What is Mathematics?

 Mathematics (from Greek máthēma, “knowledge, study, learning”)

- is the study of topics such as quantity (numbers), structure, space, and change.

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There is a range of views among mathematicians and philosophers as to the exact scope and
definition of mathematics.

 Aristotle defined mathematics as: The science of quantity.

In Aristotle's classification of the sciences, discrete quantities were studied by arithmetic,

continuous quantities by geometry.

 Auguste Comte’s definition tried to explain the role of mathematics in coordinating phenomena
in all other fields.
The science of indirect measurement.

 Galileo Galilei (1564–1642) said,

"The universe cannot be read until we have learned the language and become familiar with the
characters in which it is written. It is written in mathematical language, and the letters are triangles,
circles and other geometrical figures, without which means it is humanly impossible to comprehend a
single word. Without these, one is wandering about in a dark labyrinth.”

 Carl Friedrich Gauss (1777–1855) referred to mathematics as "the Queen of the Sciences“.
 Albert Einstein (1879–1955) stated that "as far as the laws of mathematics refer to reality, they
are not certain; and as far as they are certain, they do not refer to reality.“

 Claire Voisin , a French mathematician states "There is creative drive in mathematics, it's all
about movement trying to express itself."

 Benjamin Peirce ,1870

Mathematics is the science that draws necessary conclusions.

 Bertrand Russell ,1903

All Mathematics is Symbolic Logic.

 Intuitionism

emphasizes the construction of ideas in the mind.

 Mathematics is mental activity which consists in carrying out, one after the other, those mental
constructions which are inductive and effective.
 Formalism denies both physical and mental meaning to mathematics, making the symbols and
rules themselves the object of study.
 A formalist definition:

Mathematics is the manipulation of the meaningless symbols of a first-order language

according to explicit, syntactical rules.

 Walter Warwick Sawyer, 1955

Mathematics is the classification and study of all possible patterns.

 Bertrand Russell,1901

Bertrand Russell wrote this famous tongue-in-cheek definition, describing the way all terms in
mathematics are ultimately defined by reference to undefined terms:

The subject in which we never know what we are talking about, nor whether what we are
saying is true.

 Charles Darwin

A mathematician is a blind man in a dark room looking for a black cat which isn't there.

 G.H. Hardy, 1940

A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more
permanent than theirs, it is because they are made with ideas.

 Henri Poincaré

Mathematics is the art of giving the same name to different things.

 Paul Halmos , 1993

viewed mathematics as a creative art because mathematicians create beautiful concepts and
mathematicians think like artists.

In Halmos’eyes, mathematics is much more like a discipline with an art form.

Nature of Mathematics

 For some, the essence of mathematics lies in its beauty and its intellectual challenge.
  For others, including many scientists and engineers, the chief value of mathematics is how it
applies to their own work.

1. Mathematics is the study of patterns and relationships.

Mathematicians explore recurring ideas and of relationships between and among ideas.

2. Mathematics is a language.

Mathematics is a language using carefully defined terms and symbols. These terms and
symbols enhance our ability to communicate about science, about real –life situations and about
mathematics itself.

3. Mathematics is a mental discipline

Mathematics is a broad and deep discipline that is continuing to grow in breadth and depth.

Nowadays, a Ph.D. research dissertation in mathematics is typically narrowly focused on

definitions, theorems, and proofs related to a single problem in a narrow subfield in mathematics.

4. Mathematics is a way of thinking.

It provides us with strategies for organizing, analyzing, and synthesizing data, largely but not
exclusively numerical.

5. Mathematics is an art.

Mathematics is characterized by order and internal consistency.

6. Mathematics is a tool.

Mathematics is a tool used by mathematicians, economists, and by everyone in the course of

daily life.

 As an interdisciplinary language and tool

Like reading and writing, math is an important component of learning and "doing" (using one's
knowledge) in each academic discipline.

Mathematics is such a useful language and tool that it is considered one of the "basics" in our
formal educational system.

 As a human endeavor

Consider the math of measurement of time such as years, seasons, months, weeks, days, and so
on or the measurement of distance, and the different systems of distance measurement that
developed throughout the world.
 Think about math in art, dance, and music.

 Mathematics is universal.

It finds useful applications in business, industry, music, historical scholarship, politics, sports,
medicine, agriculture, engineering, and the social and natural sciences.

Reasons why the relationship between mathematics and the other fields of basic and applied science
is strong

 Science provides mathematics with interesting problems to investigate, and mathematics

provides science with powerful tools to use in analyzing data.
 Science and mathematics are both trying to discover general patterns and relationships, and
in this sense they are part of the same endeavor.
 Mathematics is the chief language of science.
 The symbolic language of mathematics has turned out to be extremely valuable for expressing
scientific ideas unambiguously.
 Mathematics provides the grammar of science—the rules for analyzing scientific ideas and
data rigorously.
 Mathematics and science have many features in common.

• belief in understandable order;

• an interplay of imagination and rigorous logic;

• ideals of honesty and openness;

• the critical importance of peer criticism;

• the value placed on being the first to make a key discovery;

• being international in scope; and even, with

• the development of powerful electronic computers, being able to use

technology to open up new fields of investigation.

Mathematics in Art, Architecture, and Nature

 Rectangle Pageant

 The Divine Proportion

 The Divine Proportion, better known as the Golden Ratio, is usually denoted by the Greek
letter Phi , Φ.

 Φ is defined to be the ratio obtained by dividing a line segment into two unequal pieces such
that the entire segment is to the longer piece as the longer piece is to the shorter.
 The Golden Ratio

 The definition of Φ leads to the following equation, if the line is divided into segments of
lengths a and b:

 Setting Φ equal to the quotient a/b and manipulating this equation shows that Φ satisfies
the quadratic equation:

 Applying the quadratic formula to this simple equation and taking Φ to be the positive
solution yields:

 Properties of Φ

 Φ is irrational

 Its reciprocal, 1/ Φ, is one less than Φ

 Its square, Φ 2, is one more than Φ

 Φ Is an Infinite Square Root

 Φ is an Infinite Continued Fraction

 Constructing Φ

 Begin with a 2 by 2 square. Connect the midpoint of one side of the square to a corner.
Rotate this line segment until it provides an extension of the side of the square which was
bisected. The result is called a Golden Rectangle. The ratio of its width to its height is Φ.

 Constructing Φ

 Properties of a Golden Rectangle

 If one chops off the largest possible square from a Golden Rectangle, one gets a smaller
Golden Rectangle.

 If one constructs a square on the longer side of a Golden Rectangle, one gets a larger Golden
Rectangle.

 Both constructions can go on forever.

 The Golden Spiral

 In this infinite process of chopping off squares to get smaller and smaller Golden Rectangles,
if one were to connect alternate, non-adjacent vertices of the squares, one gets a Golden
Spiral.

 The Golden Spiral

 The Golden Spiral

 The Golden Triangle

 An isosceles triangle with two base angles of 72 degrees and an apex angle of 36 degrees is
called a Golden Triangle.

 The ratio of the legs to the base is Φ.

 The regular pentagon with its diagonals is simply filled with golden ratios and triangles.

 The Golden Triangle

 A Close Relative:
Ratio of Sides to Base is 1 to Φ

 Golden Spirals From Triangles

 As with the Golden Rectangle, Golden Triangles can be cut to produce an infinite, nested set of
Golden Triangles.

 One does this by repeatedly bisecting one of the base angles.

 Also, as in the case of the Golden Rectangle, a Golden Spiral results.

 Chopping Golden Triangles

 Spirals from Triangles

 Φ In Nature

 There are physical reasons that Φ and all things golden frequently appear in nature.

 Golden Spirals are common in many plants and a few animals, as well.

 Sunflowers

 Pinecones

 Pineapples

 The Chambered Nautilus

 Angel Fish

 Tiger

 Human Face I

 Human Face II
  Le Corbusier’s Man

 A Golden Solar System?

 Φ In Art & Architecture

 For centuries, people seem to have found Φ to have a natural, nearly universal, aesthetic
appeal.

 Indeed, it has had near religious significance to some.

 Occurrences of Φ abound in art and architecture throughout the ages.

 The Pyramids of Giza

 The Pyramids and Φ

 The Pyramids were laid out in a Golden Spiral

 The Parthenon

 Cathedral of Chartres

 Cathedral of Notre Dame

 Michelangelo’s David

 Michelangelo’s Holy Family

 Rafael’s The Crucifixion

 Da Vinci’s Mona Lisa

 Da Vinci’s Study of Facial Proportions

 Da Vinci’s St. Jerome

 Da Vinci’s The Annunciation

 Da Vinci’s Study of Human Proportions

 Rembrandt’s Self Portrait

 Seurat’s Bathers

 Dali’s The Sacrament of the Last Supper

 Literally an (Almost) Golden Rectangle

Patterns Another subject common to art and mathematics is patterns.

  These usually take the form of a tiling or tessellation of the plane.

 Many artists have been fascinated by tilings, perhaps none more than M.C. Escher.

 Patterns & Other Mathematical Objects

 In addition to tilings, other mathematical connections with art include fractals, infinity and
impossible objects.

 Real fractals are infinitely self-similar objects with a fractional dimension.

 Quasi-fractals approximate real ones.

 Fractals

 Some art is actually created by mathematics.

A fractal is a natural phenomena and a mathematical set. What they have in common is a repeating
pattern that displays at every scale.

 Fractals includes the idea of a detailed pattern that repeats itself.

 The Koch Snowflake (real fractal)

 The Mandelbrot Set (Quasi)

 Blow-up 1

 Blow-up 2

 Blow-up 3

 Blow-up 4

 Blow-up 5

 Blow-up 6

 Blow-up 7

 Another Quasi-Fractal

 Yet Another Quasi-Fractal

 And Another Quasi-Fractal

 Fractals Occur in Nature (the coastline)

 Frost Patterns
 Lichtenberg figures

 Fractal Broccoli

 Tessellations

 There are many ways to tile the plane.

 One can use identical tiles, each being a regular polygon: triangles, squares and hexagons.

 Regular tilings beget new ones by making identical substitutions on corresponding edges.

 Regular Tilings

 New Tiling From Old

 Maurits Cornelis Escher

(1898-1972)

 Escher is nearly every mathematician’s favorite artist.

 Although, he himself, knew very little formal mathematics, he seemed fascinated by many of
the same things which traditionally interest mathematicians: tilings, geometry,impossible
objects and infinity.

 M.C. Escher

 A visit to the Alhambra in Granada (Spain) in 1922 made a major impression on the young
Escher.

 He found the tilings fascinating.

 The Alhambra

 An Escher Tiling

 Escher’s Butterflies

 Escher’s Lizards

 Escher’s Sky & Water

 M.C. Escher

 Escher produced many, many different types of tilings.

 He was also fascinated by impossible objects, self reference and infinity.

 Escher’s Hands
 Escher’s Circle Limit

 Escher’s Waterfall

 Escher’s Ascending & Descending

 Escher’s Belvedere

 Escher’s Impossible Box

 Rhombus Tiling

 Rhombus Tiling II

 Rhombus Tiling III

 Luca Pacioli (1445-1514)

 Pacioli was a Franciscan monk and a mathematician.

 He published De Divina Proportione in which he called Φ the Divine Proportion.

 Pacioli: “Without mathematics, there is no art.”

 Although one might argue that Pacioli somewhat overstated his case when he said that
“without mathematics, there is no art,” it should, nevertheless, be quite clear that art and
mathematics are intimately intertwined.