in a final ratio of 254/19 = 13.36842... This result is correct to
the third decimal place!
Until a short time ago the Antikythera mechanism only astounded
mathematicians, astronomers and historians. Recently John Gleave
(www.orreries.freeserve.co.uk), an English artisan, and other patient
craftsmen managed to reconstruct what are believed to be replicas of the
ancient mechanism. They work with the desired precision. Who could
be that mysterious Greek sage who constructed that original clock 2000
years ago?
 FEBRUARY 29

February 29 is a date that only comes around every 4 years. If you were
born on this date, you know that your birthday only falls in leap years,
that is, those years having 366 instead of 365 days, with an extra day in
February. Depending on your point of view, that is either an unfortunate
stroke of fate, or a reason to celebrate.
In reality, though, the situation is a bit more complicated than it
may at first appear. In some cases, people born on February 29 have
to wait eight whole years to celebrate their birthday. Years that are
divisible by 100 are an exception to the leap year rule: although they
are divisible by 4, and therefore comply with the general rule, they only
have 365 days. That is, for example, what happened in 1900, which had
365 days.
However, if you were born on February 29, you may recall that in the
year 2000 you were able to celebrate your birthday. Did somebody make
a mistake when they created the 2000 calendar, or were the guiding leap
year principles purposely ignored? Well, neither happened, as it turns
out. As it happens, the year 2000 is an exception to the exception: as it is
divisible by 400, it is still considered a leap year, as was the year 1600,
and as 2400 will be. You may rightly wonder how such complex rules
came about in the first place and whether they are necessary at all. The
truth is that our current calendar, which is based on a decree first issued
by Pope Gregory XIII in 1582, is now pretty universally observed, and
represents the culmination of a long struggle to understand the underly-
ing astronomical cycles and to devise a calendar that always keeps pace
with the seasons.

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_14, 57


C Springer-Verlag Berlin Heidelberg 2010
58 FIGURING IT OUT

The primary underlying cycle for any calendar is the solar day,
which is the most obvious and most universal measurement of time,
and certainly the first to be used. A second important cycle, which is in
fact pre-eminent in some calendars, is the lunar cycle. A third important
cycle is the solar year, which governs the annual cycle of seasons.
These cycles are not multiples of each other: there is not a whole
number of days in the lunar cycle, nor a whole number of lunar cycles
in a year, nor even a whole number of days in a solar year. It is not
possible to have a simple and perfect calendar that will always have
the same number of days in each month, will align the months with
the moon, and will also have the same number of days in a year.
Any calendar always favors one particular cycle, to the disadvantage of
the others.
The first calendars were based on the lunar cycle. Each new moon
saw the start of a new month. Early on it was realized that a purely lunar
calendar was not an ideal solution for farming communities who shaped
their lives according to the seasons. The ancient civilizations began to
complement the lunar cycle with the seasonal cycle. However, this only
made the problem worse, as a year does not contain a whole number
of lunar months. In some cases, for example in the Jewish calendar,
they decided to make the year variable, with some years containing 12
months and others 13 months, which means that some years had 353
days and others 385 days.
The Egyptians resolved the problem by creating a purely solar cal-
endar. Their year contained 365 days, which was a reasonable approxi-
mation, and the days on which it started and ended had nothing to do
with the phases of the moon. But the Egyptian civilization lasted a long
time. As their year was about 6 h too short, over time those missing
hours accumulated and became noticeable. Within a few dozen years it
was clear that the official calendar was out of phase with the flood season
on the Nile. After a period of 1460 years the calendar had gone through
the annual seasons and returned to its starting point.
The Egyptian civilization lasted more than 4000 years, and so the
astronomers in ancient Alexandria were completely aware of this error
 FEBRUARY 29 59

in their calendar. They suggested a simple solution: that every 4 years,

an extra day be inserted into the calendar. And this is exactly what we
call a leap year today.
When Julius Caesar returned from his campaign in Egypt he hadn’t
just been impressed by Cleopatra. He was also very struck by the
Egyptians’ sophisticated knowledge of astronomy. In an effort to make
the Roman calendar more orderly (at that time it was utterly chaotic),
Caesar summoned to Rome an Alexandrine named Sosigenes, and
entrusted him with this task. The new system, which later became
known as the Julian calendar in honor of its founder, sorted out the
months, established 45 BC as year 1, and stipulated that the year would
have 365 days, with an additional day every 4 years. This extra day,
which was to be added between days 23 and 24 of the month of
Februarius, was called bissextus dies ante calendas Martii [double sixth
day before the first of March].
The Julian calendar was adopted by the Catholic Church and
remained the official calendar of the church until the 16th century. The
only change made was to the starting date of the calendar. Following a
proposal by the 6th century Scythian monk Dionysius Exiguus, year 1
was changed to the assumed date of the birth of Christ. In Europe, many
kingdoms maintained the Julian calendar, others followed Dionysius’
reform.
By the end of the Middle Ages, however, it had already become clear
that the calendar was not keeping pace with the seasons. The almost
11 min difference between the Julian year and the solar year had been
accumulating for hundreds of years, and by the end of the 16th cen-
tury the calendar was 10 days behind the solar year. The spring equinox,
which should have occurred on March 21, took place on March 11. After
various attempts had been made to once again reform the calendar,
Pope Gregory XIII decided to take definitive action. In accordance with
a proposal by the astronomer Aloysius Lilius (1510–1576), and sup-
ported by the Jesuit cosmographer Christopher Clavius (1537–1612),
ten days were eliminated so that the spring equinox would again fall
on March 21.
60 FIGURING IT OUT

However, this time, in order to avoid the mistakes of the past, it was
necessary to get rid of some of the leap years. That is why years that
are multiples of 100 but not of 400 are no longer leap years. They mark
the exception to the exception. Under these new rules, our calendar will
only be one day out of step with the solar year in 4909. Finally, this gives
us a little breathing room!
 THE NONIUS SCALE

In the 16th century, sea navigation still depended on mariner’s

astrolabes and other relatively primitive instruments for measuring
astronomical altitudes. The precision of these instruments was greatly
limited by the graduated scale they used, which was normally based on a
minimum unit of one degree and could be subdivided into half-degrees
but not into smaller units, as the measurement marks engraved in the
metal instruments had to have a certain width, and began to become
indistinguishable if placed too close together.
One of the leading mathematicians of the era, Portugal’s Pedro
Nunes (1502–1578) thought of solving this problem by marking various
scales with different units. To understand his idea, you just have to think
of the subdivision of a right angle, which is equivalent to a quadrant and
is sufficient for measuring astronomical altitudes. He conceived of a sys-
tem consisting of 45 concentric scales marked in the quadrant, as can be
seen in a replica of the instrument made by James Kynuyn on display
in the Maritime Museum in Lisbon. The outer scale divided the right
angle into 90 parts, i.e., in units of one degree. Inside this was a scale
that divided the angle into 89 parts, i.e., into units of 90/89 of a degree.
Then came another scale dividing the right angle into 88 parts, and so
on until the final scale divided the angle into 46 parts. This came to be
called the nonius scale, as Nunes was called Nonius in Latin. The nonius
scale consisted of these 45 concentric scales, each with one subdivision
less than the previous one.
To measure the angular altitude of the sun or another star, an
observer had to hold the astrolabe or quadrant vertically and adjust the

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_15, 61


C Springer-Verlag Berlin Heidelberg 2010
62 FIGURING IT OUT

A modern reproduction of the Kynyun instrument, the only extant instrument from
Nunes times with the original nonius scales

alidade (the device that allows one to sight a distant object and fix that
line of sight) so that it was aligned with the position of the star. The angle
was then measured, but this measurement was not restricted to just one
scale, as in traditional instruments. The alidade passed through all the
concentric scales and the observer would select the mark that coincided
best with the position of the alidade. Let us suppose, like Pedro Nunes
in his masterpiece De crepusculis, that the mark that best coincided with
the position of the alidade was on the fourth scale from the top (the one
that divides the right angle into 87 parts). And let us imagine that it coin-
cided with mark number 30 on this scale. In this case the measured angle
would be 30/87 of the quadrant. A fraction of 30/87 is approximately
 THE NONIUS SCALE 63

equivalent to 31◦ 2 4 , a value obtained with greater precision than in

any of the individual scales used.
The nonius began to be studied all over Europe. The greatest
astronomer of the time, the Dane Tycho Brahe (1546–1601), con-
structed various quadrants using scales like the nonius scale: “I used
the subtle process presented by Nunes”, said Tycho, “and made it
more exact, increasing the number of subdivisions and calculating
tables”.1 The astronomer eventually admitted that the instrument did
not provide the precision that he wanted. So other practical solutions to
improve the system continued to be sought.
A German mathematician named Jacob Kurz, an influential figure
in central Europe and in the Vatican, proposed a modification of the
nonius scale, which would introduce simpler, more graduated scales.
Then, the Jesuit mathematician Christopher Clavius (1538–1616) took
up Kurz’s idea and further refined it by limiting the nonius scale to two
scales. In his Geometria practica published in Rome in 1604, he sug-
gested that one scale should consist of units of one degree while the
other should be marked in units of one and one-sixtieth of a degree,
i.e. of one degree and one minute (1◦ + 1◦ /60 = 1◦ 1 ).
Using this process we can directly obtain measurements of degrees
of minutes by comparing the two scales. The distance between the first
mark on the primary scale and the first mark on the secondary scale
represents one minute of one degree. The distance between the second
marks on the two scales is equivalent to two minutes of one degree,
and so on. According to Clavius, we start by using the primary scale
to measure the angle. The result is a certain number of degrees as well
as a small remainder. To measure this residual fraction of one degree,
Clavius suggested using a pair of compasses to compare it with the dif-
ferences between the marks on the two scales. If the residual amount is
equal to the difference between the tenth marks on the two scales, for
example, this would mean that that amount is equivalent to ten minutes
of one degree.

1
Letter to Christopher Rothman, January 20, 1587, as quoted by A. Estácio dos Reis, O nónio
de Pedro Nunes, Oceanos 1988, p. 72.
64 FIGURING IT OUT

This process was ingenious in its logic, but was still not very practi-
cal. Another 30 years went by before someone else came up with a quick
method for transferring measurements from one scale to the other.
This was the innovation made by a French mathematician called Pierre
Vernier (1584–1638) in 1631, and it involved adding a moveable sec-
ondary scale to the instrument, allowing the direct measurement of the
residual angle imagined by Clavius.
Vernier’s invention was very successful and was quickly adopted by
instrument-makers all over Europe. It was tailored for use in sextants,
setting circles for telescopes, callipers, gauges, and other instruments. In
fact, it is this version of the nonius scale that we use today.

A modern calliper with a moving nonius scale built according to Vernier’s idea
 PEDRO NUNES’ MAP

When you fly from, say, my home, Lisbon to New York, you usually
reach the U.S. coastline at least an hour before your plane lands. During
this hour you can typically look down and see the indented outline of
the Massachusetts coast and the island of Martha’s Vineyard, Nantucket,
and other landmarks. Then you will soon see Long Island and all its
beaches, including those of the Hamptons, as you travel northeast to
southwest towards New York City. If you were to look at a map of this
route, it would seem that the plane had taken a long way round and that,
instead of taking the shortest route across the Atlantic, it had reached the
coast of the New World farther to the north, where the Portuguese first
landed, and then followed the coast.
However, if you look at a globe, you will see that the plane did in
fact take the shortest route, which is the arc of a great circle, or in other
words of a circle that connects the points of departure and arrival and
that is centered at the center of the Earth. If you stretched an elastic band
around the Earth, making it touch Lisbon and New York, you would
see that the shortest route between the two cities does in fact touch the
Massachusetts coast. That means that to take the shortest path between
these two places, which share very similar latitudes, (Lisbon is at 39◦ ,
New York at 41◦ ) the airplane begins by flying to the west and a little
north, and ends up flying west and a little south.
All this may seem simple and obvious, but it took navigators many
years to comprehend it. The first person to understand it in all of its
implications was our friend Pedro Nunes (1502–1579), creator of the
nonius, and the Portuguese Royal Cosmographer.

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_16, 65


C Springer-Verlag Berlin Heidelberg 2010
66 FIGURING IT OUT

The shortest route

The shortest route between any two points

A and B on the globe is an arc of a great A
circle, or orthodrome. It becomes a curve B
on a map using the Mercator projection.
In this map the straight lines are loxodromes,
curves with a constant cardinal direction.

Map with the Mercator projection

Shortest route

New York Lisbon

A B
Rhumb line
(loxodrome)
Newspaper Expresso

In tackling the problem Nunes first turned it on its head. He asked:

should you always take the same direction in order to travel most
efficiently between two points? This problem had first been posed by
Martim Afonso de Sousa, the founder of Portugal’s first colonies in
Brazil. Wishing to travel from the Rio de la Plata in Brazil to Lisbon,
he ascertained that simply steering east was insufficient; that he would
also have to steer north. And he realized that it was not a simple matter
to chart the exact direction in which to travel.
The brilliance of Pedro Nunes is hard to overestimate. In 1537 he
was able to clearly distinguish two different trajectories for a ship on
the high seas. One would be a minimum-distance trajectory between
two points, which corresponds to an arc of a great circle: this is called
 PEDRO NUNES’ MAP 67

an orthodrome. The other would be the trajectory followed by a ship

that always maintained the same orientation with respect to the cardinal
points: this is called the rhumb line, later known as a loxodrome. These
two trajectories are only identical when the ship is traveling along the
Equator or along a meridian. In every other case they are different.
When he discovered the rhumb line, Pedro Nunes also proved that,
on a hypothetical planet completely covered by water, a ship that always
followed the same cardinal direction would not return to its starting
point, as was thought at that time. Instead, it would travel in an end-
less spiral, getting ever closer to one of the poles by making an infinite
number of turns around it. This curve, which is “neither a circle nor a
straight line” as Nunes said, came to be known as a loxodromic spiral.
Undoubtedly the most spectacular illustrations of loxodromic spi-
rals have been penned by Maurits Cornelis Escher (1898–1972). In
1958, Escher created some drawings of spheres with spirals, apparently
without any knowledge of their profound historical and geometrical sig-
nificance. The title of the image reproduced here is Sphere Spirals, and it
is one of his most beautiful creations. It is a woodcut in four blocks (one
for each color) with a diameter of just over 12 inches. We can imag-
ine a ship’s captain who decides to constantly steer a course of about a
60-degree angle to the north-south axis. The spirals show the path that
would be taken by such a ship. And they show that, even if the ship
started from different points, its trajectories would converge. The bands
are broader at the Equator and become narrower as they approach the
poles.
As well as showing navigators the path that they will take if they
steer a course at a constant angle to a cardinal direction, the loxo-
dromes of Pedro Nunes had a major influence on how maps began to
be drawn, and greatly contributed to the vision of the continents that
we have today. Nunes recognized that with his discovery, the old sea
charts would have to be replaced, and in 1566 he clearly explained the
precepts that should be followed in drawing the new navigational maps.
From 1569 onwards, Gerardus Mercator (1512–1594) would usher in
a revolution in cartography inspired and guided by the discoveries
of Pedro Nunes. Mercator was born in the town of Rupelmonde in
68 FIGURING IT OUT

Sphere spirals by E.C. Escher

Flanders (present-day Belgium). He was baptized as Gheert Cremer, but

his name was later Latinized to Mercator (Flemish “cremer” = English
“merchant” = Latin “mercator”). He first studied in Holland and later
in Leuven, in Flanders, where he remained and dedicated himself to
constructing globes and maps.
Mercator was, of course, aware of the work of Pedro Nunes,
as the knowledge had spread throughout Europe and had been dis-
cussed by their mutual friend John Dee (1527–1608), Queen Elizabeth’s
astronomer and astrologer. Mercator resolved to design a map that
would be of immediate use to navigators. He decided to form a grid
in which the lines of latitude would all be parallel to the Equator and
perpendicular to the meridians, which would run parallel to each other.
He also decided that the rhumb lines should appear as straight lines,
 PEDRO NUNES’ MAP 69

which had initially been proposed by Nunes. To achieve this, Mercator

progressively increased the distances between the parallels as the lines
of latitude approached the poles. In this he was again following Nunes,
who had spoken of the necessity of using “increased latitudes” for draw-
ing a map in which the loxodromes were straight lines. This is how
the “Mercator projection” was created, which even today is still the
best known and most utilized navigational mapping method. Its great
strength derives from being a conformal map, as it preserves the direc-
tion between any two points on the globe. It is ideal for planning and
plotting rhumb courses.
However, as with any map projection system, Mercator’s method
inevitably leads to distortions. If you carefully peel an orange to obtain
a whole skin and then try to flatten it on a table, it will break into pieces
and become bent. In the same way, cartographers have to take liber-
ties with the geometry of the globe in order to reproduce a spherical
surface on a plane. These deformations are of minor importance when
only a limited surface area is considered. But the world had increased
in size with the voyages undertaken in the Age of Discovery, and these
distortions soon became significant.
As Mercator’s maps are so widespread, their inevitable distortions
have molded our sense of geography. On these maps, Greenland looks
enormous, even bigger than South America, when in fact the land mass
of South America is nine times larger than Greenland’s. Mercator’s map
also continues to deceive modern travelers, who are amazed at the routes
flown by airplanes today.
 LIGHTHOUSE GEOMETRY

Strolling along the coast of the sea on a late summer evening, we can
sometimes discern the flash of a lighthouse in the distance, blinking
intermittently, as if trying to send us a signal. And indeed it is! The
lighthouse is telling us its name. It is sending us a message that is known
technically as its light characteristic.
Some lighthouses flash rapidly, while others send out prolonged
signals. Some have red lights, others white lights, and some alternate col-
ors. The pilots of ships approaching the coast are trained to read these
signals, and so to identify the lighthouses that they encounter.
The code lighthouses utilize is simple, consisting mainly of three
elements. The first is the way the light is sent. If the lighthouse sends out
short signals, they are called flashes, a term abbreviated to “Fl”. If the
lighthouse emits an almost continuous light interrupted by short peri-
ods of darkness, this is known as “occulting”, abbreviated to “Oc”. If the
light is blocked and released for equal periods, this is called “isophase”,
usually abbreviated to “Iso”.
The second element of the code is the color of the light, which is
usually designated by its initial (R = red, W = white, G = green, etc.).
The third and final component of the code is what’s called the period,
that is, the duration in seconds of a light cycle.
These characteristics are so distinctive that they can clearly identify
the lighthouse itself. For example, if you are close to the estuary of the
river Tagus near Lisbon, in Portugal, you will see the light emitted by
two different lighthouses, one at Bugio, and another at São Julião da
Barra. The former is marked on maps as “Fl G 5 s”, which means that

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_17, 71


C Springer-Verlag Berlin Heidelberg 2010
72 FIGURING IT OUT

Santa Marta Esteiro Lighthouse

Lighthouse Gibalta Lighthouse

Bugio
Lighthouse

HSA
curve

The pilot of a ship cannot calculate its position merely by observing the angle made
by two lighthouses. The set of points that comply with this angle is an arc of posi-
tion (known technically as a “horizontal sextant angle curve” or HSA curve). The pilot
requires additional information, such as a third lighthouse, to resolve the problem

its light flashes (Fl) colored green (G) at intervals of 5 seconds (5 s). The
latter is marked as “Oc WR 5 s”, which signifies that it emits an almost
continuous light that is blocked briefly (Oc), displaying a white color at
one position (W) and red in the other (R), at 5-second intervals (5 s).
Lighthouses provide invaluable information enabling a ship to
locate its position. Even today, in the age of GPS and automatic naviga-
tion, they still serve a vital function, making it easier to navigate visually
along a coastline, which provides ship captains with an additional secu-
rity check and permits them to plot the position of the boat. Lighthouses
can also be used as guides for setting a course. At the entrance to the
Tagus estuary, for example, captains can align their ships by the light-
houses at Gibalta (close to Caxias) and Esteiro (in the grove of trees
around the National Stadium), thus ensuring that they are charting a
safe course for their ships where the river is deep enough.
At times captains also take the angular height at which lighthouses
appear to them above the horizon, and knowing the actual height of
 LIGHTHOUSE GEOMETRY 73

the light beam, they can apply some elementary rules of trigonome-
try to estimate their proximity to the coast, or they can measure the
angles between various lighthouses to determine their ship’s position.
Lighthouses are useful to sailors in a thousand and one different ways.
 ASTEROIDS AND LEAST SQUARES

Carl Friedrich Gauss, the mathematician who created the “least squares” method,
which is commonly used in science

Eugene Wigner (1902–1995), who was awarded the 1963 Nobel Prize
for Physics, wrote an article in 1960 that has since become a classic,
titled: “The Unreasonable Effectiveness of Mathematics in the Natural
Sciences”. In Wigner’s words: “There is a story about two friends, who
were classmates in high school, talking about their jobs. One of them
became a statistician and was working on population trends. He showed
an scientific article to his former classmate. The article included, not
unusually, the Gaussian distribution. The statistician explained to his
former classmate the meaning of the symbols for actual population, for

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_18, 75


C Springer-Verlag Berlin Heidelberg 2010
76 FIGURING IT OUT

average population, and so on. His classmate was a bit incredulous and
was not quite sure whether the statistician was pulling his leg. ‘How can
you know that?’ was his query. ‘And what is this symbol here?’ ‘Oh’, said
the statistician, ‘this is pi.’ ‘What is that?’ ‘The ratio of the circumference
of the circle to its diameter.’ ‘Well, now you are pushing your joke too
far’, said the classmate, ‘surely the population has nothing to do with the
circumference of the circle’”.1
Experts may find it quite natural that π (pi) appears in population
studies, as it is regularly employed in normal or Gaussian distribution,
which is, in turn, frequently used in statistics. But even mathematicians
would have difficulty offering a simple and convincing explanation for
why this symbol has become indispensable in so many areas of mathe-
matics and the sciences. And the relationship between π and population
studies is even more surprising.
People have often been surprised by the extraordinary effectiveness
of mathematics in describing, comprehending and forecasting natural
phenomena.
In the early years of the 19th century the effectiveness of mathemat-
ics was demonstrated in spectacular fashion. At that time astronomers
were searching for a previously unknown planet they suspected might be
located between Mars and Jupiter. In fact they had detected a regularity
in the positions of the planets, called the Titius-Bode law, and verified
that there was a large void between Mars and Jupiter. They thought that
this void might be filled by an as yet undiscovered planet.
Astronomers cooperated internationally in a group they jokingly
termed the “Celestial Police”. This was the first time that a scientific
investigation had been so organized at the international level. Each
astronomer assumed responsibility for one sector of the sky. The zone
of the zodiac in which the sun, the moon and the planets describe their
apparent movements was divided into segments so that nothing was
left out. The sky was observed night after night by astronomers from
different countries, all looking for a luminous dot moving slowly against
the backdrop of the stars.

1
See, e.g., http://pascal.iseg.utl.pt/~ncrato/Math/Wigner.html
 ASTEROIDS AND LEAST SQUARES 77

The honor of discovery was claimed by Giuseppe Piazzi (1746–

1826), the director of the observatory in Palermo in Sicily, who detected
an unknown object, apparently at the orbit of the desperately sought
“fifth planet”. His discovery was made on January 1st, 1801, the first day
of the new century.
Piazzi was cautious and would only admit that it might be a comet.
But the members of the Celestial Police had no doubts. They were cer-
tain it was the missing planet between Mars and Jupiter. The problem
was that the object soon disappeared in the darkness, and then reap-
peared too close to the sun to be accurately observed at that time. In
the early fall of 1801, when the object ought to have reappeared in the
early-morning sky, they could not find it. Heinrich Olbers (1758–1840)
and various other astronomers did their calculations but still could not
locate the celestial body. At this point a German mathematician entered
the picture. Although only 24 years old at the time, Carl Friedrich Gauss
(1777–1855) was already considered one of the greatest geniuses in the
history of mathematics. The young Gauss exulted in the possibility of
putting his theory into practice, as he had spent several years study-
ing the problems involved in calculating orbits “without any theoretical
assumption, from observations not embracing a long period of time”,2
as he wrote.
Gauss calculated the orbit on the basis of a new method of combin-
ing observations and using these combinations to estimate the param-
eters of a function, in this case an orbit. This procedure later became
known as the method of least squares, and it solved a problem that the
best minds in Europe had been debating for decades.
The calculations made by Gauss provided estimates for the orbit
of the object that were so precise that Franz von Zach (1754–1832)
succeeded in rediscovering it on December 31st, 1801, almost exactly
1 year after Piazzi had first sighted it. He discovered it at half a degree
of angular distance from the position predicted by Gauss. The follow-
ing night Olbers saw it too. The scientific community was euphoric. It
seemed that the solar system was complete again!

2
See William Sheehan, Worlds in the Sky: Planetary Discovery from Earliest Times Through
Voyager and Magellan, Tucson and London: The University of Arizona Press, 1992, p. 105
78 FIGURING IT OUT

At that time it was thought that the object was a planet similar to the
others in our solar system, and it was given the name of Ceres. But in
1802, when its diameter was estimated, it was measured at just over 160
miles, not large enough to qualify as a planet. (Now, of course, we know
that the diameter of Ceres is about 600 miles, almost a third that of the
moon). But surprises continued apace in the following years, which saw
the discovery in the same area of Pallas (1802), Juno (1804) and Vesta
(1807), which is the most brilliant of all these small celestial bodies and
can even be observed with the naked eye. In the ensuing years many
thousands of asteroids have been discovered, and about 40,000 of them
have been cataloged to date.
How is it possible that π, used in Gaussian distribution to reduce
diverse observations to a simple equation, could have helped to discover
Ceres and many other celestial objects? The unreasonably effective sci-
ence of mathematics is the mysterious grammar of modern scientific
knowledge.
 THE USEFUL MAN AND THE GENIUS

If you took the time during the 2009 International Year of Astronomy
to observe the sun, you would likely have noticed that our star was then
without its famous dark spots. This lasted for some time. The sun was
spotless on 266 of the 366 days in 2008, and all the way up to October of
2009 there were still virtually no spots to be observed. It is not uncom-
mon for the sun to appear spotless for a brief time, but it is unusual to see
it without sunspots for such a prolonged period. That had not occurred
since 1913.
Sunspots are gigantic magnetic storms that spew forth material,
cause sudden changes in the magnetic field, and emit intense radia-
tion in the ultraviolet range. They are dark, but solar activity is most
intense at their edges, so the total radiation emitted by the sun increases.
The fact that we were registering a longer than usual period of low
activity meant that less solar radiation was reaching us, which in turn
undoubtedly had an impact on our climate. Maybe we will soon be able
to measure it.
Great caution has to be taken when observing sunspots. It is
extremely dangerous to use binoculars or a telescope to observe the sun
directly, as this can result in immediate blindness. Professional filters
must be used to ensure safety. The simplest and most practical way to
observe sunspots without endangering vision is to project an image of
the sun on a white surface (for example, by using an inverted ocular)
and to observe that image. If there are any large spots, it is easy to detect
them using this method.

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_19, 79


C Springer-Verlag Berlin Heidelberg 2010
80 FIGURING IT OUT

The discovery of sunspots at the beginning of the 17th century by

the German theologian Fabricius, the Italian physicist Galileo, the Jesuit
astronomer Scheiner, and others was for a short time a major sensation.
Decades after the excitement had abated, sunspots were relegated to the
category of a curiosity, with little continuing significance for astron-
omy. But in 1844, Heinrich Schwabe, a German pharmacist and amateur
astronomer, noticed a regular pattern: he discovered that sunspots
increased and decreased in cycles of about 10 years. Then Johann Rudolf

Johann Rudolf Wolf (7 July 1816 – 6 December 1893)

 THE USEFUL MAN AND THE GENIUS 81

Wolf, the director of the Observatory in Bern, Switzerland, took note

of Schwabe’s observations and decided to study the phenomenon more
closely. He spent the rest of his life counting sunspots, which he did day
after day whenever the meteorological conditions permitted. He col-
lected various observations made in the past, and invented a method
for quantifying sunspots, beginning to document a long sequence that
is still being quantified today, and that reveals the oscillating pattern of
solar activity. In 1852 he measured solar periodicity using an elementary
statistical method and found the period to be 11.11 years.
An entire life devoted to counting sunspots may seem like a trivial
calling, but as Wolf said later, “I have always consoled myself that he
such as I who is not a genius, can still achieve much that is useful when
he does his work right and chooses his work to suit his talents”.1
Some 62 years later, at the start of the 20th century, a young German
physicist studied the observations made by this useful man and devised
another method for estimating sunspot periodicity, one based on a
sophisticated mathematical tool called the Fourier transform. It was the
first practical application of what is today called spectral power analysis.
That young physicist was Albert Einstein, and he also calculated
the sunspot period as 11.11 years. Sunspot activity will resume and will
again fade. We know that thanks to Wolf, a remarkably humble man,
and to Einstein, a remarkable genius.

1
A.J. Izenman (1983). J.R. Wolf and H.A. Wolfer. A historical note on the Zurich sunspot relative
numbers, Journal of the Royal Statistical Society A 146, 3, 311–318.
SECRET AFFAIRS
 ALICE AND BOB1

“How am I going to tell Bob I love him?” “I can’t wait to read Alice’s letter.”

Alice and Bob live apart from each other and can only communicate by
“snail” mail. But they know that the mailman reads all their letters. Alice
has a message for Bob and doesn’t want the mailman to read it. What
can she do? She has already thought of having the message delivered in
a padlocked box. But how can she get the key to Bob? She can’t send it
inside the box, because then Bob couldn’t open the box.
After giving the problem a lot of thought, she has a brainstorm. She
does send him the padlocked box. She knows Bob well enough to know
he is intelligent and will eventually figure out her brilliant idea. With
the mail going back and forth a few times, but without ever exchanging
keys, the message arrives at its destination, where Bob is able to open the

1 This article, together with the next two articles, was awarded the first prize in the 2003
Raising Public Awareness of Mathematics competition organized by the European Mathematical
Society. They are reproduced here with some minor changes.

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_20, 85


C Springer-Verlag Berlin Heidelberg 2010
86 FIGURING IT OUT

box and read the message. How do you think they solved the problem?
If you like logical challenges, take a break now and think about it.
It is quite simple. . . Bob receives the box. When at last he under-
stands Alice’s stratagem, he locks the box using a second padlock to
which he has the key. He sends the box, now locked by two padlocks,
back to Alice by mail. She then removes her padlock using her key, and
sends the box off again by mail. When Bob receives it again, he only has
to open his padlock with his key and read the message. The mailman is
left guessing.
I have just described an old brainteaser and one of its solutions.
In 1976 it inspired three young Americans, Whitefield Diffie, Martin
Hellman and Ralph Merkle, to design a cryptographic system in which
the secret to be communicated is secured by two keys that the partici-
pants do not exchange.

THE ORIGIN OF PUBLIC KEYS

The process invented by Diffie, Hellman and Merkle marks the start
of cryptography using public keys that work in conjunction with secret
keys that are never exchanged. It is based on modular arithmetic, which
essentially consists of working with the remainders of entire division by
a specified number called the modulus. The best example is provided
by a clock. If the clock shows 10 o’clock at a given moment, what time
does it show five hours later? Obviously the answer is 3 o’clock, which
is equivalent to the remainder of the whole division of 10 + 5 = 15 by
12. In mathematics this is written 10 + 5 ≡ 3 (mod 12), as 15 ≡ 3 in the
modulus of congruence 12 in normal analog clocks. We use this nota-
tion to describe the process adopted by Alice and Bob, using an example
provided by Simon Singh. Our two friends manage to agree a common
cryptographic key without ever personally exchanging it and without
anybody else being able to discover it.

Although Alice and Bob are fictitious persons, these names have
become standard terms used by specialists in cryptography. It is more
fun to use these names than to always talk about the sender and the
recipient, or only about A and B. They are usually joined by a third
 ALICE AND BOB 87

person (in our story that was the mailman), who is normally called Eve
and plays the role of the avid listener or eavesdropper.

ALICE BOB

Alice and Bob agree on the numbers 7 and 11, so they will calculate the result
of 7x (mod 11).
(They do not bother to keep this information secret).

Alice selects 3 as her secret Bob chooses 6 as his secret

number. number
Alice calculates 73 = 343 ≡ 2 Bob calculates 76 = 117649 ≡ 4
(mod 11). (mod 11).
Alice sends the result, 2, to Bob. Bob sends the result, 4, to Alice

(This is usually a crucial moment that the participants try to keep secret.
Nevertheless, this is not a consideration in this example. Even if this
exchange of information became public knowledge, nobody would be able
to find out the secret key.)

Alice takes Bob’s result, 4, and her Bob takes Alice’s result, 2, and his
secret number, 3, and calculates secret number, 6, and calculates
43 = 64 ≡ 9 (mod 11). 26 = 64 ≡ 9 (mod 11).

Alice and Bob end up with the same number, 9, without either having
informed the other of their personal secret numbers.

Until this system was discovered by Diffie, Hellman and Merkle,

the communication of encrypted messages required the code key to
be exchanged. It was necessary that Alice and Bob met previously and
agreed on a key that only they knew. Only this permitted them to sub-
sequently exchange messages at a distance without Eve, always on the
lookout, getting to know them. This is how secret messages have been
sent and received from the time of Caesar right up to the modern era;
this is how spies, governments, generals, conspirators and even lovers
operated. The key might be simple, but it was always necessary that
Alice and Bob agreed on the entire system for coding and decoding mes-
sages from the outset. Usually though, it was possible to compress all the
information into a single number, quite possibly a very large number, so
it could be said that the key consisted only of such a number.
88 FIGURING IT OUT

So the idea of Diffie, Hellman and Merkle was revolutionary. In

accordance with the method they laid out, Alice and Bob began by
agreeing on two numbers. These numbers could even be public knowl-
edge, since even if Eve managed to get hold of them, she would not
be in a position to deduce the key. Then Alice and Bob each choose
another number that they keep entirely secret. After performing some
calculations, they both come to the same result: a number that nobody
else knows and that will become the key for encrypting their messages.
The process they invent is relatively simple, but very ingenious, and is
described briefly in the text box. It all happens as described in the story
of the two padlocks. The keys are not exchanged, but in the end both
Alice and Bob are able to open the box. The mailman (Eve) cannot.
In a highly entertaining and enlightening work entitled The Code
Book (Anchor, 2000) Simon Singh writes that it all happens as if Alice
and Bob had wanted to invent a secret paint without anyone knowing all
their ingredients. They start by choosing a certain color and then each
of them puts a quart of this color in a can. Back home, Alice adds a
quart of a secret color that she doesn’t tell anyone about, not even her
partner. Bob does the same. Then they exchange the two cans with the
mixtures, not caring whether Eve is watching or not. Each of them takes
home the can with the two quarts of color resulting from the addition of
the secret color to the other color. Back home again, Alice adds a quart
of her secret color to the can that Bob gave her. So she now has three
quarts of paint: one third is their original agreed color, one third is Bob’s
secret color and the remaining third is her own secret color. When Bob
gets home he does the same with the can that Alice gave him. The result
is identical to the color Alice obtained, as the ingredients of the paint
are the same. Neither told the other about their secret choice, but they
ended up with the same final color of paint, without anybody being able
to discover their secret, not even Eve, who was always sniffing around
and who had seen them exchange the cans.
There aren’t any colors in encryption systems, but there are num-
bers, as well as the ingenious application of a branch of mathematics
known as number theory. Without these advances in cryptography,
online commerce and communications would not be as secure as they
are today.
 INVIOLATE CYBERSECRETS

Are you apprehensive about sending your credit card number to a web-
site? Is there a CD or book you decided not to buy because the seller
required you to use Visa or MasterCard on the Internet? Well, you are
certainly not the only person who doesn’t trust the web. Many people
around the world still do not take advantage of the efficiencies of e-
commerce because they do not have confidence in the security of online
transactions. But you might be surprised to find out that sharing confi-
dential information through the web is actually one of the most secure
transactions ever devised. If you take some elementary precautions, such
as having nothing to do with sellers who are not known to you and not
sending confidential information by normal (i.e. non-encrypted) email,
the world of e-commerce is at your fingertips.
The Internet opens up possibilities that people did not even dream
of just a few years ago. It has become a giant public library offering
people everywhere rapid and secure access to international commerce.
Do you want to buy that technical manual that you just can’t find in
the bookshops (possibly because for the life of you you can’t remember
the exact wording of its title)? Do you want to acquire that Bob Dylan
CD that you have spent so much time looking for? Are you a collector
searching for a 19th century compass? The internet could help to gratify
all these wishes, as it provides access to various international chains of
second-hand bookshops. One of them might just be offering that long
out-of-print autobiography of Max Planck that you have been searching

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_21, 89


C Springer-Verlag Berlin Heidelberg 2010
90 FIGURING IT OUT

for. Who knows, you might even find a second-hand bookshop in New
Zealand selling it at a bargain basement price on the Internet, as I did.
But is it safe to send a credit card number via all those bits and bytes
that end up God knows where? How can Amazon and all those other
online stores guarantee that your data will not end up in less scrupulous
hands tapping the keyboard of some PC in Cochinchina? They may well
claim that the data is encrypted, but if my message is encoded using a
key that they send to my computer, couldn’t some scoundrel find out
the details of this key?
This question makes a lot of sense. For thousands of years secrets
that were communicated were encrypted using a system based on a
symmetric key, which allows messages to be coded and decoded. The
persons who want to share the secret agree on the key. For example,
they might agree that A is to be written as B, and B as C, and so on. So
if they wanted to write GOOD DAY, the coded version would be HPPE
EBZ. The key that allows this message to be encoded can be reversed to
decode it. The security of the message depends on keeping the key secret.
Encrypted communication on the Internet is based on an innovative
method, the asymmetric key. It is a really revolutionary step in cryptog-
raphy, maybe even the most significant advance since coded messages
first appeared. The method, which is incorporated in browsers, in some
email systems and, of course, in interbank communications, is based on
a suggestion made by Ronald Rivest, Adi Shamir and Leonard Adleman,
three scientists from the Massachusetts Institute of Technology, who
proposed an encryption method in 1977 that became known by their
initials: RSA.
Using this system, the recipient of the message, for instance the
internet vendor, creates a key consisting of two large numbers (N, e). He
sends these two numbers to his client’s computer without paying any
attention to their security. If he wants, he can even publish them in a
newspaper. Then the client’s computer rewrites the message it wants to
send as a numerical sequence (normally in accordance with the ASCII
code), obtaining a third number (M), and then applies a simple equa-
tion: it raises M to the power of e, divides the result by N and calculates
the remainder, obtaining the number C, which it sends back on the
internet.
 INVIOLATE CYBERSECRETS 91

The astounding thing is that this number C, which is the encrypted

message and contains data such as the credit card number, can be viewed
by anybody, because, even if they possess the public code (the num-
bers N, e), the message cannot be decrypted. This is due to the fact
that the mathematical function that transformed the number M into the
number C is not one-to-one: it transforms the original number into a
perfectly determined number, but other numbers could also have pro-
duced the same result, so no Internet huckster gets to know our credit
card number.
So how can the recipient of the message decrypt it? Well, the recipi-
ent, who sent the public key, knows how it was created: he selected N
as the product of two prime numbers (numbers that are only divisi-
ble by 1 and by themselves), let us call them p and q, and he did not
reveal them to anyone. Knowing them, he calculates another number, d,
in such a way that (ed – 1) is divisible by (p – 1)(q – 1). Then he raises
the encrypted number C to the power of d, divides it by N and calculates
the remainder of the entire division. This remainder is the original mes-
sage, M. A miracle? Not at all. It is simply the ingenious application of a
result from number theory known as Euler’s theorem.
What makes this method practically inviolate is that it takes an
extraordinarily long time to factorize a number that is the product of its
prime numbers if these numbers are large. It is easy to obtain the prod-
uct of two numbers. But even if you know that a given number is the
product of the two primes p and q, finding these is anything but simple:
and without knowing them, the message cannot be decrypted.
It is sufficient that the number N is large and that its two factors
p and q have been carefully selected to ensure that the time required
by a computer to factorize them is extremely lengthy, so lengthy that it
is not a practical possibility for any cyberpirate to attempt this feat. For
example, if each of the two factors has 100 digits, and a computer has the
capacity to make a trillion attempts per second, even so the estimated
period of time since the beginning of the universe would not be enough
to ensure that such a computer would succeed in finding out the prime
factors of this number.
As always, each advance in cryptographic techniques is followed by
an advance in cryptanalytical techniques. The RSA method has been
92 FIGURING IT OUT

subjected to various attacks by mathematicians who have attempted to

devise an algorithm to decrypt the private key d, in some cases by means
of the factorization of the public key N to its prime factors. The suc-
cessful outcomes that have been achieved have imposed restrictions on
the choice of the system components and have obliged the experts to
use various stratagems to optimize the security of the system, namely
large numbers for the RSA keys. Until now mathematicians have not
succeeded in developing any form of decryption whatsoever that can
break the RSA code. As it turns out, e-commerce is still much safer than
hiding money under your mattress.
 QUANTUM CRYPTOGRAPHY

The integrity of bank transactions, e-commerce and military signals is

ensured by the utilization of very secure cryptographic systems. Very
secure, however, does not mean absolutely secure. The security of the
most reliable modern cryptographic systems, the RSA system, hinges on
the difficulty of determining the prime factors of very large numbers.
The algorithms that have been devised up to now have not suc-
ceeded in performing this operation within a reasonable time, even
using the most powerful computers currently in existence. But if some
mathematician discovers an effective procedure for performing this
factorization, or if a new generation of computers is introduced (the
“quantum computers” on which many scientists are currently at work),
then the world of communications as we know it today could collapse.
If one of these revolutionary innovations was suddenly available, e-
commerce would cease to be secure, the military would have to review
all its communications systems, and banks would have to take a step
back in time and conduct their transactions at a snail s pace. It could
end the information society as we have come to know it.
What we all need is a new form of cryptography that is 100% secure.
By the time that RSA begins to be vulnerable, mathematicians, physicists
and computer scientists hope to have put in place a new system that will
take its place, and be impenetrable. This will be so because the new sys-
tem’s security will employ the most basic laws of matter, the uncertainty
principles that are at the heart of quantum physics. The absolute impos-
sibility of predicting the behavior of elementary particles will ironically
guarantee the privacy of messages under the new system.

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_22, 93


C Springer-Verlag Berlin Heidelberg 2010
94 FIGURING IT OUT

This idea has already been germinating for some time in the minds
of researchers. Charles Bennett is one such scientist. Together with a
colleague named Gilles Brassard at the IBM Center in New York, in the
1980s he finally succeeded in conceiving a quantum-based cryptography
system. But for years afterwards, in fact until the beginning of the 21st
century, this all remained in the realm of ideas and experimentation.
Recently, though, so much technical progress has been made that it has
become possible to put prototypes of truly secure cryptographic systems
into practice.
One of these cryptographic systems, which is the basis for the
scheme proposed by Charles Bennett and his collaborators, uses a ran-
dom encryption key that is as long as the message itself. The sender of
the message, Alice, starts by digitizing the text to be sent to the recipient,
Bob, by translating it into a sequence of zeros and ones (the binary lan-
guage of computers). To this sequence she adds the key, which consists
of another sequence of zeros and ones that is as long as the original mes-
sage. She transmits the result to Bob, who has the key that Alice used.
Bob subtracts the key and what is left is the original message. To read
it, it will of course be necessary to convert the series of zeros and ones
into a sequence of letters, but this is routine work for any computer. To
make this system truly private it is essential that the key is a random
sequence and that it is only used once. This means that these numbers
have to be generated in advance and that Alice has to get them to Bob.
And that is where the problems begin. If Alice and Bob never meet with
each other personally, as is normally the case with the people involved
in an e-commerce transaction, then they will have to place their trust in
the method used to transmit the key. And how is this to be done? In an
encrypted form? But for that they need to have agreed on another key,
so the problem seems to be impossible to solve. At some point, Alice and
Bob will have to meet or to entrust the information to a messenger. But
as the meddling Eve is always on the prowl, they know they will never
have absolute security.
This is where the world of quantum physics comes in. This world
has strange rules that are impossible to realize intuitively on the basis of
our daily experiences. One of them is uncertainty. And this uncertainty
 QUANTUM CRYPTOGRAPHY 95

is not based on our lack of knowledge, it is an intrinsic property of

subatomic particles.
Alice starts by sending Bob a sequence of light particles, in other
words, a series of photons. Her device contains two polarizing filters,
one that is oriented vertically and another that is at an angle of 45◦ , as
shown in the illustration. To create the key, the device alternates the
polarizing filters randomly, for example assigning the number 0 to a
photon that is polarized vertically and the number 1 to a 45◦ polarized
photon. In Bob’s device there are also two polarizing filters, one posi-
tioned horizontally and another at an angle of –45◦ . When the device
receives each photon it passes it in completely random fashion through
one of the filters.
The photons sent by Alice and received by Bob may or may not pass
through his device depending on the combination of polarizing filters.
If Alice sends a vertically polarized photon and Bob’s device makes it
pass through the horizontal filter, then the photon will be retained and
will not pass through the device. If Alice sends a 45◦ polarized photon
and Bob’s device passes it through the –45◦ filter, then the particle will
also be retained and will also not pass through. Polarizing filters that are
perpendicular to the direction of polarization retain the particles.
Surprising things happen, though, when Alice sends a vertically
polarized photon and Bob’s device passes it through the diagonal fil-
ter, or when Alice sends a diagonally polarized photon and Bob’s device
passes it through the horizontal filter, that is when the difference in the
direction of polarization between the filters in the two devices is a 45◦
angle. In this case quantum uncertainty enters the field: half of the par-
ticles pass through Bob’s filter, but the others are retained by it, and it
is impossible to know in advance which particles will pass through and
which will be retained.
At this point only Alice knows the polarization of the photons that
were sent, while only Bob knows which photons made it to their destina-
tion. Therefore Bob gets to know the polarization that Alice used for the
photons that passed through the filters, as a photon that passed through
his diagonal filter must have been polarized vertically by Alice, which
means it has the value 0. A photon that passed through his horizontal
96 FIGURING IT OUT

filter must have been polarized diagonally by Alice, so it has the value 1.
However, Bob knows nothing about the particles that were retained.
Alice now needs to know which photons passed through the filters
in Bob’s device. For this purpose she can contact Bob by means of any
channel whatsoever, however unsecured. This conversation can even be
overheard by Eve, as even if she knows which photons were received by
Bob’s device, she will not know which filter they each passed through. So
now the key has been set up using only the photons that completed their
journey and Alice and Bob can communicate in absolute security. It is
the quantum physics uncertainty principle that gives them the certainty
that nobody is eavesdropping.

Bob’s
Alice’s filters
filters

Light
source
Polarized photons

Photons

Alice’s bits 1 0 1 0 1 0
Alice’s polarization ⁄ I ⁄ I ⁄ I
Bob’s polarization \ \ – – – \
Bob’s result N N Y N N Y
Agreed key --- --- 1 --- --- 0

Only a few years ago all this would have sounded like science fic-
tion, but recent technological advances have thrust it into the realm of
possibility. As you can imagine, there are immense technical difficul-
ties still to confront. For example, how can you emit light photon by
photon? How do you ensure that these particles reach their destination?
One by one these problems have been solved. Scientists have succeeded
in putting quantum cryptography into practice via optical cables and via
air for several miles. Maybe we are not far away from being able to pro-
tect our secrets by making them travel particle by particle. At the speed
of light.
 THE FBI WAVELET

The language of mathematics can seem esoteric and purely abstract, but
many of its constructions end up having surprising applications. One
of the most recent and spectacular successes of mathematics is taking
place in the processing of signals, and in particular in the processing of
images. This new technique has a quaint name: wavelet analysis.
As always, this new tool didn’t fall from the sky: its origins can be
traced to the work of the French mathematician Jean Baptiste Joseph
Fourier (1768–1830), who created a technique known as “harmonic
analysis”, or more commonly, Fourier analysis. Jean Baptiste studied at
the military academy in Auxerre, the city of his birth, where he stayed,
drawn to mathematics. He later entered a seminary with the objec-
tive of becoming a monk, but then the French Revolution intervened.
Gradually he began to support the revolutionary movement, and even-
tually declared himself ready to fight for a free government, liberated
from kings and priests. In the turmoil of the revolution he was taken
prisoner and nearly condemned to death. Subsequently, he lectured in
Paris and accompanied Napoleon on his Egyptian campaign as a mem-
ber of his scientific committee. When he returned to France, he was
appointed prefect of the Isère department, worked in the institute of
statistics, and was elected a member of the Academy of Science in Paris.
Fourier’s most significant scientific treatise, La Théorie Analytique
de la Chaleur (The Analytical Theory of Heat) is one of the scientific
milestones of the 19th century. Fourier surprised his contemporaries by
contending that functions could be represented as the sums of waves,
that is, by the sums of the known trigonometrical functions sine and

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_23, 97


C Springer-Verlag Berlin Heidelberg 2010
98 FIGURING IT OUT

cosine, the sinusoidal waves. He believed it was often easier to mathe-

matically process the sums of waves than their original functions, and
that the sums are equivalent to the initial functions, provided a suffi-
cient (possibly infinite) number of constituents are added. It has turned
out that this seemingly odd idea is very versatile. Mathematicians have
come to rely upon Fourier analysis to resolve many problems that would
have been impossible to process by any other means.
In 1965 the work performed by James Cooley and John Tukey at
the Bell Laboratories in New Jersey transformed Fourier’s idea into an
extremely practical technique. Cooley and Tukey created a new algo-
rithm to calculate the Fourier series, calling it the Fast Fourier Transform
(FFT). Today it is used in a wide variety of areas, from the analysis
of radio signals or econometric forecasting to clinical studies of brain
waves. The mathematician Gilbert Strang, from MIT, observed that
“whole industries are changed from slow to fast by this one idea – which
is pure mathematics”.1
More recently, in the 1970s and 1980s, various engineers and math-
ematicians began to attempt to resolve certain practical limitations of
the Fourier series. For instance, if we want to codify one of Bartok’s con-
certs using sinusoidal waves, then we need a huge number of such waves
because the concert has brusque changes, whereas the waves continue
indefinitely. Mathematically, this means that a gigantic number of coef-
ficients is required to codify it properly. This is not at all practical for the
purposes of a digital recording.
Fourier’s technique, however, is perfect for capturing one specific
moment of the concert. To recreate the original sound you only have to
synthesize the different frequencies, the different notes, and reproduce
the timbres of the different instruments, which are added to these notes.
But because the notes are in constant flux, using the waves to capture the
entire concert is not a practical proposition. The idea that some people
had was to create mini-waves with a precise beginning, middle and end,
and to use those mini-waves to analyze the original signal. Yves Meyer, at

1
Wavelet transforms versus Fourier transforms, Bulletin of the American Mathematical Society,
28–2, April 1993, 288–305.
 THE FBI WAVELET 99

the École Polytechnique, and Ingrid Daubeschies, a Belgian mathemati-

cian who was working at Bell Laboratories at the time, played a major
role in the development of this idea. A new mathematical tool had been
born.
In French the new functions were called ondelettes, or small waves.
So in English they became known as wavelets.
When the FBI consulted a group of mathematicians on how best to
process the enormous archive of fingerprints held by the federal agency,
they proposed using wavelets. The FBI had been storing fingerprints
since 1924. By 1996 their archive contained 200 million files, and it
continues to grow at a rate of almost 50,000 new files every day.
When the FBI had earlier begun transmitting images from the
archive electronically, they had observed that the system they were using
was very slow, and that images had to be compressed in order to be
properly transmitted. JPEG, the most commonly used compression sys-
tem on the internet, produced a very grainy image. When a significant
compression of the image was required, the details disappeared and the
sharp transitions between lines were blurred.
After a great deal of mathematical processing and many experi-
ments by the consulting mathematicians, the FBI decided to switch to
a wavelet-based system for compressing the images. They created a new
wavelet that was uniquely suitable for reproducing fingerprints, which
made it possible to achieve significant reductions in the size of the
electronic files. Another mathematical achievement.
 THE ENIGMA MACHINE

Since ancient times men have dreamed of constructing an automatic

coding machine. As far as we know, the first attempt was made by
the Renaissance architect Leon Battista Alberti (1404–1472), who posi-
tioned one concentric disk on top of another, each one inscribed with
all the letters of the alphabet. By turning one disk to a certain position
with respect to the other, he was able to pair each letter with another,
which served to automate the task of encrypting messages by substitut-
ing letters. The device may only have mechanically reproduced actions
that could be done mentally, but it did ensure that no substitution errors
were made in the process.
The messages obtained by this simple process can be easily
decrypted by statistical analysis. In English, for example, the most
frequently occurring letter is E (with a mean frequency of 12.7%),
followed by T (9.1%) and A (8.2%), with Z at the end of the list (0.1%).
If the encrypted message is sufficiently long, it is relatively simple to
identify some of its letters. Then, half the job is done. After some letters
and parts of a message have been decrypted, it is usually simpler to
complete the task.
In 1918 the German inventor Arthur Scherbius (1878–1929) devel-
oped an automatic coding machine that used disks similar to those
Alberti had constructed but also featured some improvements. It resem-
bled a typewriter and it used electrical connections, transforming
anything typed on the keyboard into an encrypted message that was
displayed in illuminated letters. Three disks in a series, usually called
rotors, formed the core of his invention. The first disk translated the

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_24, 101


C Springer-Verlag Berlin Heidelberg 2010
102 FIGURING IT OUT

original letter into a coded letter to be transformed by the second disk

into yet another letter, which in turn was transformed by the third disk.
Scherbius’ decisive improvement was to rotate the position of the disks.
After the first disk had rotated through all of its positions, the sec-
ond disk then rotated one position. After the second disk had rotated
through all of its positions, the third disk rotated one position. By this
means, Sherbius ensured that any letter’s encryption would be repeated
only after a complete cycle of all three disks. Using our alphabet of 26
letters, this means a cycle of 26 × 26 × 26 = 17,576 positions before the
code is repeated, which is a sufficiently large number to prevent the use
of statistical analysis in discovering the frequencies.
Each of the disks in the machine invented by Scherbius had its own
corresponding letters. For example, if one of them transformed A into
S and B into F, etc., then another might transform A into H, B into Z,
and so on. By rotating the removable and permutable disks, his machine
multiplied the number of possible transformations by 3 × 2 × 1 = 6.
Finally it had a series of electrical connections that exchanged six of the
letters each time. Now, the number of possible ways of connecting six
pairs of letters in the alphabet is gigantic, more than one hundred thou-
sand million. All in all, Scherbius’ machine allowed about ten thousand
trillion codes to be created. Scherbius called his invention the Enigma
machine.
Initially Scherbius’ invention was a commercial disaster. However,
a few years later, the potential of the Enigma machine was recognized
when the German armed forces were beginning to make prepara-
tions to launch WWII. The German military ordered thousands of the
machines, convinced that by using it, their encrypted military secrets
would be kept completely safe. In this they were mistaken, as a Polish
statistician and an English mathematician would discover a way to break
the Enigma code.
Marian Rejewski was 23 years old when he was summoned to work
with the Biuro Szyfrów, the Polish General Staff’s Cipher Bureau. His
recent university training in mathematics and statistics was tailor-made
for the Bureau’s new recruitment policy, which was to concentrate on
hiring young mathematicians. Rejewski (1905–1980) proved to be a real
 THE ENIGMA MACHINE 103

catch, as he developed pioneering techniques that had been thought

impossible until then. By the early 1930s, at a time when the British
and French were despairing of ever succeeding in breaking the German
code, the Poles were able to decrypt the German messages in a mat-
ter of hours. But in December, 1938, improvements were made to the
Enigma machine which made breaking the German code more difficult:
an additional two rotors brought the total up to five, from which three

Alan Turing (1912–1954) and his colleagues at Bletchley Park in southern England
knew that the German messages had to contain some inevitable words, such as
“weather”, “attack” or “ship”. The search for sequences that might contain such
words, called cribs, was one of the techniques most often used to decrypt messages.
The film Enigma, based on the book of the same name by Robert Harris, included
some very emotional scenes of code-breakers searching for cribs. If you want to
understand the mode of operation of the Enigma machine and the story of how it
was broken, the standard reference source is David Kahn’s book, Seizing the Enigma
(published by Arrow, London, 1966). Another remarkable book that devotes many
pages to Enigma and includes a clear explanation of the principles used by the
machine and by the code-breakers is The Code Book, by Simon Singh. If you are inter-
ested in learning how the Enigma machine works, you can also look on the internet
at the website www.bletchleycovers.com, where you will find many links to many
other sites as well as a virtual Enigma machine where you can type in messages and
watch as they appear in an encrypted version
104 FIGURING IT OUT

were selected to be used, and the number of electrical connections was

expanded from six to ten. Rejewski had proved that the German system
was not invincible, but now the task had become too complex for the
computational means at his disposal. The Poles, who assumed that they
were soon going to be invaded by the Germans, decided to reveal their
knowledge of the Enigma machine to the British and French. On August
16, 1939, two replicas of the machine and two sets of decryption plans
were sent to London and Paris. Two weeks later, on September 1, Hitler
invaded Poland and the war officially began.
The British foresaw the importance of cryptography, and they
assembled an exceptional group of experts at Bletchley Park, a stately
home 50 miles to the northwest of London. One of those gifted
cryptologists was the mathematician Alan Turing (1912–1954), who
would eventually become the main protagonist in breaking the Enigma
code. Inspired by his previous studies on a theoretical computing
machine, Turing detected various weak points in the German system
and invented machines (which were called “bombes”) that could auto-
matically reproduce the sequences of the rotors in the Enigma machine.
His success would play a decisive part in the victory of the Allies.
Churchill called the cryptologists at Bletchley Park “the geese that laid
the golden eggs but never cackled”.1

1
http://infosecurity.us/?p=5735
ART AND GEOMETRY
 THE VITRUVIAN MAN

“The Vitruvian Man!” Langdon gasped. Saunière had created a life-sized

replica of Leonardo da Vinci’s most famous sketch.
This is one of the introductory scenes in The Da Vinci Code.1
The museum curator Saunière had drawn a circle around himself and
painted a five-pointed star in blood on his stomach. In the following
500 pages Professor Langdon will provide an explanation of the geome-
try of the star and the figure of the Vitruvian Man. The book is a mixture
of facts and fiction, which is perfectly acceptable as it is a novel. But as a
reader you are entitled to know which of the book’s facts have not been
embellished.
The five-pointed star, also known as a pentagram, is a geomet-
rical figure that has been a subject of investigation and curiosity for
thousands of years. It has represented a mystical symbol for various
civilizations, in some cases being associated with the supreme divin-
ity, and in others, with the Devil. Placed on its “feet” it resembles
the human figure. With one point at the bottom and two points at
the top it looks like a horned animal, naturally assumed to be of
a diabolical nature. In this magical association it is often called a
pentacle.
The ancient Greeks studied the geometry of the pentagram. The
mathematician Euclid (fl. 300 BC) showed how to draw it with a non-
graduated ruler and a pair of compasses, which gave it a classical
geometrical dignity. He also revealed some of its curious properties.

1
P. 45 of the first American edition, DoubleDay, New York, 2003.

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_25, 107


C Springer-Verlag Berlin Heidelberg 2010
108 FIGURING IT OUT

If you join one point of the star to the two points opposite, you
obtain an isosceles triangle. This triangle has two angles of 72◦ and a
third angle of 36◦ , which is half of either of the larger angles. Such a
polygon is called a golden triangle. Curiously, if we bisect one of the
larger angles, thus dividing the original triangle into two, the smaller
resulting triangle is similar to the original triangle in that it is also a
golden triangle. If we divide this angle using the same procedure, we
can construct an infinite series of golden triangles, one inside the other.

By joining the points of the pentagram we can draw another curious

geometrical structure, a regular pentagon that contains the star. Looking
at its center we discern another regular pentagon. This means that we
 THE VITRUVIAN MAN 109

The Golden Triangle …

36°

72°

72° 72°
36° 72°

1. The ratio between the 2. If you divide a golden 3. Such division can be
long and short sides of a triangle you will obtain repeated indefinitely
golden triangle is the golden another golden triangle
number Φ = 1.6180339...
… and the Pentagram

c
d

b
1. With two golden triangles 2. Two pentagons can be 3. … with an infinite series
we can construct a seen in the star … of pentagons and stars, one
pentagram inside the other, each with a
ratio that is always equal to
the golden number:
a b c
= = = ... = Φ = 1.618...
b c d

can construct an infinite series of pentagons and pentagrams, one inside

the other.
The ratio between the shorter and longer sides of a golden trian-
gle is an irrational number that is approximately equal to 1.618. This
means that the distance from one of the five points of the star to one of
the opposite points is equal to this number times the distance between
two contiguous points. And that is not the end of the surprises: the ratio
between this latter distance and the length of a segment of the star is also
this mysterious number. You may already have guessed that we are talk-
ing about the golden number or  (phi), which is referred to so many
times in Dan Brown’s novel. This is the same number that appears as the
110 FIGURING IT OUT

limit of the ratios between successive terms in a Fibonacci sequence, also

discussed in the novel. It is not surprising that Dan Brown, the author
of The Da Vinci Code, with his passion for numerology and the occult,
was so intrigued by this marvelous number.
Dan Brown finds the golden number  again in the figure of the
Vitruvian Man, Leonardo’s drawing. But in this, Brown starts to invent
relationships that do not exist. In reality the drawing’s design is based on
the human body’s simple proportions, proportions that are expressed as
integers, not as irrational numbers like .
In creating the drawing, Leonardo followed the instructions of the
Roman architect, Marcus Lucius Vitruvius Pollio (about 90–20 BC).
In his work De Architectura (known in English as The Ten Books of
Architecture), Vitruvius describes the ideal proportions between the var-
ious parts of the human body. For example he wrote that the foot should
be one sixth of the height of the body, whereas the cubit, or forearm,
should be one quarter of a person’s height. He also asserted that build-
ings should be constructed with well-defined proportions, starting with
a “perfect number”. (Subsequently he questions whether this number is
six, considered the first perfect number, or ten, the number which “Plato
held as the perfect number”. In defense of the number six, he observes
that mathematicians “have said that the perfect number is six, because
this number is composed of integral parts which are suited numerically
to their method of reckoning: thus, one is one sixth; two is one third;
three is one half; four is two thirds, five is five sixths, and six is the perfect
number” 2 ).
No traces of the golden number can be discerned in the works of
Vitruvius. Despite a very widespread but mistaken view, even in aca-
demic studies, it does not appear in Leonardo’s drawing either. Why
don’t you take the trouble to accurately measure the ratio between the
circumference of the circle and the side of the square in the drawing of
the Vitruvian Man? By using a simple ruler and tape measure you can
lay to rest many claims in the realm of numbers . . .

2
Morris Hickey Morgan translation of De Architectura in http://www.perseus.tufts.edu/cgi-
bin/ptext?doc=Perseus%3Atext%3A1999.02.0073;query=chapter%3D%2321;layout=;loc=
3.preface%201
 THE GOLDEN NUMBER

In a golden rectangle the ratio between the longer side

L and the shorter side l is the same as the ratio between
the longer and the shorter sides of the rectangle that remains
once the square has been removed from the original rectangle.
The smaller rectangle has sides l and L – l. This way, assuming
for the sake of simplicity that l = 1, then L/1 = 1/(L – 1) and
2
so L – L = 1. Solving this quadratic equation we obtain the
admissible solution L = (1 + √5)/2 = 1.6180339..., i.e., the
golden number Φ. The ratio has to be Φ.

There are numbers that surprise us. They pop up unexpectedly in all
sorts of situations. For example, take π, the number that represents the
quotient of the perimeter of a circumference by its diameter. This num-
ber also appears in equations representing the area of a circle, as well
as the surface and volume of a sphere. It is not difficult to accept this, as
the circumference must have something to do with these other measure-
ments. But it is not so easy to understand the reason why π also appears

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_26, 111


C Springer-Verlag Berlin Heidelberg 2010
112 FIGURING IT OUT

in statistics, in complex exponential functions and even in the sum of

numerical series1 such as 1 + 1/4 + 1/9 + 1/16 . . .
Another of these surprising numbers is the golden number, also
known as the golden ratio, or divine proportion. The golden number
is normally represented by the Greek capital letter  (phi), and is equiv-
alent to half the sum of one plus the square root of five. It is an irrational
number, given as the infinite non-periodic decimal 1.61803398. . .
Mario Livio, an American astronomer at the Hubble Space
Telescope Science Institute, has published a book in which he calls the
golden number the most surprising number in the world (The Golden
Ratio, Review, London). It may seem astounding that somebody could
write an entire book whose only subject is this strange number, but really
it is only the latest of several books and innumerable articles on this
topic.
Based only on the equation described above, the golden number
doesn’t immediately seem to be anything special. It’s only another num-
ber. But it does start to become more surprising the more you see where
it pops up.
Let’s begin at the beginning, or at least the first time that we know
this number was explicitly mentioned and discussed. As with many
other mathematical matters, this takes us back to the Elements of Euclid,
the most influential work in the entire history of mathematics. In Book
VI of this work, Euclid defines what he calls “cut in extreme and mean
ratio” (Definition 3). He explains that this is the division of one segment
into two unequal parts with one particular property: the quotient of the
entire segment and the larger part is equal to the quotient of the larger
and smaller parts. When you do the calculation you will see that this
ratio has to correspond precisely to , the golden number.
This subject was referred to again many times, notably in the
13th century by Leonardo of Pisa (about 1170–1240), better known as
Fibonacci, and by Fra Luca Pacioli (1445–1517), who first introduced
the expression “divine proportion”. It was only in the middle of the 19th

1
In fact, the sum of the series of the inverse squares here initiated is a fourth of the square
of pi.
 THE GOLDEN NUMBER 113

century that terms such as “golden ratio” and “golden number” were
coined. The letter phi () as a symbol for this number only appeared at
the beginning of the 20th century.
The number  appears in many geometrical constructions. For
instance, let’s take the isosceles triangle (two equal angles) in which the
smaller angle is half of either of the two larger (equal) angles, i.e., one
angle is 36◦ and the other two are each 72◦ . In this instance, the golden
number appears as the ratio between the long and short sides of the
triangle. In addition, if we divide one of the larger angles in half, we
obtain two triangles, the smaller of which is similar to the triangle that
we divided — the sides have the same ratio as the original triangle and
the angles are identical.
However, the most celebrated geometrical construction of all is the
so called “golden rectangle”, in which the ratio of the rectangle’s sides is
the number .
There has been much speculation on the aesthetic properties of this
golden rectangle. There are those who maintain that the proportions
are so perfect that they were incorporated into such ancient architec-
tural masterpieces as the façade of the Parthenon in Athens. Likewise,
there are those who claim that the golden number appears in the Great
Pyramid in Egypt, as the ratio between the height of a lateral trian-
gle and half of its base. As the mathematician George Markov, from
the University of Maine (www.umcs.maine.edu/∼markov) recently
demonstrated, though, these speculations are not based on reality. What
happens is that, as all these monuments have so many possible measure-
ments that can be compared, it only takes several attempts to find an
approximation of one interesting number or another. But what is not in
doubt is that the golden rectangle is especially beautiful and possesses
an aesthetically appealing appearance. In some studies where persons
were requested to select one of several rectangles as the most beautiful,
many of those surveyed did in fact choose the golden rectangle. This
geometrical figure seems to be more visually pleasing than, for example,
the rectangle of a sheet of A4 paper.
However, both these rectangles can be divided successively into fig-
ures that are always similar. In the A4 rectangle, the division is in half,
114 FIGURING IT OUT

creating two rectangles with the ratio of the sides being identical to those
of the original rectangle.
A different method is required if you want to successively divide a
golden rectangle to produce similar rectangles. The original golden rect-
angle is divided in such a way that a square and a rectangle are obtained.
There is only one way to do this, which is to create a square with sides
that are equal to the shorter side of the original rectangle. It just so
happens that the remainder is another golden rectangle.
If you divide a golden rectangle successively in accordance with this
rule, you will obtain ever smaller rectangles, one inside the other. We
can draw a spiral in them that converges on a point (a “pole”) that is at
the intersection of two diagonals: that of the original rectangle and that
of the golden rectangle created by the first subdivision. The spiral drawn
in a succession of golden rectangles is called a “logarithmic spiral” and is
to be found in the most varied situations, including the shells of marine
animals, the flight trajectory of falcons, flowers and galactic spirals.
One of the most amazing examples of the golden number can be
found in the arrangement of the petals of a rose. They are separated
at an angle that is a fraction of . This arrangement permits the posi-
tioning of the petals in a compact form and simultaneously maximizes
their exposure to light. Just like mathematicians, nature too seems to be
eternally fascinated by the golden number.
 THE GEOMETRY OF A4 PAPER SIZES

The paper format generally used in photocopiers and printers every-

where outside North America, and which is also generally used for
letters and writing pads, has the curious name of A4. Measuring 210 ×
297 mm (approximately 81/4 × 113/4 in.), A4 sheets are an unusual size;
it would certainly seem more logical if this measurement were a round
number. Why not 200 × 300 mm, for example?
It turns out these bizarre dimensions are actually the result of
a carefully considered convention, which has now been adopted by
most countries in the world, and is based on the German DIN sys-
tem (Deutsches Institut für Normung e. V.). The German standard was
adopted as the ISO 216 standard by the International Organization for
Standardization. In accordance with this standard, there is a series of
basic paper sizes, starting with A0, the largest, and then decreasing in
size to A1, A2, A3..., and ending in the minuscule A10, which measures
only 26 × 37 mm (about 1 × 11/2 in.). All these sizes are designed in such
a way that the size with the next highest number can be obtained by
folding the sheet in half. To take an example: if you fold an A0 sheet in
half, you have an A1 size, or folding an A4 sheet in half gives you an A5.
But there is a lot more to this standard than that. The sizes are designed
so that they will always maintain the same ratio between the length
and width of the paper. The sizes are rounded off to the nearest mil-
limeter (about one twenty-fifth of an inch), which is quite a reasonable
approximation.
This rule has many practical applications, especially in photo-
copiers. If you place two A4 sheets side by side on the photocopier and

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_27, 115


C Springer-Verlag Berlin Heidelberg 2010
116 FIGURING IT OUT

select the reduced size mode, it is possible to photocopy each of the two
original A4 sheets exactly on one half of the A4 copy produced by the
machine. It is easy to see that not every paper format would allow you to
do this. For instance, if the original sheets were square, and we wanted
to copy two of them, in reduced size mode, on to a single square sheet,
we would have to waste half of the photocopy. But with the A4 system
there is no waste, as the ratio is maintained when we fold the sheet in
half.
What format does a sheet of paper have to have in order for it to
maintain its ratio when it is divided in two? A few simple calculations
will reveal that. The sides of the rectangle have to have a ratio of one to
the square root of two (approximately 1.4142). There is no other solu-
tion. If you do the math, you will agree that 210 × 1.4142 = 296.982, or
for all practical purposes 297. Those are the proportions of the A4 series.
All this follows a perfectly logical system, but we still have to define
our starting point. How was the A0 size designed? Curiously enough,
this was not an arbitrary decision either. It was defined with the ratio
of the length and width set at one to the square root of two, as usual in
this system, but with the added restriction that the A0 size has an area
that is equivalent to one square meter (about 10.75 square feet). That
completed the definition of the A4 system. Luckily, the result was an A4
paper size that is excellent for office work.
This standard also makes it simple to calculate the weight of a ream
of paper, for example. The weight of the paper, or “grammage” as the
professionals say, is calculated in grams per square meter. Normally we
might use paper that weighs 80 g/m2 . This means that one A0 sheet of
this paper weighs 80 g. As one A0 sheet has the same surface area as 16
A4 sheets, each A4 sheet weighs 5 g and the whole ream (500 sheets) of
A4 paper weighs 2500 g or 2.5 kg (approximately 5 pounds 8 ounces).
The system defined by the ISO 216 standard also encompasses two
other series of sizes. These are the B series, used for envelopes that con-
tain the equivalent sheets of paper from the A series, and the C series,
used for slightly smaller envelopes that may contain fewer sheets of
paper. For example, if you need an envelope to mail an A4 brochure,
you can use a B4 envelope, which measures 250 × 353 mm (almost
 THE GEOMETRY OF A 4 PAPER SIZES 117

So L2/2 is W2
The L/W ratio has to be √2

The paper is designed so that the length to

width ratio, L/W, is maintained when the

paper is folded in half.

Then the longer side of the paper, L, is

reduced by half (L/2), and the shorter side,

W, becomes the longer side. The

proportions are maintained: L/W = W/(L/2)

10 × 14 in.). If you want to mail a thin A4 document, you could use a

C4 envelope, which measures 229 × 324 mm (about 9 × 123/4 in.). This
makes it simple for retailers to know what their customers need.
But the exact dimensions of the envelopes are also mathematically
logical. The geometrical mean between two consecutive A sizes was used
to define the dimensions of the B series. For example, the geometrical
mean between the dimensions of A4 and A3 paper was used to cal-
culate the dimensions of a B4 envelope. A similar procedure was also
used to calculate the C series, so that the C4 envelope is defined by the
geometrical mean between the A4 and B4 sizes.
The geometrical mean is a mean, as a pedant would state, that results
in intermediate dimensions between extreme values. But it is a special
mean. It is obtained from the square root of the products of two values.
That is why it maintains the relative proportions. So B4 is to A4 as A3 is
to B4.
118 FIGURING IT OUT

This whole complex system evolved over a period of two hundred

years, and finally it was adopted almost all over the world. As far as we
know, the first person to think of standardizing paper sizes using sim-
ilar rules was a German professor of physics called Georg Christoph
Lichtenberg (1742–1799). In a letter he wrote in 1786 to his friend
Johann Beckmann, he described the aesthetic and practical advantages
of using paper with a length to width ratio of one to the square root
of two. With regard to the practical advantages he was certainly right,
but opinions are divided about the aesthetic aspects. Graphic designers
know that the A4 system is not aesthetically advantageous for placards
or magazines—it is not used for the advertising posters you see in the
streets or for magazines. This is one of the reasons why Americans do
not want to abandon their good-looking letter format (8 × 11 in.) that
they have been using for many years.
The practical advantages of Lichtenberg’s method were seen by the
French revolutionary government when it decided to adopt it along
with the metric system. In 1794 the French Loi sur le Timbre defined
various formats equivalent to the current ISO standard sizes. The for-
mats defined were called grand registre (currently A2), grand papier
(B3), moyen papier (A3), petit papier (B4), demi feuille (B5) and effets
de commerce (1/2 B5). Only the A4 size was missing.
Today the ISO standard is used in most parts of the world except in
North America. So the next time you pick up a sheet of A4 paper, just
remember that you are holding an object with a remarkable mathemat-
ical history.
 THE STRANGE WORLDS OF ESCHER

Maurits Cornelis Escher was born in 1898, in the city of Leeuwarden in

the Netherlands. During his life he produced the most intriguing and
mathematically sophisticated woodcuts any artist has ever created.
The young Maurits had an ordinary childhood. His grades at school
were average and he showed little interest in his studies, but under pres-
sure from his family he agreed to study architecture at the Haarlem
School of Architecture and Decorative Arts, where he met a graphic
arts teacher who would transform his life. This teacher, a Dutch Jew
of Portuguese descent named Samuel Jessurun de Mesquita, taught his
N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_28, 119

C Springer-Verlag Berlin Heidelberg 2010
120 FIGURING IT OUT

student design techniques and awakened his interest in lithographs and

woodcuts. Maurits Escher soon gave up his study of architecture and
switched to graphic art, where he was tutored by Jessurun de Mesquita.
Once he had completed his studies, Escher decided to see the world.
He spent time in Spain and then Italy, where he met his future wife,
and moved to Rome. He was always able to pay his way by working
as a graphic artist. However, when the political climate in Italy became
unbearable under Mussolini, he moved to Switzerland, and later to
Belgium. In 1941 he returned to the Netherlands, where he spent the
rest of his life.
On one of Escher’s trips to Spain he visited Granada. He was
entranced by the Moorish tiles and the intricate patterns that are typ-
ical of Arab art. This experience fed his passion for geometric grids, the
division of a plane into geometrical figures, with a motif that repeats
itself, reflects itself, becomes displaced and rotates.
Escher was fascinated by the challenges and limitations of two-
dimensional representation, which is clearly evident in nearly all of
his work. One of Escher’s principal preoccupations was the creation
of illusory three-dimensional images on plane surfaces. But above all,
he was intrigued by the conflict of perspectives. The lithograph enti-
tled Drawing Hands, which he produced in 1948, depicts a hand that
is drawing a hand that is drawing a hand that is drawing a hand . . .
The hands belong to a two-dimensional world and also simultaneously
to a spatial world. They pass from one to the other. Escher argued that
we are trained to see either a plane surface or a volume in an image.
Either one or the other. He found it interesting to provoke a conflict of
representations.
But this lithograph is also interesting for another reason. It illu-
minates one of the fundamental problems of modern times. In The
First Moderns (Chicago, 1997), a study of the scientific and artistic
revolutions of the last 100 years, author William Everdell posits that
the “barber paradox”, attributed to the mathematician, logician and
philosopher Bertrand Russell (1872–1970), is one of the milestones of
the early 20th century. The paradox consists of the question “Who
 THE STRANGE WORLDS OF ESCHER 121

shaves the barber who shaves all but only those who do not shave them-
selves?” Whatever you reply, you contradict the terms of the question.
Using similar paradoxes, Russell effectively questioned the attempts by
Gottlob Frege (1848–1925) and generations of mathematicians before
and after him to construct a perfect, self-referencing logic. Using sim-
ilar arguments, Kurt Gödel (1906–1978) showed that the endeavors of
generations of mathematicians at the turn of the century, notably David
Hilbert (1862–1943), to base mathematics on an eternally perfect and
once-and-for-all complete logic were bound to fail. A hand drawing a
hand drawing a hand cannot be a real hand.

Relativity, 1953

Another interesting example of the strange world of Escher is pro-

vided in Relativity, a lithograph produced in 1953. It contains three
completely different worlds that are combined within one image. In each
one the perspective is coherent, creating its own world. But the ceiling
of one world is the wall of another. At the junction of two worlds a door
to one is also a trapdoor to another. Two of the central staircases can
122 FIGURING IT OUT

be used from either side (both upside-up and upside-down), with one
world on one side and another one on the other.
Each of these worlds has its own field of gravity, each of them pulling
in their own direction. One of the most interesting surprise effects of this
lithograph can be found in the persons who seem to be strolling through
worlds that are not their own. The man carrying a sack in the center of
the picture belongs to the world on the left, where gravity moves objects
from left to right. He appears to be walking through other worlds, yet,
for him, everything is coherent and he will be able to reach the garden
on the upper right edge of the picture by simply turning right and then
going up the stairs. On these stairs, the ones at the top of the image, are
two figures from two different worlds.
Bruno Ernst, who has studied the work of Escher, has suggested that
it might be useful for astronauts in gravity-free situations to contemplate
this image, as they need to become accustomed to using any plane of
reference in their compartments. They can even pass each other moving
in perpendicular directions. In space they might encounter the strange
worlds of Escher.
 ESCHER AND THE MÖBIUS STRIP

Escher was once quoted as saying: “In 1960 I was exhorted by an English
mathematician (whose name I do not call to mind) to make a print of a
Möbius strip. At that time I scarcely knew what it was”.1 He responded
to this challenge by producing two images that became famous: Möbius
Strip I and Möbius Strip II, which I’ve reproduced here. In the first of
these woodcuts, which seems to depict three snakes biting each others’
tails, Escher invites us to follow the line of the snakes. What we dis-
cover, to our surprise, is that the three reptiles are all on the same surface

1
Bruno Ernst, The Magic Mirrors of M.C. Escher, Taschen America, 1994, page 99.

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_29, 123


C Springer-Verlag Berlin Heidelberg 2010
124 FIGURING IT OUT

even though they appear to be following two distinct orbits. In the sec-
ond woodcut, Möbius Strip II, we see nine ants all crawling in the same
direction. This time Escher asks us to follow their path and confirm that
it is indeed a path without end, because no matter which starting point
you choose, you always end up at the same point. The ants appear to be
crawling on two separate sides of a single surface, but ultimately each of
them travels the entire length of the surface on which they are crawling.
In both these images the paths are endless.
As we’ve noted, these two woodcuts are based on what we today call
a Möbius strip, named after its discoverer, a German mathematician and
astronomer who performed important work in geometry, topology and
complex analysis. The woodcut with the ants shows an actual Möbius
strip.
August Ferdinand Möbius was born on November 7, 1790, in
Schulpforta, Germany, near both Leipzig and Jena in Saxony. He died
in Leipzig in 1868. For nearly his entire adult life he was a professor of
astronomy at the University of Leipzig. At the start of his mathematical
studies, he worked with Carl Friedrich Gauss (1777–1855), one of the
greatest mathematicians who ever lived, and the only one up to now to
be dubbed a “prince of mathematics”. He completed his studies by writ-
ing his doctoral thesis on the occultation of fixed stars, and then, like so
many scientists at that time, he dedicated his life to the study of both
mathematics and astronomy. In the course of his life the field of mathe-
matics in Germany was completely transformed. At the time of his birth
it would have been difficult to identify even two German mathemati-
cians of international renown, but by the time he died, Germany had
become one of the main centers for the teaching of and further research
into mathematical concepts, which gave it a powerful influence on scien-
tific matters throughout the world. Möbius was a key participant in this
extraordinary development, which is not surprising given the political
and social changes that were then taking place in Germany. Of course, it
was during this period that Germany ceased to be a conglomeration of
small states, and became an empire under the aegis of Prussia.
Among mathematicians, Möbius is known for many results and
constructions, principally the transformation (function) that is named
 ESCHER AND THE MÖBIUS STRIP 125

Making a Möbius strip

To make a Möbius strip, cut a long piece of
paper, twist it 180° once and stick the
two ends together.
A1 A2
A1 A2

B1 B2
B1 B2
If you cut the Möbius strip lengthwise
in the center, you will obtain another strip
that is twice as long.
B1 A2

A1 B2

Newspaper Expresso
If you cut the Möbius strip lengthwise at one-third
of its width, you will obtain two interlinked Möbius
strips, one long and one short.

after him, and which continues to play an important role in complex

analysis. He is also known for various contributions to geometry and
topology, a branch of mathematics that is, from many points of view, a
generalization of geometry.
In his topological studies Möbius was especially interested in one
particular property of surfaces, that of the possibility or impossibility of
orientation. For this study, he constructed a non-orientable surface that
became known as a Möbius strip. To make it he literally had to change
sides.
A sheet of paper has two sides (front and back), as well as a single
edge that runs around the corners of the page. Can it be that a sheet of
paper could have a single side and a single edge, so that an ant could
travel from one side to the other without crossing that edge? Today we
know that this is indeed possible: all you need to do is twist the paper
126 FIGURING IT OUT

180◦ and stick the two ends together, as shown in the illustration above.
The ants can crawl continually over the resulting surface, which is a
Möbius strip. Although it appears as if they are crawling over both sides
of the strip, going from the front to the back, in fact they are only mov-
ing on the single surface of this strange object. The Möbius strip does
not have a front or a back.
Imagine a two-dimensional being that is stuck to the strip, not
crawling over it like the ants. This being will change its parity when the
strip is rotated, i.e. it appears as a mirror image, a fact that reflects the
non-orientable character of this surface.
You can make a Möbius strip using transparent paper and then
write some words along the strip, such as ALWAYS PREPARED.
Holding the strip between two fingers, pull it to make it rotate com-
pletely. The ALWAYS PREPARED can be seen alternately in its normal
and in its inverted form.
Cut all along a strip lengthwise in the center. What you obtain is
somewhat similar to what Escher showed in his lithograph with the
snakes biting each others’ tails. The strip is not divided into two pieces,
as you might expect; rather, it remains a single strip. Stretch out that
strip and you will see that it has become thinner and longer, but it is still
one strip.
Now instead of cutting the original strip in the middle of its width,
try cutting all along it at one-third of its width. When the scissors have
finished their job, you will see that you have created two interlinked
strips, one longer and one shorter.
It is easy to see why the Möbius strip continues to fascinate math-
ematicians and artists, as well as people who are just plain curious. It
has been featured on stamps, sculptures, and even on commercial logos.
A Portuguese bank, Banco Totta & Açores, which was formed from the
merger of two banks, selected a Möbius strip as its symbol to highlight
the unity of the two original financial institutions: they were two sides
that became one. But the best and most imaginative portrayals are still
Escher’s, an artist with a profoundly intuitive geometrical knowledge of
the objects he designed.
 PICASSO, EINSTEIN AND THE FOURTH
DIMENSION

There is an amusing story I have been told about Picasso. When he was
already in his sixties, and one of the most famous artists in the world, a
very wealthy elderly lady asked him to paint her portrait. The painter did
not show any interest whatsoever, but the woman insisted, offering to
pay him whatever he wanted. Picasso, fed up with her entreaties, dashed
off half a dozen lines on a piece of paper and handed it to the lady. “That
will be ten thousand dollars”, he snapped. “Ten thousand?” she asked,
astounded. “But you didn’t even take a minute to draw that!” Picasso is
said to have retorted: “A minute! You are completely wrong. It took me
60 years.”
It really doesn’t matter, though, whether this story is true or false.
The really interesting thing is that it reminds us that an artist continues
to learn throughout this life, so it is very difficult to know the precise
origin of the ideas, the techniques and the necessary context for any
work of art. The best we can do is to look for the indirect roots of his
inspiration by studying his way of life and the intellectual climate of
the era.
Years ago art historians considered that cubism, an avant-garde art
movement that flourished between 1907 and 1919 and was clearly influ-
enced by geometrical forms, could have been inspired by the revolution
in mathematics and physics that took place at the beginning of the 20th
century. An extremely detailed discussion of this question can be found
in the classical study by Linda D. Henderson in The Fourth Dimension
and Non-Euclidian Geometry in Modern Art (Princeton, 1983) as well as
in Einstein, Picasso, by Arthur I. Miller (Basic Books, 2001). In this latter

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_30, 127


C Springer-Verlag Berlin Heidelberg 2010
128 FIGURING IT OUT

book, the author argues that mathematics and physics played a decisive
role in the development of cubism.
Steve Martin, the author of the play Picasso at the Lapin Agile, seems
to have read this latter book. His play, set in 1904, skillfully explores the
intellectual environment at the start of the 20th century. He does this by
means of dialogues between two men who never knew one another, but
who both experienced this era intensely and had the feeling that they
were on the verge of transforming the world: Albert Einstein (1879–
1955) and Pablo Picasso (1881–1973). At the age of 25, Einstein was 1
year away from publishing his famous 1905 studies, including the article
in which he set forth the theory of relativity. And Picasso, who was 2
years younger, would paint Les demoiselles d’Avignon just 3 years later.
Les demoiselles, one of Picasso’s most revolutionary paintings, was
completed in 1907. The picture, which measures approximately 8 by 7
8 , is now on display in the Museum of Modern Art in New York City
(the famous MoMA), and it is one of the star pieces in the museum’s
collection. It marks one of the decisive moments in the history of 20th
century art, and it is usually considered to have given rise to the birth of
cubism.
From Picasso’s sketchbooks and various other sources, it is known
that the artist permitted himself long periods of reflection while prepar-
ing to paint this picture. His sketches include innumerable drawings,
one of which, surprisingly, is a projection of a solid object in four dimen-
sions. All the sketches reveal continual attempts to simplify the elements
of the human form until they resemble simple geometrical figures. Later
sketches are studies from different perspectives as well as that striking
element of cubism employed by both Pablo Picasso and Georges Braque
(1882–1963) in their subsequent work: the juxtaposition of different
perspectives on the same canvas, revealing distinct points of view of the
same object.
To understand how these artistic developments could have origi-
nated in the scientific preoccupations of the era, you only have to read
the writings of the French mathematician Henri Poincaré (1854–1912),
who is considered to have paved the way for the theory of relativity.
 PICASSO , EINSTEIN AND THE FOURTH DIMENSION 129

In his Science et hypothèse, a work of philosophical reflection and sci-

entific diffusion that was published in 1902, Poincaré explains how a
four-dimensional world can be represented, starting with an analogy of
a projection of a two-dimensional picture on to our retina. We know
that objects are three-dimensional, he says, because we perceive them
sequentially from different perspectives, and we have become accus-
tomed to representing them in two dimensions. So, he continues, a
four-dimensional figure can also be represented in two dimensions. In
addition, we can select from a variety of perspectives from various points
of view, resulting in this sequence of visual perspectives that corre-
spond to different projections on two dimensions from different points
in four-dimensional space.
It is known that Einstein read a German translation of Poincaré’s
1904 book, and that what he read had an enormous influence on his
reflections about the physical world, as well as about four-dimensional
space-time. It is unlikely that Picasso would have read Poincaré, but
we do know that one of those who influenced his circle of friends was
Maurice Princet, an actuary with an extensive knowledge of mathe-
matics, and that Princet, who was later dubbed the “mathematician of
cubism”,1 spoke to Picasso often about the fourth dimension, about
non-Euclidean geometry, and about other scientific ideas that fascinated
Picasso and the members of his circle.
Les demoiselles illustrates the solution Picasso found for the prob-
lem of three-dimensional representation: the simultaneous (and not
sequential, like the representations that Poincaré talked about) painting
of different perspectives of the same object.
It is unlikely that the artist was moved to create cubism by direct
mathematical inspiration. But the similarity between Picasso’s preoccu-
pations with geometrical concepts and those of Poincaré and Einstein
with space-time are too clear-cut to be a mere coincidence.

1
A. I. Miller, Einstein, Picasso, Basic Books, 2001, p. 100
 POLLOCK’S FRACTALS

Jackson Pollock (1912–1956) is well known for his gigantic pictures that
combine colored lines, splashes of paint, extensive spirals and rhythmi-
cal tracks. But he is just as well known for the controversy his art has
generated. Some people have asserted that a monkey could paint more
interesting pictures than Pollock’s, or have commented that it is impos-
sible to tell the difference between his pictures and completely random
scrawls. How could this man have consciously created such strange,
chaotic pictures?
In 1950, a photographer from New York named Hans Namuth suc-
ceeded in obtaining Pollock’s permission to photograph him in action.
When the appointed day arrived, Namuth turned up at the painter’s stu-
dio, a barn adjacent to a farmhouse in Long Island, New York, to which
Pollock had moved in 1945 after having lived in New York City for 5
years.
When Namuth arrived at the painter’s farmhouse, Pollock informed
him that there would be nothing to photograph that day after all, as
he had finished putting the final touches to one painting and he wasn’t
ready to start another one yet. The two men headed for the barn, where
the still damp painting was lying on the floor. They paused to admire it.
Jackson Pollock began walking around the painting, and seemed obvi-
ously unsettled. Namuth couldn’t imagine why. Was something missing
or was there something out of place in that mishmash of apparently ran-
dom lines? Suddenly the painter came to a halt, shook himself, and went
off to look for a bucket of paint. He looked closely at the painting again
and then started to splash paint on it. Namuth began to take photos.

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_31, 131


C Springer-Verlag Berlin Heidelberg 2010
132 FIGURING IT OUT

Some days later, the photographer showed Pollock the images he

had taken. Pollock and his wife, the artist Lee Krasner, liked the pho-
tos so much that they gave Namuth carte blanche to continue to take
more pictures. Namuth then spent numerous long days and evenings
with the painter, taking a great many photographs. In the end, convinced
that still pictures would not convey the complexity of Pollock’s working
methods, he produced a short video documentary. This film has been
shown countless times at exhibitions, in studios and on television, and is
treasured as an invaluable documentation of the artist’s working day. It
illuminates his technique, showing how Pollock painted a picture layer
by layer, beginning with coarse applications of a background color. Next
he created finer lines by means of long arm movements, allowing rivers
and splashes of paint to fall. Finally, making shorter movements, he
threw fine lines and small splotches on to the canvas. Namuth’s film doc-
umented a very complex and not at all arbitrary technique for creating
a painting.
Recently the Australian physicist Richard Taylor decided to use
modern mathematical tools to analyze Pollock’s technique. Taylor, who
had studied art in his younger days, suspected that the visual appeal of
Pollock’s paintings had something to do with their similarity to images
from nature, formed by chaotic processes that create fractals. He imag-
ined that this similarity resulted from various idiosyncrasies in Pollock’s
technique.
In contrast to most painters, who apply paint to a canvas posi-
tioned on an easel, Pollock used a horizontal surface and let the force
of gravity act on the paint. Once again in contrast to other artists,
Pollock did not use brush strokes to produce fine controlled lines, but he
instead let paint drop onto the canvas, or, at times, even threw it down.
Taylor argued that this process is very similar to what occurs in nature,
where outlines, shapes, and vegetation are all sprinkled throughout the
landscape.
In order to achieve a better understanding of this technique, Richard
Taylor constructed a device to drop paint in a rhythmical fashion. This
device began as a simple pendulum with a nozzle. As the pendulum
oscillated, it dropped paint onto a canvas on the ground. The result is
a canvas covered by relatively simple lines.
 POLLOCK’ S FRACTALS 133

Reflection of the Big Dipper, created in 1947 by the paint-throwing method, already
shows a fractal dimension close to 1.45

The image on the left is of the paint tracks left by a normal pendulum, launched time
after time. The center image shows the tracks of a chaotic pendulum. On the right is
a detail of “Number 14”, a picture painted in 1948. The similarity between the tracks
of the chaotic pendulum and Pollock’s lines is striking

Then the Australian physicist decided to introduce a chaotic pen-

dulum to drive the nozzle. Chaotic pendulums can be seen in several
science museums. They are usually formed by two coupled pendular
systems, which move in a complex and seemingly uncontrolled pattern,
134 FIGURING IT OUT

with irregularities that are impossible to predict even though they are,
in fact, mostly controlled by known deterministic physical processes.
Sometimes these pendulums slowly achieve equilibrium, only to then
start oscillating rapidly immediately afterwards. They suddenly appear
to interrupt their motion, to the surprise of an observer. Such pen-
dulums are called “chaotic” because small modifications of the initial
conditions produce radically different motion patterns after a period of
time, which is why their future positions cannot be precisely predicted:
As it is impossible to provide an absolutely precise characterization of
the initial position and the forces acting upon the pendulum, it also
becomes impossible to predict its motion over a longer period, even if
there are no random elements in the system.

Pollock’s 1937 painting The Flame does not show any clear fractal dimension yet

This is a completely different situation than we have with normal

pendulums, like the ones used in grandfather clocks. In a typical pendu-
lum, no matter what the initial starting point is, we always have a very
precise idea of how it will move for quite some time after that initial
point. That is why these pendulums are used to measure time.
In order to endow the pendulum with a persistently chaotic
motion, Taylor constructed an electromagnetic system that pushed the
pendulum periodically, but was not synchronized with the free motion
of the pendulum. The paint tracks obtained as a result of this motion
 POLLOCK’ S FRACTALS 135

show great similarity to the paint tracks in Pollock’s pictures, as you can
see in the illustration.
The most interesting aspect is that the tracks generated by the
chaotic pendulum show fractal dimensions, in contrast to those created
by the simple pendulum. It is not easy to clearly understand the signif-
icance of this, as the mathematical concepts that make up the rigorous
definition of fractals are complex and involve the differentiation of the
so-called topological and Hausdorff-Besicovitch dimensions. But there
is one geometrical property of fractal objects that is easy to comprehend:
in these objects the patterns are repeated in similar form at different
scales. If we look at a leaf, for example, we see veins that bifurcate into
even finer veins. If we look at these very fine veins through a magnifying
glass, we note that they subdivide once again into still finer veins. If we
use a microscope, we can discern the same pattern. In other words, the
vein structure of a leaf displays fractal characteristics.
If a leaf had only a single straight vein, it would not be classified
as fractal: we can assign such a system to dimension 1. But the veins of
a leaf subdivide and multiply over its entire surface. If these veins filled
the leaf completely, we would have dimension 2, as they would cover the
surface. But instead, what we have seen is something intermediate: as
the amplification at which we observe the leaf increases, new veins pre-
viously invisible to the eye make their appearance, forming a network
that nearly, but not completely, covers the entire surface. So this system
of veins has a fractal dimension with a number between 1 and 2. The
simpler it is, the closer it is to 1; the denser the network of veins revealed
by amplification, the closer it is to 2. In statistically fractal objects, such
as those encountered in nature, it is not exactly the same patterns that
become visible when we change the scale, but they do display statistically
similar properties.
To measure the fractal dimension of objects that live on a plane,
we can divide this plane into ever-smaller squares, and then check
how the patterns are repeated as the scale is changed. This was exactly
the method used by Taylor and his colleagues to obtain very precise
estimates of the fractal dimension of Pollock’s paintings. His conclu-
sions were very clear: the artist created paintings with a markedly
136 FIGURING IT OUT

fractal dimension, and as his technique advanced, he created ever more

complex paintings, with a higher fractal dimension.
By 1943 Pollock’s paintings had a modest fractal dimension, a little
above 1. As an example, let us take The Flame, painted in 1937, which
does not possess any marked fractality. After this period, when he had
created and perfected his paint-throwing technique, he produced paint-
ings with much clearer fractal characteristics. This applies to Reflection
of the Big Dipper, painted in 1947, which has a fractal dimension of
about 1.45, a value approaching the estimated dimension for natural
structures, such as the coastline of the British Isles.
Jackson Pollock successively increased the complexity of his paint-
ings. Blue Poles, a work completed in 1952, attains about 1.72, the
highest fractal dimension of any of the paintings studied by Taylor. It
seems that the painter was exploring the limits of what the human eye
considers aesthetically pleasing. Can it be that this limit is set by nature,
but then revealed via the language of mathematics?
 VORONOI DIAGRAMS

Leonel Moura, 2004

There are mathematical concepts that emerge gradually, springing from

within a wide range of contexts that are apparently unconnected and of
limited interest. And then, all of a sudden, these same concepts begin
to attract the attention of specialists, give rise to numerous studies and
applications, and end up contributing to the development of new fields
of study. Voronoi diagrams are an example of this type of concept.
They were first described systematically in a 1908 article by the math-
ematician Georgi Voronoi, but their roots can be traced all the way
back to ideas first presented in 1644 by René Descartes on the distribu-
tion of the planets in the solar system, as well as to work undertaken

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_32, 137


C Springer-Verlag Berlin Heidelberg 2010
138 FIGURING IT OUT

in the middle of the 19th century by the German mathematician

G. L. Dirichlet.
Georgi Feodosevich Voronoi was born in 1869 in Zhuravki, (a town
that was then in the Russian Empire, but is now in the Ukraine).
Voronoi studied at the University of St. Petersburg, where he was
awarded a Ph.D. in 1897. He died in Warsaw at the age of 40 in 1908,
after spending his short life working as a university professor, mainly in
the fields of number theory and computational geometry problems. In
the last year of his life he published a study exploring and then defin-
ing a division of space that came to be called a Voronoi diagram. His
concept was picked up and reapplied in 1911 for meteorological studies,
and in 1927 once again used in the study of crystallography. Currently
the Voronoi diagram is used in a wide variety of areas, and is studied
in the field of computational geometry, a branch of mathematics that
applies algorithms to resolve geometrical problems.
To give you a sense of how Voronoi diagrams can be applied in real-
life situations, just imagine a vast forest on a plain that is monitored for
forest fires by only two park rangers, each in their own individual fire
tower. What is the best way of dividing up the area they are monitoring?
There is a simple solution to this problem. We can draw a straight
line segment joining the two towers. Then, at the midpoint of this seg-
ment, draw a line that is perpendicular to the first one. This second line
serves to divide the forest into two sections in such a way that all the
trees on one side are closer to one tower, while the trees on the other
side are closer to the second tower.
If there were three fire towers in the forest that were not all posi-
tioned on one straight line, our problem would become slightly more
difficult, although its solution remains obvious. In this case, we would
have to divide the forest into three sections bounded by half lines. It
is not too challenging to figure out how to calculate the position of
these lines. However, if there were ten towers distributed randomly
throughout the forest, it would become far more difficult to work out a
solution. Generally speaking, this situation would require that the forest
be divided into sections shaped as (closed) polygons and non-limited
figures, a type of polygon with one side missing.
 VORONOI DIAGRAMS 139

This example perfectly illustrates what a Voronoi diagram is: there

is a plane and a set of points located on it. That plane is divided into
cells, each of which is defined by grouping together all the locations
that are closer to one point than to any other point. The boundaries
between these cells form the Voronoi diagram generated by this given
set of points.
These diagrams clearly have applications in many situations. Our
fire tower example can be modified, but it will still generate exactly the
same figures. Imagine, for instance, that a fire was started at each tower
at a specified moment, and that the fires all advanced at the same speed
in every direction. After a certain time, the fronts of the fires, which
began as circumferences centered on each tower, would meet. There
would be trees that were burnt back and front by different fires com-
ing from different directions. If we then marked the positions of these
trees, we would see that they were equidistant from two or more of the
sources at the fire towers. If we then drew lines joining the positions of
these trees, we would end up with an exact duplication of the Voronoi
diagram from our previous example.
This example leads us directly to some applications of the Voronoi
diagram for crystallography. Imagine a set of dispersed points around
which a material crystallizes at a constant growth rate. What shape
would the edges of these crystals have when each of them had finished
growing, inherently limited by their collisions with the other crystals
growing beside them? The answer is a Voronoi diagram.
There are also interesting applications of Voronoi diagrams for
robotics and optimum control. Imagine a set of obstacles placed at
points dispersed on a plane a robot must cross. If we wanted to select
a trajectory on which the robot stays as far away as possible from each
obstacle (the points), such a trajectory would be part of a Voronoi dia-
gram generated by the obstacles. If we imagine that the vertices of the
resulting Voronoi diagram are locations, then we can see the similarity
with the famous problem of the traveling salesman who has to find the
shortest route enabling him to visit each of the stops on his list.
Voronoi diagrams can have countless other applications as well.
They can even become the subject of artistic endeavors, such as those
140 FIGURING IT OUT

of the Portuguese artist Leonel Moura, who painted a series of pictures

he called Algorithms on Canvas. Moura used a computer to generate
Voronoi diagrams, and then he colored the spaces in between. Does the
beauty of these figures derive from their conceptual simplicity, from the
equilibrium we see in them? At times it is not easy to discern the point
where the rigor of mathematics ceases and the hand of the artist takes
over. Maybe that is the artist’s secret, and the mathematician’s.
 THE PLATONIC SOLIDS

Have you ever looked closely at a cube? It is one of the commonest

solids. In nature it appears in crystals; in our homes in furniture design;
in casinos, cubes can be seen in the dice rolled on betting tables.
Mathematicians classify the cube as a polyhedron, a closed three-
dimensional geometric figure limited by planes. The faces of the cube
are squares, a common feature of polygons (plane figures bounded by
edges that are segments of straight lines).
Taking a closer look at the cube, we can see that it has six faces, all
of them equal, and also 12 edges, all of the same length. The 8 vertices
of a cube also have one characteristic in common: three edges are joined
at each vertex. In other words, a cube has equal faces, equal edges, and
vertices that all join the same number of edges.
There are many solids that are not as symmetrical as a cube. For
example, the pyramids in Egypt have sides formed by four triangles, but
their bases are square. But we can find other solids with the same type
of regularity as a cube; take the tetrahedron for instance, a solid formed
by four equilateral triangles.
The Ancient Greeks took a great interest in the regularity of geomet-
rical shapes. Some people even claim that it was their interest in abstract
patterns that launched the Greek intellectual miracle, which eventually
enabled western civilization to make such extraordinary progress. The
Greeks discovered a total of five solids with a degree of regularity similar
to that of the cube. Plato (about 428–348 BC) discusses them at length
in his dialogue Timaeus. That is why these five geometrical figures have
come to be known as the Platonic solids.

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_33, 141


C Springer-Verlag Berlin Heidelberg 2010
142 FIGURING IT OUT

The most curious aspect of this is that there are only five solids of
this type. If we restrict ourselves to polyhedrons (“figures with sides and
angles that are equal and equal to each other” as defined by Euclid about
300 BC), these five are the tetrahedron, the cube, the octahedron, the
dodecahedron and the icosahedron. Euclid knew of a proof of this fact,
which he included in his Elements. But it is likely that it was known
even long before Euclid’s time that it is impossible to construct any other
solids with these precise characteristics.
For followers of Plato and of Pythagoras, these polyhedrons pro-
vided proof of the intrinsic harmony of the world. These devotees of
Plato and Pythagoras studied many of the aspects of polyhedrons and
discerned a series of curious relationships: while the cube has 6 faces and
8 vertices, the octahedron has 6 vertices and 8 faces. While the dodec-
ahedron has 12 faces and 20 vertices, the icosahedron has 12 vertices
and 20 faces. This means that we can assign these four solids into two
sets of pairs, the cube with the octahedron and the dodecahedron with
the icosahedron. Such pairs are called duals. The tetrahedron, however,
remains alone, with its 4 faces and 4 vertices it is its own dual.
Some of these solids are rigid. If we construct the edges of a cube
using strong rods and flexible adhesive, we will see that it can be
destabilized by twisting or bending it. This does not happen with the
octahedron, the dual of the cube. Once we have joined the rods that form
its sides, the octahedron cannot be manipulated. You can only twist it
by first dislodging its rods altogether. This also occurs with the other
pair of dual solids: the icosahedron is rigid, but the dodecahedron can
be bent. Our single, the tetrahedron, is rigid.
The Greeks were awed by these relationships, and believed that they
revealed hidden secrets of the universe, even of the physical and contin-
gent universe. Plato contended that earth is represented by the cube, fire
by the tetrahedron, air by the octahedron and water by the icosahedron.
The outsider was the dodecahedron, which represented the cosmos. We
may today smile at these suppositions, but the truth is they still reveal a
profound knowledge of three-dimensional geometry.
Another characteristic of the Platonic solids is their geometrical
symmetry with respect to a center. Any one of these solids can be
 THE PLATONIC SOLIDS 143

inscribed in a sphere, with all its vertices touching the surface of the
sphere. This means that all the vertices are at the same distance from
the central point of the solid. But, in addition to an outer sphere that cir-
cumscribes the polyhedron, we can draw an inscribed inner sphere that
touches the central point of each of the faces of the solid.
The Greeks were not the only ones to become fascinated by the sym-
metry of the Platonic solids. Almost 2000 years after Plato, Johannes
Kepler (1571–1630), one of the greatest scientists ever, thought that he
had found the secret of the world’s harmony in the perfection of geom-
etry. This German astronomer constructed a model of the solar system
using spheres circumscribed in Platonic solids. Everything seemed to be
in perfect harmony. The sun was at the center, itself forming a sphere.
Around it was another sphere inscribed inside an imaginary octahe-
dron, touching the centers of its faces. Mercury, the nearest planet to the
sun, moved within this sphere. Another sphere, touching the vertices
of the polyhedron, was inscribed around this octahedron. The planet
Venus moved within this second sphere, which was inscribed inside an
144 FIGURING IT OUT

icosahedron, touching the centers of its faces. The icosahedron was, in

turn, circumscribed around another sphere, where the Earth orbited and
so on. As only six planets had thus far been discovered, the six spheres
separated by their five solids seemed to reveal the hidden harmony of
the universe.
However, Kepler loved the truth, and never settled for mere assump-
tions. Although he was convinced for some time that he had indeed
found the secret of the cosmos, he came to accept the factual evidence.
After spending many years studying the observations collected by his
master, Tycho Brahe (1546–1601), he perceived that his model did not
correspond to the facts. The relative distances between the planets did
not agree with the requirements imposed by his theory of the spheres.
And later Kepler discovered that the planetary orbits were not circular,
but elliptical, a fact that he revealed to the world in what came to be
known as Kepler’s First Law.
A seventh planet in the solar system was discovered in 1781 by the
astronomer William Herschel (1738–1822), an event that once again
demolished the model of the solar system based on Platonic solids. But
not everyone learned from the humility shown by Kepler. Two hundred
years later, the philosopher William Georg Friedrich Hegel (1770–1831)
remained convinced that the solar system consisted of only seven plan-
ets, as seven was a divine number and the universe was the material
revelation of spiritual harmony. Hegel was still alive when the first
minor planet, the asteroid Ceres, was discovered in 1801 by the Italian
astronomer Giuseppe Piazzi (1746–1826). At that time this eighth solar
satellite was considered to be a planet, and its discovery considerably tar-
nished the scientific reputation of Hegel, a philosopher who delighted in
speaking of mathematical and physical concepts that he did not under-
stand (a defect that may still be noted in certain modern-day intellectual
impostors). In 1846 the planet Neptune was discovered; and then Pluto
in 1930. Science may have learned to study the geometry and symme-
try of solids, but it also learned to be wary of speculative theories of the
Platonic harmony of the universe.
 PYTHAGOREAN MOSQUITOES

Before people have regular exposure to other cultures, there is a natural

tendency to assume that everything that happens in the world mirrors
our own experiences. We might think, for example, that everyone every-
where else eats a fried breakfast, or that there is no city on earth more
hospitable than our own. But then one day we encounter sushi, or read
the work of writer Machado de Assis, and begin to see that there are
many ways of living life that are different from our own way.
The discovery of such a variety of cultures, criteria, and creeds was
a very positive development in the 20th century. Many western thinkers
began to see that embracing cultural differences was both necessary and
beneficial. Artists and musicians also agreed, thinking up new ways to
paint and creating musical compositions that, in the old days, would
have been dismissed as noise. We came to see that aesthetic criteria can
vary according to a person’s culture and education.
One school of thought whose influence spread from literary criti-
cism to sociology and philosophy took this perspective to an extreme:
it maintained that there is nothing in art, ethics, politics, and even in
science that is absolute, that does not depend for its relevance on the
culture through which it is being viewed. In other words, everything
would be a “social construction”, as they began to call it.
Can this be right? Does this mean science is not an objective disci-
pline? That there is no confrontation with external reality? And even in
art? Excrement repels us and beautiful sunsets delight us. Is this simply
due to a social construction? Consider the passion Pythagoras felt when

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_34, 145


C Springer-Verlag Berlin Heidelberg 2010
146 FIGURING IT OUT

he considered the interplay of numbers in musical harmony. Is this just

another social construction?
Legend has it that Pythagoras studied the proportions between
the various sizes of blacksmith’s hammers and ascertained that some
produced harmonies while others created dissonance. Pythagoreans
were later amazed when it was found that this discovery could be
applied more generally. For example, they deduced that the harmonic
proportions in string instruments correspond to strings with length
proportions that can be expressed as ratios of whole numbers.
Today when we speak of sound frequency, we measure it in cycles
per second or Hertz (Hz). When the frequency of a sound is doubled,
for instance, it becomes higher, going up one octave in the musical scale.
When a frequency is multiplied by 3/2 it goes up by a fifth, and so on.
The resulting chords are pleasing to the ear. The amazing thing about
this is that nature apparently thinks so too.
Four scientists from Cornell recently published an article in Science
about their investigation into the mating songs of the yellow fever
mosquito (Aedes aegypti).1 They did not actually hear “amore mio” in
insect language, but they did manage to establish that the mating ritual
of these mosquitoes is preceded by a tuning of frequencies in a common
harmony.
Here’s how it works: the males start by using their wings to produce
sounds at a frequency of 600 Hz – six hundred cycles per second –, or
about the same as G in our scale. But when they begin to really pur-
sue the females, they double the frequency, producing a note similar to
G one octave higher. The females, who normally produce sounds at a
frequency of 400 Hz (close to C on a piano), respond to the males by
tripling the frequency of their own wing beats so that they tune in with
the males at 1200 Hz, which is the smallest common multiple of the two
primary frequencies. The resulting note, G, is the mosquito’s music of
love. Pythagoras himself could not have done better.

1
L. J. Cator, B. J. Arthur, L. C. Harrington, and R. R. Hoy, Harmonic convergence in the love songs
of the dengue vector mosquito, Science 323, 1077–1079, 20 February 2009.
 THE MOST BEAUTIFUL OF ALL

How can we find beauty in an equation? Readers will certainly have

divided opinions about this. Some people will assume the question is
ironic: what possible beauty could there be in those incomprehensible
squiggles that filled our schoolbooks? But in the view of others who
carried on with math after leaving school, and came to enjoy this sub-
ject, even making it their life’s work, the simplicity and elegance of
certain equations make them beautiful. Quite beautiful, in fact, though
in some cases it is difficult to explain the reasons for their beauty. One
is surely the strange condensation of reality they conveyed, reality that
may be geometrical, physical, biological, or purely ideal. And then there
is their flexibility, their applicability to infinite numbers of unexpected
situations, as well as their graphic representation.
In mathematics, nobody is surprised by the existence of equations;
the condensation of relationships by means of symbols seems to define
science itself. But they also illuminate the enormous explanatory and
predictive power of science.
For example, the brilliance of physics is that it uses equations to
interpret very general laws, and permits forecasts to be made that reveal
unsuspected aspects of reality. In many cases, for instance in Einstein’s
most celebrated equation, E = mc 2 , the quest to explore mathematical
relationships led scientists to formulate new questions and conclusions.
This seems to signify that equations contain more than their creators put
into them, that perhaps they are “wiser than their discoverers”,1 as the

1
E.T. Bell, Men of Mathematics, New York, Simon & Schuster, 1937, p. 16.

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_35, 147


C Springer-Verlag Berlin Heidelberg 2010
148 FIGURING IT OUT

German physicist Heinrich Hertz once put it. Graham Farmelo, the edi-
tor of It Must Be Beautiful: Great Equations in Modern Science, observed
in this book that an equation is an expression of perfect equilibrium, and
what makes it beautiful is its synthesis of truth without a single wasted
symbol. The various essays that comprise the book discuss the extent
to which aesthetics is a factor in the work of mathematicians and sci-
entists. For example, Einstein declared that “the only physical theories
we are willing to accept are the beautiful ones”.2 Another physicist, Paul
Dirac (1902–1984), went farther, and used a bit of a hyperbole when he
claimed that “it is more important to have beauty in one’s equations than
to have them fit experiment”.3
In the aftermath of Graham Farmelo’s book, Robert P. Crease, a
columnist for the journal Physics World, initiated an opinion poll among
his readers, asking them to list the equations they thought were most sig-
nificant in the whole history of science, taking into consideration their
reach, depth and aesthetic appeal. Their responses are revealing.
His readers tended to favor equations that are at once important
and also notably simple. The most popular was a purely mathemati-
cal equation formulated by the Swiss mathematician Leonhard Euler
(1707–1783). Following closely behind came a series of equations cre-
ated by the Scottish physicist James Clerk Maxwell (1831–1879) that
describe the behavior of an electromagnetic field.
Mathematicians were not surprised to learn that Euler’s equation
had come out on top. Using only seven symbols, it includes three basic
operations and states the relationship between the five most impor-
tant numbers in mathematics. Its beauty derives from its simultaneous
simplicity, and its profundity. Have a look:

eiπ + 1 = 0

2
Graham Farmelo, It Must Be Beautiful, London, Granta, 2002, p. xiii.
3
Paul Dirac, The Evolution of the Physicist s Picture of Nature , Scientific American, May 1963,
208, p. 47.
 THE MOST BEAUTIFUL OF ALL 149

Euler’s equation contains the two most important integers, 1 (which

is the unit from which it is possible to find all the integers and ratio-
nal numbers by using only four elementary operations) and 0 (which
unleashed a revolution in mathematics as it constitutes a higher abstrac-
tion). Then comes π, the quotient of the circumference and its diameter,
an irrational number that, even today, has not yet revealed all its mys-
teries, as well as two other numbers that are ubiquitous in advanced
mathematics.
One of these numbers is the “imaginary unit”, the square root of –1,
a number that was created in order to resolve algebraic problems and has
since been shown to have a right to exist. “Complex numbers”, which
encompass real numbers, can be written as sums of real and imaginary
numbers. Complex number have tremendous interest for mathematics,
and have direct applications in physics and in such practical areas as
electronic engineering.
The other fundamental number included in Euler’s equation is the
base of natural logarithms, the number e = 2.71828. . . This is such an
important number that it plays a role in such disparate topics as com-
pound interest, population growth, and radioactive decay, as well as in
the spirals found in flowers and galaxies. In pure mathematics, it can be
found in the definition of angles, and as the limit of important series and
sequences; it also appears in derivation, integration and other analytical
operations. As far as we can judge, it came to light for the first time in
1618, seemingly emerging from a simple problem in compound interest
computation. It is truly astonishing that the very same number appears
in such diverse fields and has so many practical applications. Maybe it
is even more amazing that it can equal the unit after being raised to the
imaginary unit multiplied by π. Euler’s equation is a celebration of the
unity of mathematics, and of the power of science. It is not at all sur-
prising that many people consider it the most beautiful equation that
has ever been written.
MATHEMATICAL OBJECTS
 THE POWER OF MATH

“How can it be” wondered Einstein, “that mathematics, being after all
a product of human thought which is independent of experience, is
so admirably appropriate to the objects of reality? Is human reason,
then, without experience, merely by taking thought, able to fathom the
properties of real things?”1
This question is neither naïve nor ingenuous. As the great physi-
cist observed in a 1921 lecture on “Geometry and Experience”, “At this
point an enigma presents itself, which in all ages has agitated inquir-
ing minds.”2 The response to this enigma has divided mathematicians
and philosophers. While some regard the applicability of mathematics
as the natural product of its roots in our experience, others consider
that its success corresponds necessarily to the real world with the logi-
cal premises on which mathematics is based. Einstein relativized these
matters in a point of view that is supported by many mathematicians,
scientists and philosophers. “As far as the propositions of mathematics
refer to reality, they are not certain; and as far as they are certain, they
do not refer to reality”3 he said.
According to Einstein, modern mathematics, founded on a logical-
formal deduction based on axioms, has succeeded in separating its
logical-formal aspect from its objective and intuitive content. The

1
The English translation of Einstein’s 1921 lecture is available online at http://pascal.iseg.utl.
pt/∼ncrato/Math/Einstein.htm
2
Ibid.
3
Ibid.

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_36, 153


C Springer-Verlag Berlin Heidelberg 2010
154 FIGURING IT OUT

correspondence of mathematical conclusions to physical reality is

merely approximate, and is derived from the possible approximation of
the axioms to fundamental natural laws.
Ian Stewart, a prolific English mathematician who has also spe-
cialized, very successfully, in the field of science communication, does
have an answer to these questions. In his book Nature’s Numbers, pub-
lished by Basic Books, he recognizes that there are various theories that
explain the usefulness of mathematics, “ranging from the structure of
the human mind to the idea that the universe is somehow built from lit-
tle bits of mathematics”.4 But his response is “quite simple: mathematics
is the science of patterns, and nature exploits just about every pattern
that there is”.5
In his book Stewart attempts to show how the mathematical inves-
tigation of patterns can explain many phenomena found in nature. For
instance, he brings up the old question regarding the spiral pattern of
the shells of snails, whelks and similar creatures that was posed at the
beginning of the century by the Scottish zoologist D’Arcy Thompson in
his book On Growth and Form, a classical work written in 1917 and still
in print today.
The spiral structure of the shells of these animals may be explained
by means of geometry. If we accept that the shell develops as the animal
grows, that it always develops in a similar way, and that the width of
the developing shell tube depends on the size of the animal at any given
moment, then it is natural that the rings that are produced take the form
of the spirals that really occur in nature. The relationship between the
width of the rings and the dimensions of the animal at the time it is
creating them gives rise to various types of spirals that can be described
via well-established mathematical equations.
This example permits us to return to Einstein’s argument. The per-
fect geometrical spirals resulting from mathematical functions are not
found in the natural world. The ones found in nature are imperfect.
As is sometimes said, perfect points, straight lines and triangles do not

4
I. Stewart, Nature’s Numbers, Basic Books, New York, NY, 1995, pages 18
5
Ibid.
 THE POWER OF MATH 155

exist outside our minds, but analyzing these perfect objects using pre-
cise logic helps us to draw conclusions with respect to the imperfect and
approximate points, straight lines and triangles that do exist in nature.
Another interesting example of a mathematical pattern found in
the real world is the arrangement of petals and florets (the small rudi-
mentary flowers that are found, easily visible, in the center of some
flowers such as sunflowers). In some species these florets are distributed
in groups of spirals that curl in different directions and intercept each
other. Often the number of elements that curl in one direction is 34,
while the number of elements curling in the opposite direction is 55.
Other cases involve pairs of spirals consisting of 55 and 89, or even 89
and 144 elements.
This may seem to be just another curious fact, but mathematicians
look at these numbers and recognize that they are consecutive terms in
the sequence 1, 1, 2, 3, 5, . . ., 34, 55, 89, 144, 233, . . . As we have seen
before, this is a numerical sequence constructed in 1202 by Leonardo of
Pisa (1170–1250), also known as Fibonacci, in a discussion of a prob-
lem involving the population growth of rabbits. In this sequence all
the terms after the first one are obtained by adding together the two
previous terms (2 = 1 + 1, 3 = 1 + 2, 5 = 3 + 2, etc.).
Why do these numbers, created in response to such a different prob-
lem, recur in the elements of flowers? Biologists may be tempted to
say that they are numbers that are found in the genes of the plants,
but mathematicians search for other reasons. The genes determine
how a being develops, but it develops in the context of a physical
and geometrical world where restrictions also exist. Mathematicians
have succeeded in demonstrating that elements that develop around
a central point in such a way that they occupy the surface area in
the most compact manner possible achieve this by means of a precise
angle of divergence, the “golden angle” (approx. 137.5◦ ). Well, circu-
lar elements that develop in such a way that they are always separated
by this angle tend to form spirals in which the Fibonacci numbers
occur. So it is not surprising that this angle and these numbers are
found with great regularity in flowers, both in their petals and their
florets.
156 FIGURING IT OUT

Mathematics succeeds in explaining the geometrical and numerical

regularity on the basis of very simple principles of growth that may be
determined by the genes. But the living world does not need to have
all the rules of mathematics inscribed in a DNA code. Such rules occur
naturally, based on even simpler rules of growth. After all, these math-
ematical patterns are patterns that are necessary in nature. Could this
explain why mathematics is “so admirably appropriate to the objects of
reality”?
 DOUBTS IN THE REALM OF CERTAINTY

In the early 1980s Philip Davis and Reuben Hersh wrote a bestselling
book entitled The Mathematical Experience. The book’s philosophical
message is found throughout the narrative of the intellectual adven-
ture that has given rise to modern mathematics. In another book by
Hersh published in 1997, called What is Mathematics, Really?, the author
makes his philosophical leanings clear. It was enough to re-ignite a
lively debate on the fundamental aspects of this mysterious entity that is
mathematics.
Everybody knows, or thinks they know, what mathematics is: a
game of numbers, unknowns and relationships used to pay the bill
at the supermarket, calculate taxes, design bridges, determine the tra-
jectory of the space shuttle and reassure us about the future orbits
of asteroids. But philosophers, just like mathematicians, have never
agreed on the roots of the miraculous power of these numbers and
equations.
Throughout the centuries, two main currents of thought have vied
for supremacy: Platonism and formalism. It is difficult to reduce these
currents to more general schools of philosophy, such as materialism or
idealism, for it is a debate that is unique to the world of mathematics.
Platonism, sometimes also referred to as realism, asserts that mathe-
matical entities exist outside space and time, outside the human domain,
independently of our existence. Accordingly, the objective of mathe-
matics would be to uncover the various aspects of a grandiose abstract
structure that is composed of objective, unquestionable and immortal
truths.

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_37, 157


C Springer-Verlag Berlin Heidelberg 2010
158 FIGURING IT OUT

The Hungarian mathematician Paul Erdős (1913–1996) used to say

that God had a book in which the best proofs of all the theorems were
written – the perfect proofs. He considered that his task as a mathemati-
cian was to discover what was in that book, although God did not always
play along. Of course Erdős was being ironic, but his comment describes
the Platonic point of view.
Formalism, on the other hand, views mathematics as an axiomatic
and logical construction, as a “game” of no significance. Therefore it
should be possible to construct various truths depending on the presup-
positions (the axioms) that are made. As an external reality approaches
the presuppositions of a theory, so the results of that theory can be
applied to that reality. But always only approximately.
Einstein was not far from this point of view. “In my opinion”, the
physicist said in defense of the axiomatic approach, “the answer to this
question is, briefly, this: as far as the propositions of mathematics refer
to reality, they are not certain; and as far as they are certain, they do not
refer to reality.”1 So where does the power of mathematics come from?
Two and two seem to make four, both in the mental addition derived
from the rules of algebra as well as when counting and adding up real
oranges.
The greatest apostle of formalism was David Hilbert (1862–1943),
a German mathematician who taught in Göttingen. At the turn of the
20th century he proposed that the basic tenets of mathematics should be
subjected to absolutely rigorous revision. Hilbert urged that the whole
of mathematics should be based on a finite set of axioms from which
all results should be derived by logical rules. “The goal of my theory”,
he stated, “is to establish once and for all the certitude of mathematical
methods.”2
In 1931, Kurt Gödel (1906–1978), a logician of Austrian origin who
worked in Vienna and at Princeton, was able to demonstrate that such

1
See http://pascal.iseg.utl.pt/∼ncrato/Math/Einstein.htm
2
D. Hilbert, “On the infinite”, 1925 address, as translated and reproduced in P. Benacerraf and
H. Putman, editors, Philosophy of Mathematics: Selected Readings (2nd edition), Cambridge
University Press, 1983, p.184
 DOUBTS IN THE REALM OF CERTAINTY 159

an outcome was unattainable. Given any set of axioms, there are always
mathematical results that cannot be proven using it. But Gödel and
his famous theorems of incompleteness did not annihilate the formalist
vision, although they did limit its dream of perfection.
Reuben Hersh quips that most “working mathematicians” vacillate
between these two currents of thinking, being Platonists on weekdays
and formalists on weekends. “On weekdays, when doing mathematics,
he’s a Platonist” writes Hersh, “he is convinced he’s dealing with an
objective reality whose properties he’s trying to determine. On week-
ends, if challenged to give a philosophical account of this reality, it’s
easiest to pretend he doesn’t believe in it.”3
Hersh endeavors to bypass this dilemma by promoting a new the-
ory, one of mathematics as a social construct. He supports it by invoking
the theses of Imre Lakatos (1922–1974), a philosopher of Hungarian
origin who worked with Karl Popper (1902–1995) but later disagreed
with him, and who also worked with the mathematician George Pólya
(1887–1985). Lakatos emphasized the fallible nature of mathemat-
ics, observing that it takes problems and conjectures as its starting
points and then flourishes by means of the criticism and correction
of theories, which are always susceptible to ambiguity and error. It
is not by chance that the magnum opus of Lakatos is entitled Proofs
and Refutations.
In his search for a new philosophy of mathematics, Hersh explores
its framework, i.e., the manner in which it is constructed. It would seem
to be a perfect and precise construction, and this is the image purveyed
by scientific articles, books and textbooks. But this framework conceals a
construction that at times vacillates, is subject to advances and reverses,
uncovers errors, and leads to innumerable blind alleys. Hersh empha-
sizes that, in fact, many truths that were once considered absolute have
since been shown to be imperfect. Many of the steps in mathematical
reasoning suffered at the hands of subtle errors, discovered only at a
later date.

3
P. 39.
160 FIGURING IT OUT

Nobody doubts that the construction of contemporary math can

contain errors and imprecisions. But this human construction may
reveal something about our universe that cannot be explained simply
by the actions of society. Hersh’s detractors accuse him of confusing the
construction process of mathematics with the discussion of its reality
and applicability. There is no doubt that there are errors and restrictions
in the construction process, as there are in every activity undertaken
by human beings. But it is the confrontation with reality that gradually
corrects these restrictions and errors. Mathematics, like science, cannot
only be a social construct. Logic, coherence and reality are its points of
reference. There is indeed a reason for the unreasonable effectiveness of
mathematics.
WHEN CHANCE ENHANCES RELIABILITY

Randomized algorithms have revolutionized the way mathematics oper-

ates. Surprisingly, making use of chance may be the fastest way to obtain
the solution to a problem, and not only when it comes to math.
You will certainly have experienced something like this: you’re call-
ing a friend and suddenly the connection is lost. You call your friend
back but now the line is busy, as he is also trying to call you. You
hang up and wait a few seconds to see if the phone will ring, but of
course it doesn’t, as your friend is also waiting to see what will hap-
pen. So you call again, but once more the line is blocked. . . and so it
goes on until you and your friend stop making the same decisions at the
same time, and then finally one of you gets through. This is a typical
situation involving a conflict of rational patterns, and a successful out-
come is achieved only by the chance occurrence of the decisions being
taken out of phase. It is difficult to find a solution for this problem,
since only two contrary solutions that have to occur simultaneously (one
person waits and the other calls) breaks the impasse. If there is no agree-
ment between the two persons from the start that upsets the symmetry
between their actions, then there is no optimum process for solving the
problem. In the end it is chance that allows the two friends to finally
reconnect.
The same thing is true for many mathematical algorithms, that is for
many finite sequences of rules that lead to a solution, whether exact or
approximate, of a clearly formulated problem. The deterministic algo-
rithms used to solve many such problems have limitations that can only
be overcome by chance.

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_38, 161


C Springer-Verlag Berlin Heidelberg 2010
162 FIGURING IT OUT

One example that illustrates this point is the QuickSort algorithm, a

process used to sort a series of elements, such as a set of names that are to
be listed in alphabetical order. This algorithm is very common in com-
puter programs. It compares the elements of the list, starting with the
first, by making pairwise sequential comparisons, correcting any incor-
rect sequences that it finds, and then proceeding to the next element on
the list. If the elements in the series are randomly mixed, the algorithm
is very efficient, as it can minimize the number of comparisons.
The efficiency of a sorting algorithm can be measured by the num-
ber of comparisons it makes. If we are unlucky, or the algorithm is not
very efficient, we can end up having to compare each element one by
one with every other one, which is the most inefficient method of all.
Surprisingly, this is what we find when the list has already been
sorted and we want to sort it again or double check the order. In this
situation the QuickSort method is the worst possible solution. The algo-
rithm will compare the first name with each of the others and then leave
it in its original position. It will then compare the second name with all
the subsequent names, etc., eventually comparing every possible pair of
names.
It seems difficult to find a better procedure. Beginning with the first
name is the worst possible way to begin. What about starting with the
last name? In that case we potentially encounter the same problem.
What if we were to start in the middle? It would not work well if the
original list has blocks with ordered names. . .
It seems we are facing the same dilemma as before: do we call or
wait to be called? If our decision is always the same as the one our friend
makes at the same time, we will never get to talk to one another.
What we need is a method of selecting an initial element that has the
lowest probability of coinciding with a sorting element of the list. So this
problem does not have a deterministic solution. Nothing is better than
simply tossing a coin and allowing chance to determine our choice.
A modified version of this algorithm, the Random QuickSort, does
exactly that. It starts by randomly selecting an element from the list
that is to be compared with all the others, effectively dividing the list
into two subgroups. Within each of the two subgroups the algorithm
 WHEN CHANCE ENHANCES RELIABILITY 163

makes a random selection of the element that it will use as its reference
point, and keeps repeating this process until the list has been completely
sorted. The probability that the elements selected will systematically be
the last elements in each subgroup is so small that usually the algorithm
is extremely efficient.
Randomized algorithms like this that always find a solution to the
problem are known as Las Vegas type algorithms. These algorithms have
their origin in another older method called the Monte Carlo method.
Both names refer to the famous gambling centers. Unlike Las Vegas
algorithms, Monte Carlo algorithms are not guaranteed to always find
a solution, but they do affect the desired degree of reduction of the
probability and magnitude of an error.

Mary Diana Diana

Hank Hank
Pauline
Mary Mary
Diana
Pauline Norm
Hank
Norm
Norm Pauline

This basic idea was the brainchild of Stanislaw Ulam (1909–1984),

a mathematician of Polish origin who worked in the U.S. from 1936
onwards, and was involved in some of the greatest intellectual adven-
tures of the century. Ulam belonged to the famous Los Alamos group
that included Oppenheimer, Teller, Fermi, von Neumann, Feynman
and so many other celebrated physicists and mathematicians. It was
this group that developed the atomic bomb and pioneered automatic
computation.
164 FIGURING IT OUT

As Ulam relates in his autobiography, Adventures of a

Mathematician, “the idea for what was later called the Monte Carlo
method occurred to me when I was playing solitaire during my illness.
I noticed that it may be much more practical to get an idea of the
probability of the successful outcome of a solitaire game [. . .] by
laying down the cards, experimenting with the process and merely
noticing what proportion comes out successful, rather than to try to
compute all the combinatorial possibilities, which are an exponentially
increasing number [. . .] This is intellectually surprising, and if not
exactly humiliating, gives one a feeling of modesty about the limits of
rational or traditional thinking”.1

PUTTING THE HOUSE IN ORDER

Let us suppose that we have to sort a list of names: Mary, Pauline, Diana,
Hank and Norm. We start with the first two names, comparing Mary
with Pauline, and keep their order, as M comes before P in the alphabet.
Then we compare Mary with Diana and put Diana before Mary, as D
comes before M. And we keep on doing this.
At the end of this step, in which we compared Mary with all the other
names, we have created two groups: one group of names that come before
Mary (in our case Diana and Hank), and another group of names that
come after Mary (in this case Pauline and Norm). The order of each
group is not important. From this point on we compare the names in
each group. The names in the first group do not have to be compared
with the names in the second group, as all the names in the first group
are before Mary and all those in the second group come after Mary. Two
more comparisons (Diana and Hank in the first group, and Pauline and
Norm in the second) are sufficient to sort our list completely. We have
made a total of 6 comparisons to sort the list. If we had needed to com-
pare each name with every other one we would have had to make 10
comparisons.

1
S.M. Ulan, Adventures of a Mathematician, University of California Press, Berkeley, CA, 1991,
p. 196–197.
 WHEN CHANCE ENHANCES RELIABILITY 165

This example shows how QuickSort works. If the list had been longer,
we would have had to create subgroups from the first two groups, and
then more subgroups from the first subgroups, and so on until each of
the final subgroups contained a single name, which would mean that the
list had been completely sorted.
In our example the difference between 6 and 10 comparisons does
not seem very substantial. But just imagine if you had a list of twenty
thousand names of subscribers to a newspaper. An inefficient algorithm
would have to make more than 200 million comparisons, so any time
saved would be very welcome indeed.

It was 1946. Stanislaw Ulam was discussing his idea with John von
Neumann (1903–1957), a mathematician of Hungarian origin who had
settled in the U.S. and was also working at the Los Alamos laboratory
at that time. Von Neumann, the architect of the electronic computer,
immediately recognized the potential of the idea, and the two men
started to apply the method to difficult calculations for the construc-
tion of the nuclear bomb. Today, innumerable simulation methods are
based on the Monte Carlo method.
It is impossible to enumerate all the types of problem that are
resolved today by randomized methods: calculating areas and vol-
umes, studying engineering projects, forecasting the weather, modeling
the behavior of markets, an endless number of theoretical or applied
problems.
If chance often enhances reliability in scientific applications, why
don’t we use it in our daily lives as well? Well, now you know. . . the
next time you lose a phone connection, why not just toss a coin?
 THE DIFFICULTY OF CHANCE

At first glance nothing would seem to be as natural and easy as chance. It

is orderliness and organization that seem difficult to achieve. But reality
is very different. Things frequently take on lives of their own, organizing
themselves and creating patterns. In a forest, for example, each type of
tree tends to appear in specific areas, simply because the greatest con-
centration of the seeds of that tree end up in those areas. In the oceans,
waves seem to move in concert, in groups, in consecutive lines, because
the water masses move in coordination. In the universe, the stars appear
to be grouped into galaxies, which in turn gather in groups, and then
those groups cluster as superconglomerates.
This natural tendency to organize prompts us to think that pure
chance must be something completely disorganized, in which there are
no discernible patterns. But it is difficult to reproduce pure chance. Ask
people to speak syllables haphazardly, as if inventing a new language.
After just a few “words”, everybody will no doubt begin to repeat the
same syllables, as it is very difficult to produce really random sounds.
In one famous experiment psychologists asked people to write
sequences of zeros and ones by imagining they were tossing a coin and
writing 0 every time heads appeared and 1 every time tails appeared.
Then they asked the same participants to write another sequence of ones
and zeros, but this time after tossing a coin and each time recording the
results. It turns out that if the sequences are sufficiently long, let’s say,
containing 20 or more digits, statistical analysis can normally find out if
the result is a purely random sequence, or if the outcome was imagined.
You can try this by writing a sequence of zeros and ones on a piece of

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_39, 167


C Springer-Verlag Berlin Heidelberg 2010
168 FIGURING IT OUT

paper. If you want the experiment to be really conclusive, be patient and

write out about 250 digits and then toss a coin 250 times and write down
each of the results.
Various methods can be used to assess whether the sequences are
random. For example, we can count the number of zeros, which ought
to be approximately half of the total. That is, if we calculate the mean
value of the digits, we should obtain a result of 0.5, or something very
close to that. When a coin is tossed, it has been shown that this is what
happens, i.e., “heads” and “tails” are obtained in very similar numbers.
In the 18th century the French naturalist Georges Louis Leclerc (1707–
1788), known to mathematicians as Count de Buffon, decided to try this
experiment. He, or perhaps one of his servants, tossed a coin 4040 times,
and obtained heads 2084 times, which is an average of 0.5069. Later,
in the 20th century the English statistician Karl Pearson (1857–1936)
repeated this experiment, tossing a coin 24,000 times, and obtaining
heads 12,012 times, or an average of 0.5005. Then during the Second
World War, an English prisoner of war occupied his time in the same
way, obtaining heads 5067 times in 10,000 attempts, for an average of
0.5067.
These results suggest that a coin can be a reasonably random instru-
ment when the two possible results are equally balanced. If you wish to
repeat these experiments, you will have to make sure that you catch the
coin when it is still in the air – if you let the coin roll on the ground
before it falls to one side, then the different designs on the two faces
normally favor one side or the other.
If you count the relative numbers of zeros and ones in an imagined
sequence you will be surprised. You will see that their sequence has an
average that may be quite far from 0.5, while the sequence generated
by coins is very close to this value. But where the human experimenters
usually leave their mark is in the sequences of consecutive ones or zeros,
the “runs”. People think it is more realistic to create runs of ones or zeros
that are short, resulting in frequent changes between the two numbers.
Our intuition tells us that if we toss coins we are more likely to obtain the
sequence 0100101101, as an example, than the sequence 0000011111, to
give another example. But in reality both these sequences are equally
 THE DIFFICULTY OF CHANCE 169

likely to occur, though patterns similar to the first example are more
frequent than those similar to the second. Human beings have difficulty
in intuitively estimating the length and number of runs of the same digit.
If you have taken the time to create a sequence of 250 digits, now
count the number of runs with three or more identical digits. It is very
likely that you will have created very few, as your intuition has likely
told you that such runs are not very likely. However, in the real world,
a completely random sequence of ones and zeros (that is, the zeros and
ones occur with equal probability and independently of the values that
have already occurred) should have about thirty-two runs of three or
more identical digits. Was your sequence like that? Quite frankly, it is
highly unlikely. And what happens with runs of four or more identical
digits? Does your sequence contain any of them? Perhaps it will surprise
you to know there should be about sixteen. And what about sequences
of five or more identical elements? Again, you probably didn’t predict
that there should be about eight, just as there ought to be four sequences
of six digits, and two with at least seven elements.
If you now take the trouble to toss a coin 250 times, you will find
that the runs described above will occur at, or very close to, the frequen-
cies stated here. If you prefer to toss the coin fewer times to create a
shorter sequence, you can compare your self-created sequence with that
produced by your coin tossing experiment. Now that you know people
tend to avoid runs of one number, as they associate alternating numbers
with randomness and runs with a deliberate pattern, perhaps you are no
longer the ideal person to do this experiment. Persuade a friend to join
you in the experiment, asking them to write a random list of zeros and
ones. Then toss the coin once for each of the zeros and ones on the list
and count the number of runs. You will undoubtedly see that the coin
tosses have produced more runs than your friend’s list contains.
All of this may seem like a trivial game, but the reality is that cre-
ating a series of random numbers is a very important matter in science.
For instance, in statistics it is known that random samples have very
desirable properties that make them excellent candidates to “represent”
a population. When organizing experiments, for example, to test drugs
that are administered to one group of people but not to another group
170 FIGURING IT OUT

(who form the control group), it is important to know how to make a

random selection of the elements that form part of each group, in order
to avoid bias due to the subjectivity of the analyst. In computational
science and in all areas that use computer simulations, it is important
to make use of random numbers that allow algorithms to simulate the
variability of real processes. In all these cases, the objective is to obtain
sequences that have properties similar to those of sequences resulting
from the toss of a coin, and that avoid the inherent subjectivity of human
evaluation.
In the past, scientists made use of tables of random numbers that
were expressly created for this purpose via very laborious procedures.
Nowadays, though, these random numbers are obtained from a com-
puter, by means of algebraic processes that produce random numerical
sequences. In reality, these numbers are generated by deterministic pro-
cesses, so they are “pseudo-random” numbers. But, as in chaos, the
sequences obtained are such good reproductions of completely random
sequences that they pass all possible tests, and so can be considered ran-
dom for all intents and purposes. As the mathematician Donald Knuth,
one of today’s most renowned computational scientists, said, “random
numbers should not be generated with a method chosen at random.
Some theory should be used”.1 In other words, chance is too important
to be left to chance.

1
D. E. Knuth, The Art of Computer Programming, vol. II, Addison-Wesley, 1969, p. 6. The italics
are in the original.
 CONJECTURES AND PROOFS

The so-called Collatz conjecture was formulated in 1937 by the German

mathematician Lothar Collatz. A conjecture concerns a mathematical
assumption, something that we think is true, but that has never been
finally proved or disproved. Like some of the most famous mathemati-
cal assumptions, these conjectures are easy to understand and they pass
the common sense test, but they are tremendously difficult to prove or
disprove.
Collatz’ conjecture states that the number 1 is always obtained after
making certain sequential operations starting from any natural number
(positive integer). It works this way. We start with a positive integer. If
this number is even, divide it by 2. If it is odd, multiply it by 3 and add
1. At the end of this process, a new number is produced. We take that
number and repeat the whole process. It was Collatz’s conjecture that
if this sequence of operations were repeated indefinitely, the final result
would inevitably be 1.
There is nothing to match an example. Let’s start with 6. As it is
even, divide it by 2 to obtain 3. As this number is odd, multiply it by
3, then add 1, and the result is 10. Continue this process. . . If you got
your math right, you will find that the numbers obtained are: 6, 3, 10,
5, 16, 8, 4, 2, 1. We do end up with 1, after all. You can try this with
other numbers, but you will almost certainly end up with 1, as countless
others have tried this and always ended up with 1.
The Portuguese scientist Tomás Oliveira e Silva explored a great
number of hypotheses, starting with the number 1 and getting past the
number 27 thousand trillion. He did not come up with one case in which

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_40, 171


C Springer-Verlag Berlin Heidelberg 2010
172 FIGURING IT OUT

he did not end up with the number 1. While this is an important result,
it is not conclusive enough for mathematicians. It may well still be pos-
sible to find a number that has not yet been checked, and that would
disprove the conjecture. In the absence of a rigorous proof or an exam-
ple that refutes the conjecture, we cannot be sure whether the conjecture
is definitively true or false.
 MR. BENFORD

Would you like to win a bet? Let’s see if you can manage to guess the
first digit of a number or at least whether that number is less than 4. Tell
somebody to think of everyday-life numbers, such as the street number
of a house or the amount of money they have in their checking account,
or a percentage rate they commonly use such as a mortgage interest rate,
or, perhaps, a physical or mathematical constant such as pi. If your bet
is that this number begins with 1, 2, or 3, you will be right more than
half the time, in fact, you will be right 60.2% of the time. Your betting
partner will no doubt be surprised by your success rate, for at first sight
there is no reason at all for the laws of chance to produce such a result.
Many people, even those who know probability theory quite well,
are often still amazed by this fact, and cannot see any mathematical basis
for such a deviation from common sense. In reality, any number can
start with any one of the digits from 1 to 9 (we exclude 0, as it not the
first significant digit of any number). But will any one of these nine digits
have the same probable frequency? Why would a number that is selected
at random be more likely to start with 1, 2, or 3? Shouldn’t there be a
uniform probability distribution, as they say, with a probability of 1/9
for each digit? And isn’t 3/9 less than half, which would mean our wager
should be a losing bet?
In reality the first digits of the numbers we encounter in everyday
life do not all occur with the same probability. Number 1 is the most
frequent, about 30.1% of the time, followed by number 2, with a 17.6%
probability of occurrence, and so on, with the probability decreasing
systematically until number 9, which only occurs in 4.6% of cases.

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_41, 173


C Springer-Verlag Berlin Heidelberg 2010
174 FIGURING IT OUT

The first person to discover this strange fact appears to have been
the American astronomer Simon Newcomb, who published his findings
in 1881. But the first to make a thorough study of the phenomenon
was the American physicist Frank Benford, who worked for General
Electric. He published his findings in 1938, in the periodical Proceedings
of the American Philosophical Society (78, 551–572), but his work never
became as well-known as it deserved. William Feller (1906–1970), who
was a professor at Princeton for many years, was one of the few who rec-
ognized the significance of Benford’s work, which he noted in his famous
manual on probability theory.1 Benford had noticed that the first pages
of the logarithmic tables were much more worn than subsequent pages,
meaning that pages with tables for numbers starting with 1 were more
worn than the other pages. The tables seemed to be used less and less as
they progressed to numbers that started with higher digits.
Nowadays, we rarely use logarithmic tables, as calculators and com-
puters work much more precisely and rapidly. But these tables were used
intensively over a period of several 100 years, from the start of the 17th
century, when logarithms were created and popularized by the Scottish
mathematician John Napier (1550–1617), until just a few decades ago,
when electronic calculators first appeared. While those tables may have
fallen into disuse, logarithms are still used extensively in the fields of
mathematics and computation.
Surprised by his discovery, Benford wondered whether this could
also happen in other circumstances, so he analyzed a vast variety of
numbers, including the surface areas of rivers, sports statistics, numbers
published in the press, and the addresses of celebrities, all in all, more
than 20,000 sets of numbers. It turned out that in each case he encoun-
tered the same “anomaly”, as he called it, verifying that more than 30%
of the numbers started with 1, whereas barely 5% began with 9. What
was the explanation?

1
W. Feller, An Introduction to Probability Theory and its Applications, Vol. 2, 3rd edition, Wiley,
New York, NY, 1968
 MR . BENFORD 175

The phenomenon is difficult to comprehend, but there are some

examples that help make it clearer. For instance, let’s take the house
numbers in a street address. How many houses are there on average in a
street? Exactly 999? That’s not very likely. Exactly 99? Not very probable
either. So let’s agree on 50 as an example. In this case, it is quite easy
to work out how many house numbers begin with 1. There are eleven of
them: house number 1, then 10, 11, 12, and so on until 19. The probabil-
ity that a house chosen at random will have a number starting with 1 is
11/50 or 22%. On the other hand, there would only be one house with a
number beginning with 9, i.e., house number 9 itself. So the probability
of a house having a number starting with 9 is 1/50, which is only 2%.

Frequency of first significant digits

%
,1
30

Benford’s Law
%

Numbers in
,6

newspapers
17

Dow Jones
%
,5
12

7%

Newspaper Expresso
9,

9%

7%
7,

8%

1%
6,

6%
5,

5,

4,

1 2 3 4 5 6 7 8 9

According to Benford’s Law, the probability for a digit k to be the first significant
digit in a number is not 1/9, but the log to the base 10 of the quantity (k + 1)/k.
Therefore the probability that 1 is the first significant digit is about 30.1%, whereas
the probability for 9 is only about 4.6%. The logarithm is an exponent for a given
base number, and any positive number can be expressed as a power of a specific
base number. For example, 1000 can also be expressed as 103 , so its logarithm to
the base 10 is 3.
176 FIGURING IT OUT

The bar chart compares the frequencies of the first significant digits
forecast by Benford’s Law with those observed on the first pages of news-
papers and with the values in the Dow Jones stock index. There is great
correlation between the forecast frequencies and those actually observed.

If we think about it carefully, for any one of the nine digits 0–9 to
have the same probability of being the first significant digit in a door
number, the street has to have exactly 9 houses or 99, or 999 . . . which
certainly will not occur very often. In any other context there is a bias
against the higher digits, while the lower digits appear more frequently.
Another clear example is provided by growth rates. Just think of a
stock exchange index, or a price index, or a production index. When
the index was first created, it started with a value of 100 (it could have
started with any other number, but the problem would still be identical).
We know that this type of index has a tendency to increase. Let’s suppose
that the growth rate is constant and that the value of the index doubles
every year. During its first year, the index remains in the 100 s, and by
the beginning of the second year, it has risen to 200, then to 400 at the
start of the third year, and so on. The value of the index starts with 1 for
12 months. Then the periods during which it starts with 2 and 3 each last
for about 6 months. When the first significant digit at last reaches 9, this
only holds true for about 1 month! It is easy to see that, even if we select
a growth rate at random, it is most probable that a low digit will still be
the first significant digit. If you bet that the digit will be less than 4, your
chance of being right is 60.2%.
This case is particularly significant, as it illustrates the fact that it is
an exponential and not a linear behavior that is the dominating factor in
many natural and social phenomena. The behavior of the first significant
digits reflects this fact.
Frank Benford understood this, and formulated what he called the
“law of anomalous numbers”, today usually referred to as Benford’s
Law. Although his formulation was ignored for many years, it is once
more attracting the attention of mathematicians and statisticians. As
 MR . BENFORD 177

often happens in mathematics, it is sometimes only years later that the

possible applications and relevance of some results are seen. Benford’s
Law, for example, is beginning to be applied in a surprising domain,
the detection of tax fraud. Mark Nigrini, an accountant with a statistical
background who worked in Cape Town, New York, Kansas, and Texas
before becoming a professor in The College of New Jersey, has created
a method for analyzing tax declarations by checking the frequency of
occurrence of the first significant digits.
People who systematically falsify a tax declaration by inventing non-
existent expenses try to use numbers that seem to be as completely
random and different as possible. Tax officials may have their suspicions
if, for example, exactly the same quantity of expenses is claimed for each
business trip. Well, the smartest cheats try to make the numbers as uni-
formly random as possible, starting as often with a 9 as with a 1. It is
at this point that the new technique pioneered by Mark Nigrini comes
into play. Many of the tests he has devised are quite simple; they only
count the frequency of occurrence of the first significant digits. Some of
the other tests are more complex.
By the way, if you are thinking about using Benford’s Law to help
you find winning numbers in the lottery, forget it. The distribution of
lottery numbers is uniform, and does not follow Benford’s Law. But you
can certainly use it to win bets with your friends.
 FINANCIAL FRACTALS

The mathematician Benoît Mandelbrot once said that his childhood

ambition was to become the “Newton” of a specific mathematical field,
no matter how small, in other words to propose something creative and
completely innovative in a particular area of mathematics. Ultimately
he chose what came to be called fractal objects, at that time relatively
unknown geometrical figures with curious properties.
It is very difficult, if not impossible, to define fractals in non-
technical terms, but the basic idea is that such objects contain a structure
that is reproduced at every scale. For example, a square is not a frac-
tal object – if part of one of a square’s edges is amplified, what we see
is a segment of a completely smooth straight line without any of the
outlines that characterize a square. But a snowflake can have a fractal
structure. When its branch-like aspects are observed under a micro-
scope, we see within them other, similar branch-like structures. If we
observe these through an even more powerful microscope, we can dis-
cern even more similar structures, and this pattern can go on and
on. The snowflake has a structure that repeats itself at various scales.
Clearly, this degree of similarity is not present at the molecular scale,
but we can imagine geometric models in which patterns are repeated
infinitely.
Mandelbrot called such objects fractals and he recognized that their
dimension, at least in a certain technical sense, could have a fractional
value. A segment of a straight line has one dimension, a square has two
dimensions, a cube three dimensions. But objects such as snowflakes can
have a fractional dimension, such as two and a half.

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_42, 179


C Springer-Verlag Berlin Heidelberg 2010
180 FIGURING IT OUT

Mandelbrot offers fascinating examples of the strange properties of

fractal objects measured at different scales. For example, what is the
length of the coastline of Great Britain? If we use a good map to mea-
sure it, we will find a numerical result; but if we then use an even more
detailed map, we will count more bays and capes, so the result will be a
larger number than before; if we then decide to measure the rocks and
coves, the number increases even further.
Mandelbrot concluded that the measurement of the length of cer-
tain irregular objects depends on the gauge that is used. The smaller
and more precise the gauge, the greater the length that is measured.
The border between Spain and Portugal is shown differently on Spanish
and on Portuguese maps. When measured and officially documented by
the Portuguese it is almost 20% longer than according to the Spanish
records. This is also the case with the border between Belgium and the
Netherlands, for example, where the difference is also about 20%, with
the smaller country always being the one that reports the larger value.
Mandelbrot has an explanation for this: the smaller country measures
its border more carefully, using a more precise gauge. The length of the
border depends on the scale that is used.
For more than 40 years, Mandelbrot has consistently emphasized
that the phenomena of scale and of self-reproduction are equally appli-
cable in terms of economic and financial matters as well. The setting
of prices is the result of various individual choices, decisions made by
many buyers and sellers. And the variation over a long period is the
result of many shorter-term variations. Large movements result from
small movements, which in turn result from even smaller movements,
all of which have similar properties. They are the snowflakes whirled
around by the economy.
Mandelbrot derived two properties as the mathematical result of
this hypothesis, which he metaphorically called the “Noah effect” and
the “Joseph effect”. By referencing the Biblical flood, Mandelbrot asserts
that extremely large variations may be produced occasionally, i.e., very
extreme values occur with a significant probability. Using the seven fat
years and the seven lean years Joseph experienced in Egypt as a ref-
erence point, Mandelbrot asserts that economic phenomena exhibit a
 FINANCIAL FRACTALS 181

degree of persistence that is technically known as “long-term memory”,

prolonging for long periods the effects of “shocks” or irregular
variations.
What is absolutely amazing about this is that these two basic ideas,
which were not taken seriously at the time, have come to dominate much
of the econometric debate today.
The extreme irregularity of financial series, the drastic collapses
such as occurred on “Black Monday” in New York in October 1987,
and also more recently at the start of the current financial crisis, and the
subsequent spectacular recoveries, are all phenomena that are difficult
to explain by means of the probability distributions normally used in
econometrics, like the normal or Gaussian curve. Extreme movements
seem to indicate that the financial series do not have a finite variance
or volatility, as it can be technically called. Even today this problem is a
point for debate between economists and statisticians: every year dozens
and dozens of new studies are published, re-evaluating the hypotheses
of infinite variance introduced by Mandelbrot.
On the other hand, the persistence of variability in financial series
has been given more credence by recent econometric studies, some-
thing Mandelbrot foresaw more than 40 years ago, which is called
“volatility persistence” or “contagious speculation”. What happens here
is apparently paradoxical. Essentially, it holds that movements in finan-
cial markets are impossible to foresee – nobody knows whether the
stock market is going to rise or fall tomorrow. But we can make reason-
able statistical forecasts for the magnitude of the changes, the so-called
volatility. If the market is nervous, if it has experienced great fluctu-
ations, then it is natural to predict that tomorrow will bring a major
fluctuation. If the market is calm and stable, then it is likely that tomor-
row it will remain calm, and that any price changes will be minor. High
volatility tends to be persistent, as does low volatility. This “long-term
memory” of volatility in financial markets, which represents the Joseph
effect proposed by Mandelbrot more than 40 years ago, only began
to be statistically evaluated in the 1990s, and it became an interesting
model for the economic sciences. Clearly, Mandelbrot had much more
important things to tell economists than many believed at the time.
 TURING’S TEST

Will we succeed one day in constructing machines capable of thinking?

This is a more profound question than it at first seems, and has been one
of the most hotly debated topics among philosophers over the centuries.
When computers first appeared, these debates flared up again and their
tone became more pressing. Nowadays, with computer programs capa-
ble of beating the world’s chess champion, it makes a lot of sense to ask
whether, when all is said and done, artificial intelligence is not in fact
true intelligence.
The modern debate about the possibility of creating intelligent
machines started in 1950, when the first computers became available.
In England the émigré Hungarian chemist Michael Polanyi (1891–1976)
went to great lengths to show that the human mind could not be reduced
to a simple mechanical system. He based his arguments on a theory
propounded by Kurt Gödel (1906–1978), a mathematician and logician
who had demonstrated that it is impossible to construct the whole field
of mathematics on the basis of a fixed system of axioms. The philoso-
pher Karl Popper (1902–1994) also contributed to the debate, affirming
that only the human brain can make sense of a machine’s meaningless

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_43, 183


C Springer-Verlag Berlin Heidelberg 2010
184 FIGURING IT OUT

capacity to produce truth. In other words, computers cannot see any

significance in any true propositions that they may construct, as only
the human mind can make sense of the instructions provided to the
machines and to the results of their operations.
At this juncture the debate was joined by Alan Turing (1912–1954),
the British mathematician who had managed to break the German
encoding machine during the Second World War. In 1937 Turing had
shown that there are mathematical problems that are impossible to
resolve by means of automatic computation, and accordingly he did
not believe in the absolute power of the computer. Nor, however, did he
believe in the non-material nature of the mind. In Turing’s view, intel-
ligence only exists and becomes manifest in dialog and interaction with
the environment, most commonly among people. That is, comparison
with the self is the only basis a human being can use to evaluate whether
a person or a machine is, or is not, intelligent.
Turing suggested a practical test to characterize intelligence. In a
study published in 1950 in the philosophical journal Mind, he imagined
a game in which a person communicated in writing with another per-
son or with a machine. Using a series of questions, the inquirer had to
try to find out whether he or she was writing to a human or a machine. A
machine’s intelligence would be measured by the extent to which it man-
aged to fool the inquirer. So, for Turing, a computer could be intelligent
to the extent that humans do not unmask it.
Turing’s test had an enormous influence on the development of a
branch of computational science known as artificial intelligence. First
specialists tried to understand the way the human brain functioned by
reproducing it, as far as was possible, in a computer. Later they per-
ceived that there are some areas in which machines are much superior
to humans, and others in which humans defeat machines emphatically.
They also observed that these two aspects can be differentiated as fol-
lows: whereas human beings are capable of distinguishing or creating
patterns based on a variety of information, machines are capable of
systematic and repetitive searches. So improvements made to the capac-
ity of computers would result from algorithmic improvements for the
systematic search for solutions.
 TURING’ S TEST 185

When the Internet arrived, the distinction between human intelli-

gence and artificial intelligence became a pressing problem that had to
be resolved in practice. Some computer programs had infiltrated into
chats on Yahoo and other systems, collecting information on the human
participants and inserting commercial references to companies or prod-
ucts. Other programs had created problems for the major portals by
automatically registering thousands of non-existent users, thereby cre-
ating innumerable email accounts that were then used to send spam,
massive commercial mail shots that slow down servers and accumulate
in our mailboxes, while others have succeeded in worming their way
into individual accounts, and trying out keywords repeatedly until they
gain access to the account. The security systems devised to prevent peo-
ple from violating systems can be rendered powerless when confronted
by machines.
To resolve this problem, technicians at Yahoo set up a cooperative
program with scientists at the Carnegie Mellon University in Pittsburgh.
The so-called CAPTCHA project (www.captcha.net) attempts to gener-
ate tests that only human beings can easily pass, and that computers
usually fail. Some of these tests are based on the distortion of words or
phrases, which OCR (optical character recognition) programs find dif-
ficult to decipher. The ability to recognize visual patterns is the result
of an evolution spanning millions of years and of very long training in
the case of each individual. It is not easy to figure out exactly how this
capability developed, much less reduce it to a set of instructions that can
be imparted to a computer. However, it is simple to write a computer
program for operations that we were taught rationally, such as logical or
arithmetical functions, which is why machines can easily handle prob-
lems that can be reduced to these formal operations, though they cannot
recognize subtler patterns in messages containing a lot of visual noise.
Various companies are now engaged in exploring this limitation of
computers in order to find ways of protecting their own services. Yahoo’s
CAPTCHA project involves a process that restricts the automatic
registration of users. If you want to open an email account in this portal
(www.yahoo.com), then at one stage of the procedure you will have to
read some letters written in a bizarre script and then type the letters into
186 FIGURING IT OUT

the registration form. If you don’t read the letters correctly (and OCR
programs have great difficulty with this), you won’t be given access to
the next stage of the procedure, and so you will not be permitted to
open the email account. In this case human beings succeed in doing
something machines can’t do.
The CAPTCHA concept is now widely used. Turing’s test gave rise
to a flourishing investigation involving various applications in the com-
puter industry. Because of this, today we can provide multiple responses
to the question: “Do machines think?” Fortunately for us, there are still
occasions when they don’t know how to think.
 DNA COMPUTERS

On April 25, 1953, James D. Watson and Francis Crick published an

article in the journal Nature that was less than two pages long. In this
study they presented the famous double-helix structure of DNA, and
as a consequence the world today has changed. The discovery of DNA
has opened up new paths in biology, medicine, agriculture, forensic
science, and in numerous other technical and scientific fields. The dou-
ble helix has appeared in works of art, and the abbreviation has even
been adopted as the name for perfumes. And now something that peo-
ple have been talking about for some time seems to be coming true:
DNA could begin to be used for computation. In the near future PCs
might no longer use silicone, but instead an aqueous solution of DNA
molecules.
In 1994 Leonard M. Adleman was the first person to show how
DNA could be used to resolve computational problems. The problem
he solved using DNA could today be solved by a computer in a fraction
of a second. It took Adleman 7 days to accomplish, but in the process he
demonstrated how DNA could be used as a basis for calculation.
Adleman was faced with a “traveling salesman” problem. He imag-
ined seven cities and a salesman who had to visit each one of them,
starting from a specific city, and never visiting any city twice. This might
seem simple, but when we increase the number of cities the problem
rapidly becomes impossible to tackle. We may be forced to investigate
each possible solution, which would require tremendous computational
resources. This method came to be called the “brute force method”, and
is the one Adleman employed.

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_44, 187


C Springer-Verlag Berlin Heidelberg 2010
188 FIGURING IT OUT

First Adleman generated all the possible routes. He then isolated

those that originated in the designated city. He subsequently coordi-
nated the itineraries with the exact number of cities. And finally he
selected the paths that only visited each city once. To do all that, he made
use of the way in which DNA transmits information.
Whereas computers store data in sequences of zeros and ones, DNA
codes them in sequences of four bases, represented by the letters A, T,
C and G, aligned sequentially in chains. The double helix is formed
by two complementary chains that are joined in such a way that A on
one chain is linked to T on the other, just as C is linked to G; the two
chains are connected via bonds between these pairs. For example, this
means that the complement of the sequence ATCAG is TAGTC. One
chain can only be associated (“hybridized”) with another if they both
have complementary sequences.
Adleman encoded each city as a particular sequence of these four
letters. He then did the same for the connections between the cities. For
example, Denver could be represented by CTACGG and Salt Lake City
by ATGCCG. The route connecting these two cities could be CGGATG,
combining the last three letters of Denver with the first three letters of
Salt Lake City. The complement of this route sequence is GCCTAC,
which could be associated with the sequences for the two cities, as the
letter G is associated with C (the fourth letter of Denver), C with G (the
fifth letter of that city), and so on until the last letter, C, which is asso-
ciated with G (the third letter of Salt Lake City). In other words, within
a long DNA sequence, the appearance of GCCTAC is associated with a
sequence containing Denver followed by Salt Lake City.
Adleman then constructed DNA fragments containing the codes
of the cities in random order, which is a relatively simple task using
a DNA synthesizer. He also constructed sequences that contained the
codes for the connections between the cities, also in random order.
Then, using the PCR (polymer chain reaction) technique that has since
been completely mastered, he produced many copies of the sequence of
the connections that began with the departure city, such as Denver, and
ended with the destination city, for example Santa Fe.
 DNA COMPUTERS 189

Next, Adleman used another well-known technique to select DNA

fragments that contained exactly 30 bases (5 cities × 6 bases) in such
a way that none of the selected routes passed through any city twice.
Finally he used a well-known chemical technique (affinity purification)
to review the residual groupings and select the molecules that contained
all the cities. The final groupings, linking the chains of cities to those of
the connections, constitute a solution, or solutions, to the problem.
The problem that Adleman solved was just an initial demonstration
of the computational power of DNA, but since that time progress has
been rapid. In 2003, a team from the Weizmann Institute in Israel suc-
ceeded in creating a system in which propulsion energy itself is provided
by the DNA molecule. The Guinness Book of World Records recognized
this system as the world’s smallest biological computational device to
date.
Doubts have been expressed about the feasibility of creating opera-
tional computers based on DNA: the calculations are performed rapidly,
but the preparation and reading processes are very slow. Nevertheless,
this miraculous molecule has alluring properties. It possesses a capac-
ity for storing information that electronic systems can only dream of: a
single cubic centimeter (about one-sixteenth of a cubic inch) of DNA
can store as much data as a trillion CDs. Even more importantly, it
is a multi-processor: the molecules work on parallel tracks, and so are
able to produce millions of answers simultaneously, and thus incredibly
quickly. For the problems that it can resolve, this system first con-
structed in Israel is already one hundred thousand times faster than
today’s PCs.
 MAGICAL MULTIPLICATION

1 3

2 7 3

The video shows what it claims is a new method of multiplying. It is

simple, somewhat strange, but it always seems to work. It starts by
showing us how to multiply 21 by 13. To do this we have to draw two
horizontal lines, which represent the number of tens (2) in the first
number. Below we draw a single line that represents the number of
units (1) it contains. Then we draw the second number using vertical
lines: one on the left for the number of tens (1), and three on the right
for the number of units (3).
Then we count the points at which the vertical and horizontal lines
intersect. There are four corners, each with its points. At the top left

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_45, 191


C Springer-Verlag Berlin Heidelberg 2010
192 FIGURING IT OUT

there are two intersecting points, so we write a 2 for the hundreds. At

the bottom right there are three intersection points, so we write 3 for
the units. The remaining two corners have a total of seven points, so we
write 7 for the tens. That’s it: 21 × 13 = 273.
The video continues with a more complicated example to show that
the method always works: 123 × 321 = 39,483. In this case there are five
types of intersection points that provide the five digits in the answer.
There is one more step that has to be taken: “and carry over”, as in the
usual multiplication algorithm.
In fact, strange as it may seem, these drawn lines simply represent
a method that is similar to the way we normally multiply two numbers.
In the end, what is 21 × 13? We just have to note that the numbers 21
and 13 simply represent our decimal notation for the quantities. So we
see that 21 = 2 × 10 + 1 and that 13 = 1 × 10 + 3. The multiplication
can be done with the two numbers so partitioned and adding the partial
results (20 × 10 + 20 × 3 + 1 × 10 + 1 × 3). This is easy and it works. It
must work! The intersection of x lines with y lines has to be x × y points.
That is exactly what multiplication is.
Having arrived at this point, you may be tempted to ask: why didn’t
they teach me that in school? Wouldn’t it have been easier and more
fun? You are right: it would be easier and more fun, but only for very
basic numbers. Try multiplying 99 by 99 and you will see that it is nei-
ther easy nor fun to arrive at the answer by drawing lines. In fact, you
will find that it is easier and less error-prone to multiply these num-
bers as we were originally taught. The multiplication algorithm that we
learned (or should have learned) in school is the result of a series of tri-
als and improvements that have lasted for hundreds of years. It is worth
taking the trouble to master it.
 π DAY

We all know certain commemorative dates by heart. In the U.S., for

example, Mother’s Day falls on the second Sunday in May, Father’s
Day on the third Sunday in June, Labor Day on the first Monday in
September. And π day? Do you know when that is?
This date was not chosen at random. To commemorate the num-
ber 3.14, the 3rd month and the 14th day were chosen, so π day falls on
March 14. In schools the date has been observed for some decades. But
now the U.S. House of Representatives has made it official: on March 12,
2009 with 391 Yeas and only 10 Nays the House supported “the designa-
tion of a ‘π Day’ and its celebration around the world”. The resolution
further explained that “π can be approximated as 3.14, and thus March
14” is an appropriate day for such celebration.
Of course π is not exactly 3.14. It is a number with many decimal
places. Some people write it as 3.1416. The more precise calculators note
it as 3.1415926535. And the number does not stop at that point. As a
matter of fact, where does it end?
It has been known for a long time that π is a number that does not
have a finite decimal form. It cannot be written as a finite decimal num-
ber, nor as a periodical infinite decimal. It is an irrational number, which
means it cannot be written as the ratio of two integers, so its decimal
expression is endless and does not repeat itself. No matter how we write
π, we always have to deal with an approximation.
An initial approximation to π was the number 3, as is implicit in
the Bible, in the Book of Kings, written in about the 6th century BC.
Even before that, the Babylonians had a better approximation, 3.125,

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_46, 193


C Springer-Verlag Berlin Heidelberg 2010
194 FIGURING IT OUT

and the Egyptians implicitly used the square of 16/9, which is 3.16049. . .
In the 3rd century BC Archimedes discovered a rigorous method for
calculating π, and obtained an approximation that was correct to the
second decimal place.
Archimedes’s method entails fitting a circumference into inscribed
and circumscribed polygons. The perimeters of the polygons are simple
to calculate, and the perimeter of the circumference is fitted between
them. It is an ingenious method and it allows us to obtain any desired
approximation simply by using polygons with more sides.
During the Middle Ages Archimedes’s work was rediscovered by
the Arabs, and later by the Europeans. A race to calculate the value of
π developed. In the 9th century al-Khwarizmi calculated it to 4 deci-
mal places. By the end of the 16th century mathematicians had achieved
twenty decimal places, and then by the close of the 18th century they had
arrived at 140. By the beginning of the 20th century more than 500 dec-
imal places had been achieved. Somewhat later computers made their
appearance. Thousands of decimal places were calculated, then mil-
lions, and later thousands of millions. . . The record is currently held by
the Japanese mathematician Yasumasa Kanada, who managed to obtain
1,241,100,000,000 digits by the end of 2002, which is more than a trillion
decimal places.
Apart from having started a race to make an ever more precise cal-
culation of the value of π, the method devised by Archimedes allowed
this calculation to become autonomous from geometry. In fact, when
we write the equations for the perimeters of the polygons, π appears as
a pure number.
In the 17th century various mathematicians, including Leibniz and
Newton, devised equations that were more efficient at calculating π, and
it came to be observed that this number pops up in the most unex-
pected circumstances. The strangest process for calculating π, however,
may well be the probabilistic method proposed by the French natu-
ralist Count de Buffon (1707–1788), a curious fellow we have already
encountered. You can try it out at home.
Begin by getting a sewing needle. Then draw a series of parallel
straight lines right across a large sheet of paper. The straight lines must
 π DAY 195

always be the same distance apart: to make things simple, separate them
by twice the length of the needle. Then toss the needle repeatedly on
to the paper. Count the number of times the needle intersects a straight
line. Divide the total number of times you tossed the needle by the num-
ber of times it touched a straight line, and the result is an approximation
of π. This method is probabilistic and converges very gradually. You
may well need to toss the needle hundreds of times before you get the
first decimal place correct. But the experiment is so strange and curious
that it is worth trying it out. It is a good way of commemorating π day.
 THE BEST JOB IN THE WORLD

Many scientists will secretly admit that they enjoy their jobs so much
they would pay to do what they are paid for. Of course, this is something
of an exaggeration. Very few of them could actually keep up a full-time
job in a scientific field if they were not paid. The “good old days” when
aristocrats pursued science as a hobby are long past. Today, science is a
job. A good job.
In the United States in particular, careers in science are well remu-
nerated. According to a survey published recently by CareerCast.com,
a business organization, science is the top profession. That’s right:
the best.
This survey contradicts the disdain for studying and obtaining qual-
ifications that is often to be read in public comments made thoughtlessly
by people in responsible positions. Young people have to be reminded
that studying is a precious investment in the future. That has to be stated
repeatedly. A qualification in the sciences or the arts brings various
professional benefits. It is not just a question of salary. It involves job
quality, quality of life and cultural richness.
The CareerCast study presents a ranking list (with all the simplifica-
tions typical of a list of this type) along with a good deal of comparative
information also extracted from the survey. In it, 200 types of jobs were
reviewed and then classified according to five basic criteria: work envi-
ronment, remuneration, future prospects, physical requirements and
emotional stress.
The survey endeavors to use objective criteria. Job elements such
as the necessity to adopt uncomfortable positions, put up with possibly

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_47, 197


C Springer-Verlag Berlin Heidelberg 2010
198 FIGURING IT OUT

toxic conditions, or accept confinement in small spaces are among the

data used to assess the work environment. The factors used to classify
compensation include the dates paychecks are issued, and any addi-
tional bonuses or perks. Regarding future prospects, some of the criteria
are the prospective risk of unemployment, the possibility of future salary
increases, and chances of promotion.
The fourth criterion, physical requirements, is a domain loved by
physicists, who have assisted official statisticians in measuring this area
by taking various jobs and evaluating them on the basis of such factors as
the weight that has to be lifted, types of movements required (kneeling,
bending, etc.), and the work actually performed (in the physical sense of
the word).
Finally, the study assesses emotional stress, considering such factors
as relative competitiveness, risks, and pressure to respond.
After all these factors are considered, it turns out the worst job
(200th in the list) according to this study, is lumberjacking, which
involves tremendous physical risks for a low average pay of less than
$32,000 a year. Other jobs at the bottom of the list include taxi-drivers
(198), fishermen (197), garbage collectors (195) and fire fighters (182).
At the other extreme, the top-rated jobs include meteorologists (15),
physicists (13), accountants (10), systems analysts (6) and IT engineers
(5). These are surpassed by biologists (4).
The best three jobs in the survey are statisticians (that is, math spe-
cialists in data processing and inference), actuaries (that is, specialists in
the mathematical calculation of the risks involved in the insurance and
investment businesses), and finally, in first place, mathematicians!
OUT OF THIS WORLD
 ELECTORAL PARADOXES

Voting in elections is one of humanity’s great achievements. No bet-

ter method has yet been invented to achieve a system of government
that guarantees liberty and progress. But would it be possible to invent
a better one?
Democracies adopt a system usually known as “one man, one vote”.
This is overall a fair system, though it is not free of paradoxes or con-
tradictions, as has been noted since ancient times. Apparently Pliny the
Younger, a lawyer and magistrate of Ancient Rome (this name is used
to distinguish him from his uncle Pliny the Elder) was the first to reveal
some of the paradoxes involving voting. In the 2nd century BC only an
elite group was permitted to vote, but their problems were largely the
same as ours today. Electoral paradoxes first began to be discussed sys-
tematically in the 17th century, with the objective of devising a perfect
and rational electoral system, but soon issues began to arise.
The French mathematician Jean-Charles de Borda (1733–1799) was
the first person to study these electoral issues systematically. Regarding
the electoral system as a means of aggregating opinions to determine the
collective choice, he noted that different voting methods lead to differ-
ent results. This became known as Borda’s paradox and was a popular
topic of discussion at the time, although no satisfactory resolution to the
paradox emerged.
On June 16, 1770, Borda formally presented the problem to the
Royal French Academy citing an example in which 21 voters had to
choose from among 3 candidates. He took under consideration the rela-
tive preferences of each voter, that is the way in which each voter ranked

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_48, 201


C Springer-Verlag Berlin Heidelberg 2010
202 FIGURING IT OUT

the candidates. What he noticed was that, surprisingly, it was possi-

ble to elect a candidate the majority of voters had ranked last. This
only required that the votes for the other candidates were sufficiently
split. Nowadays this phenomenon is referred to as the election of a
“Condorcet loser”, that is of a candidate who loses in direct comparison
with all the others. In Borda’s example candidate A would lose the elec-
tion if his only opponent was candidate B, and would also be defeated if
his only opponent was candidate C, but wins the election if both B and
C are competing against him.
To resolve this paradox, Borda proposed a system that came to be
known as the Borda count. Instead of the “one man, one vote” sys-
tem, Borda gave each voter the opportunity to award points to each
candidate. With a total of three points at their disposal, voters could
assign two points to their preferred candidate, one point to their second-
preference candidate, and zero points to the third candidate. The points
would then be added up, and the candidate with the most points
would win.

BORDA’S PARADOX

Preference 1 Voter 7 Voters 7 Voters 6 Voters

1st choice A A B C
2nd choice B C C B
3rd choice C B A A

Each preference profile is shown in one of the columns together with the
number of voters who chose it. So, for example, only one person voted for
candidates A, B and C as their first, second and third choices respectively.
In the second column we see that 7 voters ranked the candidates in the
order A, C, B. In this example provided by Borda, the candidate with the
most votes according to the one-man-one-vote system is A with 8 votes,
compared with 7 for B and 6 for C. Nevertheless, A is the candidate most
disliked by the majority of the electorate, since 13 out of 21 voters place
him last in their ranking.
 ELECTORAL PARADOXES 203

This system at first appears perfect, but it also has certain prob-
lems. First, why do the points awarded to each preference have to be
successive whole numbers? Giving zero points to the least liked candi-
date, one point to the next candidate, and so on, might not result in
an exact reflection of the voters’ preferences. Why couldn’t a voter give
zero points to one candidate, half a point to another, and one and a half
points to the preferred candidate? The curious thing about this is that
under such a system it would be possible for a different candidate to win.
Even if the ranking of preferences remained identical, which candidate
wins still hinges on the weighting system being used. In other words, the
Borda count system will not always produce the same result.
As another French mathematician and philosopher, the Marquis de
Condorcet (1743–1794), showed, an even more serious electoral prob-
lem is that it is not always possible to aggregate the voters’ preferences in
a coherent manner. The ranking of the preferences of each voter should
be transitive: if a voter ranks candidate A ahead of B, and B ahead of C,
then he or she will also rank A ahead of C. However, this is not neces-
sarily reflected in the total number of votes in an election. More people
might vote for A than B, or for B than C, but still prefer C to A! How can
this problem be resolved?

CONDORCET’S PARADOX

Preference Group 1 Group 2 Group 3

1st choice A B C
2nd choice B C A
3rd choice C A B

The first group of voters prefers candidate A, followed by B and then by

C. There is a transitive logic in the choices expressed by this group. If
A is preferred to B and B is preferred to C, then A is also preferred to
C. But this transitivity property is not passed to the final election results.
Let us suppose that each group is comprised of an identical number of
voters. Then A will beat B, as this is the ranking preferred by groups 1
204 FIGURING IT OUT

and 3. This is the majority opinion of the voters. On the other hand, B
will also beat C, as this is the ranking preferred by groups 1 and 2. It
would seem logical that, as A beats B and B beats C, then A should also
beat C. Nevertheless C beats A, as you will see using the same reasoning:
both group 2 and group 3 rank C ahead of A.

Donald Saari, a mathematician from the University of California at

Irvine who has devoted himself to the study of electoral problems, has
illustrated how minor changes in any electoral system can lead to major
changes in election results. Saari is among the mathematicians and polit-
ical scientists who have studied problems related to “public choice”, a
field that experienced major developments in the second half of the 20th
century.
At this stage you may already suspect that a perfect system is very
hard to come by. But the problem is even more complicated than it
appears. Kenneth J. Arrow, a Nobel-winning American mathemati-
cian and economist, examined a set of electoral conditions that were
apparently reasonable, such as the transitivity of preferences mentioned
above, and demonstrated that it is not possible to have a single electoral
system that satisfies all these conditions at the same time.
So what can be done? Mathematically, there is no solution to this
problem, but society does not really need systems that are perfect.
Rather, it needs rules that lead to choices that are acceptable to society,
even if they are fallible and approximate. Mathematics can help people
to perceive the problems of various electoral systems, but it does not go
so far as to question the democratic system itself, as this is a social moral
choice validated by history.
 THE MELON PARADOX

This is a curious problem that comes up regularly in math competitions.

You will see it in one form or another in published collections of prob-
lems from these competitions. It’s not that the math itself is difficult.
The difficulty lies in believing the results. As an example, let’s start with
a melon weighing 50 ounces. Only 1% of the mass of the melon is made
up of solid matter, while the remaining 99% is water. The melon is left
in the sun and dehydrates to such an extent that it now only contains
98% water. The question is: how much does the melon weigh now? The
answer is easy, provided you do your sums properly. But let’s start by
guessing the weight. Will it be about 49 ounces? Or even 49 and a half
ounces? Or just 45 ounces?
Most people do guess somewhere in the region of these numbers,
perhaps figuring that 50 ounces × 98/99 is the solution, which would
mean the melon weighs 49 and 49/99 ounces, which we can round off
to 49 and 1/2 ounces. The surprise though, is none of these answers are
correct. They are, in fact, completely wrong! It turns out the melon has
lost half its weight, and now weighs just 25 ounces!
The calculation is simple: if 99% of the original melon was water,
the solid content was 1/100th of 50 ounces. If the water content of
the dehydrated melon was then reduced to 98%, the solid part is now
2%. This means that the total weight is now 50 times the weight of
the solid content, which did not change. Therefore the weight has
dropped to 25 ounces. It has been reduced by half. This problem is

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_49, 205


C Springer-Verlag Berlin Heidelberg 2010
206 FIGURING IT OUT

educational, and almost a paradox. It shows how misleading intuition

and simple proportions can be when we are dealing with relative mea-
surements. But to be surprised, you have to start out by guessing the
answer.
 THE CUPCAKE PARADOX

When a fair-minded group of friends shares a plate of cupcakes, each

one takes one and eats it, taking care to leave a cupcake for the next
person. However, if the cupcakes are especially tasty and everyone is
hungrier than usual, what happens when there is only one cupcake left
on the plate? Carefully, one of the friends cuts the cake in half and takes
one of the halves. A second person can’t resist, and cuts the remaining
half in half. Then a third friend comes forward and cuts the remainder
in half. And so it goes on . . . Theoretically we could imagine a virtual
cake that is infinitely divisible, and a group of friends with all the time
in the world to go on eating half of whatever was left of the cake.
At the end of time would all the cake be eaten? That is, does the sum
1/2 + 1/4 + 1/8 + . . . equal 1? It is not difficult to see that the answer
is yes, but it is easier to deduce this by subtraction than by addition. In
other words, it is easier to see that what remains tends towards zero. In
sums like this, calculators are not really much help.
Try out another problem that will make you aware of the limita-
tions of the calculator. See if you can find out where this infinite sum
converges: 1/2 + 1/3 + 1/4 + . . . A reasonably accurate programmable
calculator will make about one hundred million calculations and end
up with a value in the vicinity of 18. No matter how good the numeri-
cal calculator is, the answer will always end up around a relatively small
finite value. Nevertheless, this series actually does not have a fixed total;
it increases infinitely. See if you can regroup these fractions and com-
pare them with 1/2 + 1/2 + 1/2 + . . . Obviously this sum is infinite. It
will not stop at 18.

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_50, 207


C Springer-Verlag Berlin Heidelberg 2010
208 FIGURING IT OUT

Provided you add a sufficient number of fractions, you will end up

with a number greater than a million, or a billion, or a trillion tril-
lions . . . whatever you like. However, after a certain time a calculator
stops at a certain number, depending on the calculator, and this number
will not increase no matter how many more fractions you add. To fully
appreciate the problem, imagine that the calculator operates with a five-
digit internal precision. In that case, if the calculator added 10–0.0001,
it would not come up with 10.0001, but would continue to record the
answer as 10 or 10.000, as it could not distinguish between 10.0001 and
10.000. In fact, calculators do have greater precision, but this problem
always remains. It is this limitation that prevents the total from prop-
erly increasing when very small fractions are added; the machine just
does not know any other way to do it.
Why don’t you try another example? Enter 10 on your calculator.
Press the square root key and then the square key. The answer will be
10, as you expected. But now enter 10, press the square root key 25 times
and then press the square key 25 times. You will see that the answer
is not 10, but something like 9.99239. . . depending on the precision of
your calculator. The error is minute, but if you press the keys 33 times
instead of 25 times, the answer you get should be 5.5732. . . which dif-
fers markedly from the original 10. It turns out that when you multiply
rounded-off numbers over and over again, the result can be a disaster.
 INFINITY

Galileo, whose scientific activities were celebrated during the Inter-

national Year of Astronomy, considered various paradoxes having to do
with infinity. One of the simplest and most illustrative paradoxes con-
cerns two sets, one of the natural numbers (1, 2, 3, . . .), and one of their
doubles (2, 4, 6, . . .). We can establish a one-to-one (bijection) corre-
spondence between the two sets: 1 corresponds to 2, 2 corresponds to
4, 3 corresponds to 6, and so on. The first set seems to contain twice as
many elements as the second set, because it contains both odd and even
numbers. But doesn’t the fact that we can establish a one-to-one corre-
spondence between each number and its double indicate that each set
has the same number of elements?
Seemingly, this was how human beings learned to count. Thousands
of years ago, before writing had been invented, people counted sheep, or
whatever, by collecting as many pebbles, or making as many notches, as
there were sheep. The one-to-one correspondence between sheep and
pebbles ensured that the two sets had the same number of elements.
It then seems clear that the sets {1, 2, 3, . . .} and {2, 4, 6, . . .} both
have the same number of elements. At the same time, it seems clear that
the first has twice the number of elements. . .
From this paradox, Galileo (1638) concluded that “the attributes
of greater, lesser, and equal do not suit infinities, of which it can-
not be said that one is greater, or less than, or equal to another”.1

1
G. Galilei, Two New Sciences, translated and edited by Stillman Drake. University of Wisconsin
Press, Madison, WI, 1974, p. 40.

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_51, 209


C Springer-Verlag Berlin Heidelberg 2010
210 FIGURING IT OUT

Two hundred and fifty years later, the German mathematician Julius
Dedekind (1831–1916) took this idea as his starting point to define
mathematical infinity. According to Dedekind, a set is infinite if it is
possible to establish a one-to-one correspondence between it and one of
its subsets. This is so in the case of the correspondence between the set
of natural numbers and the subset of even numbers. Even numbers are
also natural numbers, but there are natural numbers that are not even
numbers, as is well known.
One of the most delightful paradoxes concerning infinity is
“Hilbert’s Hotel”, usually attributed to the German mathematician
David Hilbert, though it is more likely a product of physicist George
Gamow’s imagination. Gamow was the first person to chronicle
“Hilbert’s Hotel” in writing. It goes like this: imagine that a hotel with
an infinite number of rooms is fully booked. In this imaginary hotel, it
is always possible to make room for an additional guest. This is accom-
plished by the receptionist simply asking the guest in room 1 to move
to room 2, the guest in room 2 to move to room 3, and so on. The guest
who has just arrived is given room 1 and nobody is left out. But if there is
always room for one more guest, then there is also always room for two
more guests. And if there is always room for two more, there is always
room for three more. The clients only have to move to a room with the
corresponding number. Having an infinite number of guests arrive at
the same time makes it more difficult, but if you think about it I am sure
you will find a solution.
Hilbert’s hotels would be ideal for a hotel chain. Not so for the
guests. It cannot be pleasant to spend all night moving from one room
to another.
 UNFAIR GAMES

Imagine that we are in a casino that is promoting the following game:

We put 100 dollars on the table, and win or lose by tossing a coin. If it
is heads, we win 40 dollars, and if it is tails we lose 30 dollars. Should we
join the game?
If the coin being used is perfectly balanced and tossed correctly,
heads and tails have the same probability of coming up. This means we
have a 1 in 2 chance of winning 40 dollars and a 1 in 2 chance of los-
ing 30 dollars if we join the game. The expected value of this game is
40/2 – 30/2, or 5 dollars. This means that if we put 100 dollars on the
table many times and play the game for a long time, we will win approx-
imately 5 dollars each time we toss the coin. After tossing the coin one
thousand times we should have won about 5000 dollars. This means it
would pay us to go to such a casino. It would be like having our own
printing press for dollar bills.
But the croupier knows how annoying it is to have to keep on
putting 100 dollar bills on the table, and decides to simplify the game.
Now instead of winning 40% or losing 30% of 100 dollars each time we
toss the coin, we have to place 100 dollars on the table and then we can
play the game, but this time we win 40% or lose 30% of whatever sum is
on the table. For example, if the results of three tosses are heads, again
heads, and then tails, our 100 dollars at first increase by 40% of 100–140,
then by another 40%, now of 140, making the total 196. Following the
third toss, that 196 is reduced by 30%, leaving 137 dollars and 20 cents
on the table.

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_52, 211


C Springer-Verlag Berlin Heidelberg 2010
212 FIGURING IT OUT

It does seem as if the croupier is making things simpler for us. We

start the printing press rolling and enjoy our good fortune by going for
a stroll. We take the opportunity to wine and dine well. Why not? Our
printing press is rolling!
Two hours later we pass by the table to collect our winnings. During
this time, the coin has been tossed one hundred times. How much
money will we have won? A few thousand dollars, we anticipate.
So we are astounded when the croupier only hands us 36 dol-
lars. And people who stayed at the table assure us that, oddly enough,
heads and tails each came up exactly 50 times. The chances were exactly
balanced. So how could we have lost money?
Very simply! As the game was played sequentially, the result is the
product of 100 dollars multiplied 50 times by 140% and another 50 times
by 70%. Calculate the result – it is 36 dollars. The fact is that 140% times
70% is 98%, which means that we were losing 2% of the money on the
table for each heads and tails sequence.
This surprising result, which you can only obtain once you have
done your sums carefully, is a curiosity arising from a field known as
recreational mathematics, which is particularly useful for its applica-
tions in other areas. The lesson? It is one thing to add up your expected
winnings, it is quite another matter to multiply them.
Take another example regarding the calculation of bank fees. If a
bank makes you a loan, charging fees of 4% per quarter but paying 8%
interest per annum on your long-term deposits, who ends up winning?
 MONSIEUR BERTRAND

We expect to receive two Olympic medals, and we know that neither of

them is bronze. There are three boxes in front of us, each containing two
medals. One contains two gold medals (GG), another two silver medals
(SS), and the third one gold and one silver medal (GS). The boxes are
indistinguishable from one another, each with two drawers containing
one medal. This is all the information we know. We select a box at ran-
dom, open one of the drawers and find a silver medal inside. What is
the probability that there will be a gold medal in the other drawer of this
box? That seems easy. We have eliminated the possibility of the box con-
taining two gold medals (GG), so there are two hypotheses: we selected
the box with two silver medals (SS) or the box with one gold and one
silver medal (GS). That seems to be it – the probability of finding gold
in the other drawer of the box is 1 in 2.
Or is it? Let’s look at the problem from a different perspective. The
way the problem is set, the probability of choosing any drawer would be
the same. This means that we could have selected a drawer in the box
with two silver medals (SS), the other drawer in the same box (SS), or
the drawer with the silver medal in the mixed box (GS). So we have three
hypotheses. We find gold in the other drawer in only one of those three
hypotheses. Which means that the probability of finding gold after silver
is only 1 in 3.
If this has confused you, don’t despair. Just think a little and check
that the second version is right. This puzzle was invented 110 years ago
by the French mathematician Joseph Bertrand and many people still find

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_53, 213


C Springer-Verlag Berlin Heidelberg 2010
214 FIGURING IT OUT

it confusing. It has nothing to do with the Olympics, where little is left

to chance.

Joseph Louis François Bertrand (March 11, 1822 – April 5, 1900)

 BOY OR GIRL?

Mary and John have two children. Their first-born is a boy named Jack.
What is the probability that the couple have two children of different
genders? This seems at first to be a ridiculously simple question. If we
concede that it is just as probable for a boy or a girl to be born, and if
we also concede that this has nothing to do with the gender of the first-
born baby, then there is no doubt that the probability that the second
child will be a girl (therefore not the same gender as the first-born) is 1
in 2. And that is the answer: 1 in 2.
Simple, right? But now let us consider another couple, Josephine
and Joseph, who also have two and only two children. We know that
one of them is a boy, but we don’t know if he is their oldest or youngest
child. The question is: what is the probability that this couple has a boy
and a girl? Lulled by our previous findings, we predict that their chance
is also 1 in 2. And that’s that. Except that, surprisingly, this answer is
wrong. The right answer is 2 in 3.
In actual fact, we know that Josephine and Joseph could have two
boys (BB), two girls (GG), a boy followed by a girl (BG) or a girl fol-
lowed by a boy (GB). The only thing we know for sure is that one of
their two children is a boy, so the two girls (GG) hypothesis is ruled out.
Of the three remaining hypotheses (all of which are equally probable)
two (BG and GB) involve children of different genders. Therefore the
probability is 2 in 3. Hard to believe, isn’t it? That is why the paradox is
so entertaining. We have to think a little to understand it.

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_54, 215


C Springer-Verlag Berlin Heidelberg 2010
 A PUZZLE FOR CHRISTMAS

Everyone knows that Santa Claus likes to please people. But he doesn’t
like to waste presents. He put money in my stocking. But we came to
an agreement, he and I. Or rather, he explained the rules of the game
to me.
He appeared at my house after I had spent an evening out with
friends. He said: “My dear Nuno, you have worked hard this year, and
I want to reward you. But as mathematicians think they know every-
thing, I am going to give you a lesson in humility, with a lot of cash on
the side.”
I rubbed my eyes, dumbfounded (maybe I had drunk more wine at
dinner than I’d realized). I looked up again, but the old man with the
white beard was still there.
“Hang two stockings beside the chimney” he instructed. “I’ll put a
thousand dollars in the stocking on the left, but I’m not sure about the
other stocking; I’ll leave either a million dollars or nothing in it. You’ll
have two choices: you can either choose the stocking on the right and
throw away the stocking on the left, or you can choose both stockings.
If you choose the stocking on the right, you will find a million dollars in
it. If you choose both stockings, there will be nothing in the one on the
right, so you’ll just have a thousand dollars.”
I was confused: “Santa Claus! I guess I understand the rules, but
there’s one thing I don’t get. You’re going to put the money in the stock-
ings before I decide which to choose. How will you know how much
money to put in the stocking on the right if you don’t know what I’m
going to do?”

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_55, 217


C Springer-Verlag Berlin Heidelberg 2010
218 FIGURING IT OUT

“Well, that’s where you’re wrong. I have a tremendous capacity for

telling the future. I am almost absolutely certain that I know what you’ll
choose. I’ve already played this game with people who are much cleverer
than you, and I’ve always guessed right. I have even played this game
with Archimedes, Al-Khwarizmi, Pedro Nunes, and other mathematical
geniuses, except it wasn’t in dollars because they used other currencies
in those days.”
I thought and thought about it before I fell asleep. The old guy with
the white beard didn’t turn up again. On Christmas Eve I hung my two
stockings beside the chimney. I went to bed happy, thinking I had found
the answer. It was so simple. I just had to choose the second stocking,
the one on the right, with the million dollars in it. How lucky I was! The
old guy would play along!
The following morning the two stockings bulged. Santa Claus had
kept his word. Had he doubted that I was going to choose the stocking
on the right and throw away the one on the left?
Just at that moment, however, something stopped me. I’ve always
been a bit of a cheapskate and now I couldn’t bring myself to throw
away the stocking on the left that was surely stuffed with 1000 dollars.
What if I were to just take both stockings?
I was absorbed in this internal debate when I heard Santa Claus’s
voice in my head telling me “I’ve already played this game with peo-
ple who are much cleverer than you”... And then I reminded myself, he
could predict the future. If I chose both the stockings, he would have
foreseen that and would not have left a million dollars in the stocking
on the right. It was better not to be greedy and to just throw away the
stocking on the left.
But then a thunderbolt hit me! The stockings were there in front
of me with the money inside. The one on the left definitely contained a
thousand dollars. The one on the right either had a million or zilch. It
was best to grab them both.
I couldn’t make up my mind. If the most logical choice was to take
both, Santa Claus would have known that this is what I would do and he
would have put nothing in the stocking on the right. But how could the
 A PUZZLE FOR CHRISTMAS 219

logical choice leave me with only a thousand dollars when a million was
a real possibility?
Christmas Day passed, the New Year came, and I still don’t know
what to do about the stockings. The old guy’s thunderbolt really got me
mixed up!
∗∗∗
The Santa Claus in this story is actually a physicist named William
Newcomb. His paradox involving a highly skilled Predictor was pub-
lished in 1969 by the U.S. philosopher Robert Nozick (1938–2002), and
immediately gave rise to a heated debate.1 Logicians, mathematicians,
economists and theologians have discussed possible solutions for this
paradox, without coming to any consensus.
The problem seems to be related to the famous prisoner’s dilemma,
to the theory of free will when confronted by an omniscient being, as
well as to the irreversibility of time, and other crucial questions in the
fields of logic and philosophy. A paradox’s possible ramifications, and
the difficulties in solving them is what make them so fascinating. Don’t
worry if they stumped you too. Have a good one!

1
R. Nozick, Newcomb’s Problem and Two principles of Choice, in N. Rescher and S. Library,
Editors, Essays in Honor of Carl G. Hempel, D. Reidel, Dordrecht, 1969, p. 115.
 CRISIS TIME FOR EASTER EGGS

Every year the Easter Bunny has an infinite number of eggs available
for distribution. Nobody knows where he gets them from, or how he
manages to get them to all the children on Easter morning.
But this year the bunny was anxious, as he had heard talk of the
economic crisis and thought this might affect the supply of eggs. If there
wouldn’t be enough eggs to go round, how could he fulfill his mission
this year and the next? He decided the best thing would be to start
saving eggs.
He made his preparations. He took his magic bag with room for an
infinite number of eggs, and made up an identical bag. He cut a hole
in the bottom of the first bag and told his assistant Tricksy to get into
the bag and stay there to help. As the Easter Bunny and his assistant are
infinitely quick and infinitely efficient, they can easily handle an infinite
number of eggs in a short time.
Now, and this is a fact known only to a very few people, when the
Easter Bunny receives Easter eggs they are already numbered: 1, 2, 3, . . .
The Easter Bunny decided to place the eggs in groups of ten into the first
bag: first the eggs numbered from 1 to 10, then those numbered from
11 to 20, then those numbered from 21 to 30, and so on. His assistant,
Tricksy, hiding in the bottom of the bag, was tasked with taking the first
egg from each group of ten and placing it into the second bag. So when
the group of eggs numbered from 1 to 10 were put in the bag, Tricksy
removed egg number 1 and placed it in the second bag. When the group
of eggs numbered from 11 to 20 arrived, Tricksy grabbed number 11
and put it in the second bag. And the two of them did this again and
again . . .

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3_56, 221


C Springer-Verlag Berlin Heidelberg 2010
222 FIGURING IT OUT

The Easter Bunny knew that by using his method they would end
up with an infinite number of eggs in the first bag to be used this year
at Easter, and with an infinite number of eggs also in the second bag,
to be kept for the following year. There would be one egg in the second
bag for every nine eggs in the first bag. But after repeating this process
an infinite number of times there would be an infinite number of eggs
in each bag. Infinity is like that, full of surprises. Although you might
think that the second bag contained nine times fewer eggs than the first,
actually they both contained an infinite number of eggs. So any prob-
lems with next year’s egg supply that might be caused by the crisis had
been solved.
But the Easter Bunny hadn’t reckoned with the tricks perpetrated
by his assistant Tricksy, who’d switched the numbers by sticking the
number 11 label on egg number 2, the 21 label on egg number 3, and so
on. This way, Tricksy grabbed egg number 1 when the Easter Bunny put
the group numbered 1–10 in the bag, took out egg number 2 when the
second group was placed in the bag, removed egg number 3 when the
third group was put in, and so on.
But Easter Bunny, who is used to Tricksy’s foolishness, didn’t really
mind. He kept on putting eggs into the first bag in groups of ten, while
Tricksy continued to take one egg from each group and put it into the
second bag, until their task was completed. However, that was when the
Easter Bunny realized that there was nothing left in the first bag, all the
eggs had ended up in the second bag. How could that be possible, when
only one egg was placed in the second bag for every nine put into the
first bag?
But it is possible! Think of any egg whatsoever; let’s say number 27
or even number 10145? Well, these eggs disappeared from the first bag
when the Easter Bunny put the corresponding groups of ten into the
bag. If you think it through, this means that every single egg eventually
ends up in the second bag.
Are you confused? So was the Easter Bunny! Although he thought
there were nine times more eggs in the first bag than in the second, he
realized that all the eggs were in the second bag. So this year’s eggs will
be from the stash intended for next year, and that is the crisis.
 INDEX

Note: The letters ‘b’, ‘f ’ and ‘n’ followed by locators refers to boxes, figures and note numbers cited
in the text

φ (phi), 109, 112–113 Binary logic, v

π (pi), 76, 112n1, 173 Borda count, 203
Borda’s paradox, 201–202, 202b
Braque, Georges, 128
A Brassard, Gilles, 94
A4, 113, 115–118 Brown, Dan, 109–110
Academy of Science in Paris, 97
Adleman, Leonard, 90, 187–189
Alberti, Leon Battista, 101 C
Algorithm, 3–5, 7–9, 18, 54–55, 92, 93, 98, 138, Caeser, Julius, 59
140, 161–163, 165, 170, 184, 192 Calendar, 57–60
Angular distance, 77 CAPTCHA, 185–186
Annals of Mathematics, 13 Carnegie Mellon University, 185
Antikythera, 53f, 55–56 Chance (article), 24
Appel, Kenneth, 12 Chance, laws of, 161–165, 167–170, 173, 176,
Archimedes, 194, 218 193, 211–212, 214, 215
Argonne laboratory, 12 Clavius, Christopher, 59, 63–64
Arrow, Kenneth J., 204 Code, 40, 71, 87–88, 90–92, 102–104, 103f, 107,
Artificial intelligence, 183–185 110, 156, 188
ASCII code, 90 Commutator, 39
Asteroids, 75–78 Computational geometry, 138
Astronomical cycles, 57 Computational sciences, 170, 184
Astronomy, 59, 79–80, 124, 209 Computer, 11–13, 187–189, 194
Asymmetric encrypting key, 90 Computer program, 4–6, 18n2, 32, 162,
183, 185
Condorcet’s paradox, 203–204b
B Conformal map, 69
Banach, Stefan, 8 Conjecture
Beckmann, Johann, 118 of Euler, 33, 91, 148–149
Bell, Alexander Graham, 40 of Kepler, 12, 144
Bell laboratories, 98–99 of Robbins, 12
Benford, Frank, 173–174, 176–177 Cooley, James, 98
Benford’s law, 175f, 176b, 176–177 Cornell University, 22
Bennett, Charles, 94 Crick, Francis, 187
Bertrand, Joseph, 213, 214f Cryptography, 86b, 88, 90, 93–96, 104
Bijection, 209 Cubism, v, 127–129
Binary language, 94 Cupcake paradox, 207–208

N. Crato, Figuring It Out, DOI 10.1007/978-3-642-04833-3, 223


C Springer-Verlag Berlin Heidelberg 2010
224 INDEX

D Formalism, 157–158
Daubeschies, Ingrid, 99 Four colors theorem, 12
Da Vinci Code (Dan Brown), 110 Fourier analysis, 97–98
Da Vinci, Leonardo, 107, 110 Fourier, Jean Baptiste Joseph, 97
Davis, Philip, 157 Fourier transform, 81, 98n1
de Borda, Jean-Charles, 201 Frege, Gottlob, 44–45, 121
de Buffon, Count, 168, 194
de Condorcet, Marquis, 203
G
Dedekind, Julius, 210
Galileo, 80, 209
de Mesquita, Samuel Jesserun, 119–120
Gamow, George, 210
Democracy, 17, 201, 204
Gardner, Howard, 20
Department of Defense, 50
Gauss, Carl Friedrich, 75, 77, 124
de Solla Price, Derek J., 55
Gauss curve, 20, 21f, 181
de Sousa, Martim Afonso, 66
Gauss distribution, 75–76, 78
Diffie, Whitefield, 86–88, 86b
General g factor, 19–20
Dirac, Paul, 148
Geometrical mean, 117
Discoveries (Age of Discovery), 69
Geometry, 69, 71–73, 107, 115–118, 124–125,
DNA, 36, 156, 187–189
129, 138, 142–144, 153–154, 194
Gleave, John, 53f, 56
E
Gödel, Kurt, 121, 158–159, 183
École Polytechnique, 99
Golden number, 109f, 110, 111–114, 111f
Ecological fallacy, 16
Golden rectangle, 111f, 113–114
Ecological inference, 15–17, 18n2
Golden triangle, 108–109, 109f
e-commerce, 89, 92, 93–94
Goltra, Inez, 15, 15n1
Einstein, Albert, 81, 127–130, 147–148,
Gosnell prize, 18
153–154, 158
Gould, Wayne, 32
Elections, 201
GPS (global positioning system), 49–52, 72
Electromagnetic waves, 51
Great circle, 65
Elements (Euclid), 112, 142
Great circle arc, 66–67
Encrypted communication, 90
Gregory XIII, Pope, 57, 59
Enigma machine, the, 101–104
Group theory, v
Epidemiology, 16
Erdös, Paul, 158
Ernst, Bruno, 122, 123n1 H
Escher, Mauritus Cornelius, 67, 68f, 119–122, Haken, Wolfgang, 12
123–126 Hales, Thomas C, 12–13
Eubulides, 43–45 Halton, John, 27
Euclid, 107, 112, 142 Harriot, Thomas, 11–12
Euler, Leonard, 33, 91, 148–149 Harris, Robert, 103
Euler’s theorem, 91 Harvard University, 4, 17
Hellman, Martin, 86–88, 86b
F Henderson, Linda, D., 127
FBI, 97–99 Hersh, Rueben, 127
Feller, William, 174, 174n1 Hilbert, David, v, 121, 158, 210
Fibonacci, 112, 155 Hilbert’s Hotel paradox, 210
Fibonacci sequence, 11 Hitler, Adolf, vi, 16, 104
Fink, Thomas, 29 Hubble Space Telescope Science
Fisher, Ronald, 33–34 Institute, 112
 INDEX 225

I Meyer, Yves, 98–99

IBM, 4–5, 94 Miller, Arthur I., 127
Indifference curves, 45 Möbius, August Ferdinand, 124
Intelligence quotient (IQ), 19–22, 21f Möbius strip, 123–126, 125f
Internet, 18n2, 40, 89–91, 103, 185 Modular arithmetic, 86b
MoMA (Picasso’s painting), 128
J Monte Carlo method, 163–165
Juno (celestial body), 78 Morse code, 40
Moving knife method, 8, 9f
K
Kahn, David, 103 N
Karp, Richard, 4–5, 5n1 Napier, John, 174
Kepler, Joanhannes, 143 National Science Foundation (NSF), 18
Kepler’s conjecture, 12, 144 Nature (article), 24
King, Gary, 17–18, 17f, 18n2 Navigation, 49, 51, 61, 67, 69, 72
Knaster, Bronislaw, 8 Neisser, Ulric, 22
Knuth, Donald, 170, 170n1 Newcomb, Simon, 184
Newcomb, William, 219, 219n1
L Nigrini, Mark, 177
Lakatos, Imre, 159 Nobel prize, 75, 209
Las Vegas algorithms, 163 Northern Ireland (Ulster) University, 22
Latin square, 31f, 34 Nozick, Robert, 219, 219n1
Leclerc, Georges Louis, 168 NP-Complete, 5, 34
Leibniz, Gottfried, 194 Number theory, 88, 91, 138
Les demoiselles d’Avignon (painting), 128 Nunes, Pedro, 61–63, 62f, 65–69, 218
Lichtenberg, Georg Christoph, 118
Lilius, Aloysius, 59 O
Livio, Mario, 112 Ogburn, William, 15, 15n1
Logic, v, 12, 32, 45, 64, 121, 155, 160, 203, 219 Olbers, Heinrich, 77
Loxodrome, 66f, 67, 69 Operations research, v
Loxodromic spiral, 67 Optical signals, 41
Lynn, Richard, 22 Orthodrome, 66f, 67
Orthogonal squares, 33
M
Magic squares, 33 P
Mandelbrot, Benoît, 179–181 Pacioli, Fra Luca, 112
Mao, Yong, 29 Palermo’s observatory, 77
Marketing, 16, 45 Pallas (celestial body), 78
Markov, George, 113 Paradox, 23, 44–45, 120–121, 181, 201–204,
Martin, Steve, 128 205–206, 207–208, 209–210, 215, 219
Massachusetts Institute of Technology, 90 See also specific paradoxes
Mathematical model, 24 Park, Bletchley, 103–104
McCune, William, 12 Peer reviewed, 13
Melon paradox, 205–206 Pentacle, 107
Mercator, Gerardus, 67 Pentagram, 107–108, 109f
Mercator’s map, 69 Physics, 75, 93–94, 96, 118, 127–128, 147–149
Mercator’s projection, 66f, 67–69 Piazzi, Giuseppi, 77, 144
Merkle, Ralph, 86–88, 86b Picasso, Pablo, 127–129
226 INDEX

Planet, 56, 67, 76–78, 137, 143–144 Scherbius, Arthur, 101–102

Platonism, 157 Shamir, Adi, 90
Pliny, the Younger, 201 Singh, Simon, 86b, 88, 103
Poincaré, Henri, 128 Sinusoidals, 98
Polanyi, Michael, 183 Social sciences, 16
Pole star, 79 Solar system, 77–78, 137–138, 143–144
Polster, Burkard, 27–29 Sorites paradox, 44
Pólya, George, 159 Sosigenes, 59
Popper, Karl, 159, 183–184 Southern Cross, the (satellite), 49
Prime numbers, 55, 91 Spearman, Charles, 19
Princet, Maurice, 129 Stanford University, 24
Princeton University, 127, 158–159, 174 Statistics, 15–16, 33, 76, 97, 102, 112, 169, 174
Probability theory, 173–174n1 Steinhaus, Hugo, 9–10
Proof (mathematical), 11–13, 142, 158, Strang, Gilbert, 98
171–172 Sudoku, 31–34
Proofs and Refutations (Lakatos), 159 Sun, 49, 55, 61, 76–77, 79, 143, 205
P-type algorithms and problems, 5–6 Switch, 39–41
Public keys, 86b Szyfrów, Biuro, 102
Pyramidal pile, 11
T
Q Telegraph, 40
Quantum computers, 93 Telephone, v, 39–40
Quantum cryptography, 93–96 Theoretical economics, 45
Quantum uncertainty, 95 Théorie Analytique de la Chaleur,
La (Fourrier), 97
QuickSort (algorithm), 162, 165f
Theory of relativity, 128–129
Thompson, D’Arcy, 154
R Tibshirani, Robert J., 24–25
Raleigh, Sir Walter, 11 Times, 32
Raven’s progressive matrices, 21–22 Titius-Bode law, 76
Redelmeier, Donald A., 24–25 Trigonometry, 73
Rejewski, Marian, 102–104 Tukey, John, 98
Rhumb line, 66, 67–69 Turing, Alan, 4–5, 103–104, 184
Rivest, Ronald, 90 Turing’s test, 183–186
Robbin’s conjecture, 12
Robinson, Williams, 16 U
Royal French Academy, 201 Ulam, Stanislaw, 163–165
RSA, 90–92, 93 University of California, 204
Russell, Bertrand, 44, 120–121 University of Illinois, 12
Rybczynski, Witold, 39 University of Maine, 113
University of Michigan, 12
S University of Toronto, 24
Saari, Donald, 204
Salesman problem, the, 27, 139, 187 V
Satellites (artificial), 49–52, 55, 144 Vitruvian man, 107–110
SAT (satisfiability problems), 4 Vitruvius (Pollio Marcus Lucius Vitruvius),
SAT (Scholastic Aptitude Test), 20–21 110
 INDEX 227

Volatility and volatility persistence, 181 Wigner, Eugene, 75

von Neumann, John, vi, 163, 165 Wos, Larry, 12
von Zach, Franz, 77
Vote, 15, 17, 201–203, 202f Y
Yale University, 55
X
Watson, James D., 187 Z
Weizmann Institute, 189 Zodiac, 76