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Chapter 1

Nature of Mathematics

LEARNING OUTCOME(S):
● Appreciate the patterns, beauty and practical uses of mathematics in everyday life
● Identify Patterns in Nature and regularities in the world.
● Appreciate the nature, beauty and uses of mathematics in everyday life.

TIME FRAME: ​3 hours

LESSON 1.1: Patterns and Numbers in Nature and the World

A hike in the woods or a walk along the road to our houses reveals an endless variety of forms.
Nature abounds in spectral colors and intricate shapes - the rainbow mosaic of a butterfly's
wing; the delicate curlicue of a vine tendril; the undulating ripples of the raindrops. But these
miraculous creations not only delight the imagination, they also challenge our understanding.
How do these patterns develop? What sorts of rules and guidelines, shape the patterns in the
world around us?

butterfly's wing vine tendril ripples of the raindrops

https://www.flickr.com https://www.tumblr.com https://www.masterfile.com
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Some patterns are molded with a strict regularity. Thousands of times over, the cells of a
honeycomb repeat their hexagonal symmetry. The honeybee is a skilled and tireless artisan
with an innate ability to measure the width and to gauge the thickness of the honeycomb it
builds. Although the workings of an insect's mind may baffle biologists, the regularity of the
honeycomb attests to the honey bee's remarkable architectural abilities.

The nautilus is another meticulous craftsman, who designs its shell in a shape called a
logarithmic or equiangular spiral (explained ahead). This precise curve develops naturally as the
shell increases in size but does not change its shape, ever growing but never changing its
proportions. The process of self-similar growth yields a logarithmic spiral. We find the same
spiral in the path traced by a moth drawn towards a light.

honeycomb nautilus shell

https://slate.com https://www.shutterstock.com

Patterns in nature are visible regularities of form found in the natural world. These patterns
recur in different contexts and can sometimes be modelled mathematically. Natural patterns
include symmetries, trees, spirals, meanders, waves, foams, tessellations, cracks and stripes.
Early Greek philosophers studied pattern, with Plato, Pythagoras and Empedocles attempting to
explain order in nature. The modern understanding of visible patterns developed gradually over
time.

As a practical matter, mathematics is a science of pattern and order. Its domain is not
molecules or cells, but numbers, chance, form, algorithms, and change. As a science of abstract
objects, mathematics relies on logic rather than on observation as its standard of truth, yet
employs observation, simulation, and even experimentation as means of discovering truth.

Let's take a look at some of the different types of patterns to help you appreciate them as well.

1. S​ ymmetry ​is when different sides of something are alike. These reflections may be mirror
images with only two sides like the two sides of our bodies, they may be symmetrical on several
sides like the inside of an apple sliced in half, or they might be symmetrical on all sides like the
different face of a cube.
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We understand symmetry quite well in living organisms because it is a function of their

environment. In order to balance, we need to have symmetrical body structure so we don't fall
over from imbalanced weight.

What we don't understand very well is symmetry in non-living things. Snowflakes have six-fold
symmetry but it is unclear why this occurs. Crystals like diamond are likewise constructed with
mathematical regularity. A chemist could readily explain how positively and negatively charged
sodium and chloride ions arrange themselves neatly in a crystal lattice, resulting in salt crystals
with a perfect cubic structure, and while beautiful it is still somewhat of a mystery.

face of Dr. Gellor Snowflake diamond

Alphaeus images https://www.quora.com https://www.bishopsjewelry.com

2. ​Fractals are the 'never-ending' patterns that repeat indefinitely as the pattern is iterated on
an infinitely smaller scale. Infinite iteration is not possible in nature so all 'fractal' patterns are
only approximate We see this type of pattern in trees, rivers, mountains, shells, clouds, leaves,
lightning, and more.

Ferns are a common example of a self-similar

set, meaning that their pattern can be
mathematically generated and reproduced at
any magnification or reduction. The
mathematical formula that describes ferns,
named after Michael Barnsley. In other words,
random numbers generated over and over
using Barnsley's Fern formula ultimately
produce a unique fern-shaped object
https://www.fibrex.co.uk

This variant form of cauliflower is the ultimate

fractal vegetable. Its pattern is a natural
representation of the Fibonacci or ​golden
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spiral​, a logarithmic spiral where every quarter

turn is farther from the origin by a factor of
phi, the golden ratio.

https://www.flickr.com
3. ​Spirals are another common pattern in nature that we see more often in living things. Think
of the horns of a sheep, the shell of a nautilus, and the placement of leaves around a stem. A
special type of spiral, the logarithmic spiral, is one that gets smaller as it goes. We see this
pattern in galaxies, hurricanes, and some seashells

The Galaxy hurricane Yolanda (Nov. 2013)

https://www.bbc.com http://edition.cnn.com

LESSON 1.2: THE FIBONACCI SEQUENCE

The ​Fibonacci Sequence​ is the series of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
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The next number is found by adding up the two numbers before it. The number 2 is found by
adding the two numbers before it (1+1). 3 is found by adding the two numbers before it (1+2),
and 5 is (2+3), and so on

When we make squares with those widths, we get a nice spiral:

Do you see how the squares fit neatly together?

For example 5 and 8 make 13, 8 and 13 make 21,
and so on

A nautilus is a cephalopod mollusk with a spiral

shell and numerous short tentacles around its
mouth.
A nautilus shell is grown in a Fibonacci spiral. The
spiral occurs as the shell grows outwards and tries
to maintain its proportional shape.

https://www.shutterstock.com

Other Amazing Examples of the Fibonacci Sequence in Nature

Sunflowers boast radial symmetry of the Fibonacci

sequence. This is not uncommon since many plants
produce leaves, petals and seeds in the Fibonacci sequence.
Sunflowers and other plants abide by mathematical rules
for efficiency. In simple terms, sunflowers can pack in the
maximum number of seeds if each seed is separated by an
irrational-numbered angle.
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Here are the three most natural ways to find spirals

in this pattern. Note that the black pattern is
identical in all the images on this page. Only the
colored lines indicating the selected spirals are
different. The 1​st set of lines show 34 spirals of
seeds.

Choosing another slope,

these set of lines And choosing a very shallow
slope, these
show 55 spirals of seeds. set of lines show 21 spirals of seeds
LESSON 1.3: THE GOLDEN RATIO

1. The ​Golden Ratio ​is a special number found by dividing a line into two parts so that the
longer part divided by the smaller part is also equal to the whole length divided by the
longer part. It is often symbolized using phi, after the 21st letter of the Greek alphabet. In
an equation form, it looks like this:

a (a+b)
b
= a
= 1.6180339887498948420 …

Each section of your index

finger, from the tip to the
base of the wrist, is larger
than the preceding one by
about the Fibonacci ratio
of 1.618, also fitting the
Fibonacci numbers 2, 3, 5
and 8.
By this scale, your fingernail is 1 unit in length. Curiously enough, you also have 2 hands, each
with 5 digits, and your 8 fingers are each comprised of 3 sections. All Fibonacci numbers!
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Here are other ways of expressing the same basic relationship in its connection to the golden
ratio and golden spiral

Engr. Lynn M. Remo

https://www.goldennumber.net
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WORKSHEET 1

NAME: SCORE:

SECTION CODE OR CLASS SCHEDULE: DATE:

● Pattern recognition in local community

● Take at least 5 pictures (with ​selfie​) of any example of patterns (Symmetry, Spiral, Fractal,
Fibonacci Sequence, etc.) that you can find from your local community and paste it in the
space provided below.
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WORKSHEET 2

NAME: SCORE:

SECTION CODE OR CLASS SCHEDULE: DATE:

Based on your reading or videos you have seen (Science Documentary 2016 - The Math Mystery
Mathematics in Nature and Universe – YouTube and Nature by Numbers) write an essay for
each of the following questions. ​Make sure to also discuss the connection of the topic to your
essay writing.
1. What is mathematics all about?
- In order to understand the universe you must know the language in which it is
written and that language is Mathematics. Math is all around us, in everything
we do. Math is used in everyday life. Math helps us discover new stuff, learn
from new things and to know that everything around us is organized orderly
because it was made by Math. Mathematics may not teach us how to add
happiness or how to minus sadness , However, it teaches us one important thing.
It is about that every problem always has a solution. God made us with a sense
of curiosity so due to that, we can learn and do the things we love. Math has
things that connect everything together. Such as Music, cooking and so on. Math
is all about Everything in the universe.

2. What is the importance of mathematics in your life?

- Life is the only question to which answers are different, One's life is the answer
that one has evaluated to this question called life. Math is all around the
universe. Mathematics taught me that every problem has a solution and
mistakes is the proof that you’re trying. In other words, Math taught us that
there’s an answer for everything. Everyday in my life math is always with me.
Math portrays reality because everything we have here in the universe, Math
will follow. Math has helped me to contact my family and friends. Every device I
used was made by Math. And Math plays the rule of everything.
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WORKSHEET 3

NAME: SCORE:

SECTION CODE OR CLASS SCHEDULE: DATE:

1. Continue The ​Fibonacci Sequence ​by completing the 1​st​ table below.
0 1 1 2 3 5 8 13 21 34
55 89 144 233 377 610 987 1,597 2,584 4,181

6,765 10,946 17,711 28,657 46,368 75,025 121,393 196,418 317,811 514,229

2. Complete the 2​nd table below by taking the ratio of the two successive Fibonacci
Sequence from the 1​st​ table. Round off your answer to the nearest thousandths.
x x 1.000 2.000 1.500

3. Measure your arm from A to B and from A to C as shown in the figure below. And
write your measurement ( cm ) in the given table below. Get the ratio of the two
M easure f rom A to B
measurements, i.e., M easure f rom A to C
. Look for other measurements from your friends
or family and get the average.

Name Measure Measure M easure f rom A to B

from A to from A to M easure f rom A to C
= (cm)

B ( cm ) C ( cm )
1.
2.
3.
4.
5.

Average of the last column: _______________

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