The Nature of Mathematics

STARTING ACCURATELY (Introduction)
OBJECTIVES:
CHAPTER 1
The Nature of Mathematics
Have you ever stopped to look around and notice all the amazing shapes and
patterns we see in the world around us? Have you tried to ask where did
mathematics came from? If not, then this chapter will enlighten you on the
connection of mathematics to the world.
“All things in nature occur mathematically.” This is according to René
Descartes, the father of modern mathematics. Cliché as it is, but mathematics is all
around us. As we discover more and more about our environment and our
surroundings, we see that nature can be described mathematically. The beauty of
a flower, the majesty of a tree, and even the rocks upon which we walk can exhibit
natures sense of symmetry. Although there are other examples to be found in
crystallography or even at a microscopic level of nature.
At the end of the lesson, you should be able to:
1. identify patterns in nature and regularities in the world;
2. articulate the importance of mathematics in one’s life;
3. express appreciation for mathematics as a human endeavor.

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Mathematics in the Modern World
University of Antique
College of Arts and Sciences
Chapter 1: The Nature of Mathematics
Lesson 1: Mathematics in our World
Lesson 1 – Mathematics in Our World
Stimulating Learning (Motivation)
Inculcating Concepts (Input/Lesson Proper)
Mathematics forms the building blocks of the natural world and can be seen in stunning
ways. It reveals hidden patterns that help us understand the world around us. Nothing in nature
happens without a reason, all of these patterns have an important reason to exist and they also
happen to be beautiful to watch. Check out examples of some of these patterns and you may be
able to spot a few the next time you go for a walk.
PATTERNS
~ 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 sequences, symmetries, fractals, spirals, meanders, waves, foams, tessellations,
cracks, and stripes.
Traditionally, we think of patterns as something that just repeats again and again
throughout space in an identical way, sort of like a wallpaper pattern. But many patterns that we
see in nature aren't quite like that. We sense that there is something regular or at least not
random about them, but that doesn't mean that all the elements are identical. I think a very
familiar example of that would be the zebra's stripes. Everyone can recognize that as a pattern,
but no stripe is like any other stripe.
Here are a few examples of mathematical patterns in nature;
1. Sequence
A sequence is an enumerated collection of objects in which repetitions are allowed
and order matters. One of the most well-known type of sequence that mostly occurs in
nature is the Fibonacci sequence.
The Fibonacci Sequence
“A certain man put a pair of rabbits in a place surrounded on all sides by a wall.
How many pairs of rabbits can be produced from that pair in six months if it is supposed
that every month each pair begets a new pair which from the second month on becomes
productive?”
Can you
identify some
patterns
around you?
Do these patterns
play an important
role in learning
Mathematics?

Page | 3
The figure below shows the number of pairs of rabbits on the first day of each of
the first six months. The larger the rabbits represent mature rabbits that produce another
pair of rabbits each month. It will produce a sequence of 1, 1, 2, 3, 5, 8 which is the first
6 terms of a Fibonacci sequence.
This is a number sequence, it begins with the numbers 1 and 1, and then each
subsequent number is found by adding the two previous numbers. Therefore, after 1 and
1, the next number is 2. The next is 3 and then 5 and so on.
1, 1, 2, 3, 5, 8, 13, 21, 34, …
1 + 1 = 2
1 + 2 = 3
2 + 3 = 5
3 + 5 = 8
5 + 8 = 13
8 + 13 = 21
13 + 21 = 34 …
The list is named after Leonardo Pisano Bigollo (also
Leonardo Fibonacci), an Italian mathematician who wrote the book Liber Abaci in 1202.
The book contained a problem that concerns the birth rate of rabbits. It is this problem
which rise to the Fibonacci numbers.
What’s remarkable is that the numbers in the sequence are often seen in
nature. A few examples include the number of spirals in a pine cone, pineapple or seeds
in a sunflower, or the number of petals on a flower. The numbers in this sequence also
form a unique shape known as a Fibonacci spiral which we see in nature in the form of
shells and the shape of hurricanes/typhoon/storm.
Note that the ratios of successive Fibonacci numbers approach the number Φ (Phi),
also known as the Golden Ratio. This approximately equal to 1.618 and equal to the
irrational number
1+√5
2
.
The Golden Ratio can also be expressed as the ratio between two numbers, if the
latter is also the ratio between the sum and the larger of the two numbers. Geometrically,

Page | 4
it can be visualized as a rectangle perfectly formed by a square and another rectangle,
which can be repeated infinitely inside each section.
Fibonacci numbers are intimately connected with the golden ratio. To demonstrate
this, try to divide any number in the Fibonacci sequence by the one before it, and tabulate
the results.
1
1
= 1
2
1
= 2
3
2
= 1.5
5
3
= 1.6666 …
8
5
= 1.6
13
8
= 1.625
21
13
= 1.615385 …
34
21
= 1.61904
55
34
= 1.617647 …
Each ratio converges to 1.618 …
The Golden ratio is visible in many works of art and architecture such as in the Mona
Lisa, the Notre Dame Cathedral, and the Parthenon. In fact, the human DNA molecule also
contains Fibonacci numbers.
Da Vinci’s Mona Lisa
Notre Dame Cathedral
Parthenon
The Human DNA Molecule

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The Golden ratio in you.
You can find a number of instances in your own body that
approximate 𝜙 (phi).
 The lengths of your finger joints
 The distance from the floor to your navel relative to
your height
 Front two incisors height to width
 Ratio of forearm and hand length
Fibonacci numbers
in nature
Flower petals
Lilies and iris have 3 petals; buttercups have 5
petals; some delphiniums have 8; corn marigolds
have 13 petals; some asters have 21 whereas daisies
can be found with 34, 55 or even 89 petals.
Seed Heads
Counting along the spirals of seed heads normally
leads to a Fibonacci number.
Pine Cones
Pine cone scales are also normally arranged in a
Fibonacci spiral
Number of Leaves
Number of Branches

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Scales of a Pineapple
2. Symmetry
Symmetry in everyday language refers to a sense of harmonious and beautiful
proportion and balance. Symmetry occurs when two or more parts are identical after a flip,
slide or turn.

Page | 7
Types of symmetry
 Reflection – (or a flip) can be thought of as getting a mirror
image. It has a line of reflection or mirror line where the
distance between the image and the mirror line is the same as
that between the original figure and the mirror line. The line of
symmetry is any line that divides the figure into two congruent
parts. This line is unmoved by the reflection. This is called as Line
of or bilateral Symmetry or “Mirror Symmetry”.
 Translation – (or slide) moves a shape in a given direction by sliding it up, down,
sideways, or diagonally.
 Rotation – (or a turn) has a point about which the rotation is made and an angle that
says how far to rotate. The rotation is completely determined by its center and angle of
rotation. When an image is rotated (around a central point) it appears 2 or more times
(order). Angle of Rotation=
360°
𝑛
Line of Symmetry

Page | 8
 Dilation – a transformation which changes the size of an object.
Example of symmetry in our locality.
3. Fractals
Fractals are another intriguing mathematical shape that we see in nature. A fractal
is a self-similar, repeating shape, meaning the same basic shape is seen again and again in
the shape itself. In other words, if you were to zoom in or zoom out, the same shape is
seen throughout.
Fractals make-up many aspects of our world, included the leaves of ferns, tree
branches, the branching of neurons in our brain, waterfalls and coastlines.
A fractal processes the following characteristics: self-similarity, fractional dimension,
and formation by iteration.
Antique’s Traditional Pot or popularly
known as “Kuron”
Matryoshka doll

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Self-similarity
Geometric figures are similar if they have the same shape. Self-similar objects
appear the same under magnification. They are, in some fashion, composed of smaller
copies of themselves.
Notice that the outline of the figure is an equilateral triangle. Now, look at all the
equilateral triangles inside. All of these are similar to each other and to the original
triangle. There are infinitely many smaller and smaller triangles inside. This figure is called
the Sierpinski Triangle.
Iterative formation
This self-similar behavior can be replicated through recursion:
repeating a process over and over. Each smaller triangle formed in
the Sierpinski triangle is an iteration of the base equilateral triangle.
Fractional dimension
Notice that each step of the Sierpinski gasket iteration removes one-quarter of the
remaining area. If this process continued indefinitely, we would end up essentially
removing all the area, meaning we started with a 2 −dimensional area, and somehow end
up with something less than that, but seemingly more than just a 1 −dimensional line.
Examples of Fractal patterns.
Koch Snowflakes
Sierpinski Square
Sierpinski Triangle

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4. Tessellation
A tessellation or tiling of a flat surface is the covering of a plane using one or more
geometric shapes, called tiles, with no overlaps and no gaps. Or a pattern of polygons
fitted together to cover an entire plane without over lapping.
Regular Tessellations
A regular tessellation is a pattern made by repeating a regular and congruent
polygon, with common vertices.
Look at a Vertex ...
A vertex is just a "corner point".
What shapes meet here?
Three hexagons meet at this vertex,
and a hexagon has 6 sides.
So this is called a {6, 3} tessellation.
{𝑛𝑜. 𝑜𝑓 𝑠𝑖𝑑𝑒𝑠 𝑜𝑓 𝑎 𝑝𝑜𝑙𝑦𝑔𝑜𝑛, 𝑛𝑜. 𝑜𝑓 𝑝𝑜𝑙𝑦𝑔𝑜𝑛𝑠 𝑚𝑒𝑒𝑡 𝑖𝑛 𝑡ℎ𝑒 𝑣𝑒𝑟𝑡𝑒𝑥}
For a regular tessellation, the pattern is identical at each vertex!
Semi-regular Tessellations
A semi-regular tessellation is made of two or more regular polygons. The pattern
at each vertex must be the same.
There are only 8 semi-regular tessellations:

Page | 11
Example of tessellation in our locality.
Hexagons in Nature
Another of nature’s geometric wonders is the hexagonal pattern. A regular hexagon
has 6 sides of equal length, and this shape is seen again and again in the world around us.
The most common example of nature using hexagons is in a bee hive. Bees build
their hives using tessellation of hexagons. We also see hexagons in the bubbles that make
up a raft bubble. Although we usually think of bubbles round, when many bubbles get
pushed together on the surface of water, they take the shape of hexagons.
The bees have instinctively found the best solution, evident in the hexagonal
construction of their hives. Cells of honeycombs in the shape of hexagons allow bees to
store the largest quantity of honey given a limited amount of beeswax. These geometric
patterns are not only simple and beautiful, but also optimally functional.
Optimization
It is a field of mathematical investigation used to aid decision making in business
and industrial engineering. The goal of optimization is to maximize (or minimize) the
quantity of an output, while at the same time minimizing the quantity of resources needed
to produce it.
Pappus of Alexandria (last Ancient Greek Mathematician) pointed out that triangles,
squares, and hexagons are the three regular polygons that can tile a plane without gaps.
For choosing the design of hexagons, honeybees seemed imbued with natural wisdom,
because it is the polygon that which holds the largest quantity of honey given a limited
amount of beeswax.
Native “Banig” from Libertad, Antique
“Kawayan” basket weave pattern

Page | 12
5. Concentric Circles in Nature
Concentric means the circles all share the same center, but have different radii.
The region between two concentric circles of different radii is called an annulus. This
means the circles are all different sizes, one inside the other. The common example is in
the ripples of a pond when something hits the surface of the water. But we also see
concentric circles in the layers of an onion and the rings of trees that form as it grows and
ages.
The Earth has three major concentric shells or chemically distinct materials: the
crust, the mantle, and the core.
To better understand the lesson, watch the video about “Patterns and Numbers in
Nature and the World||Mathematics in the Modern World.”
(https://www.youtube.com/watch?v=PApce5Q3U0k&t=214s )

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Using/Applying Knowledge (Application/Integration)
Direction: Match each of the descriptions in Column A with corresponding patterns in Column
B by connecting the dots.
Column A Column B
1. It is a self-similar, repeating shape,
meaning the same basic shape is seen
again and again in the shape itself
  Sequence
2. It is tiling of a flat surface is the
covering of a plane using one or more
geometric shapes, called tiles, with no
overlaps and no gaps.
  Fibonacci sequence
3. The circles all share the same center,
but have different radii
  Symmetry
4. The sum of two consecutive numbers
is equal to succeeding number
  Fractals
5. When two or more parts are identical
after a flip, slide or turn
  Tessellation
6. It is an enumerated collection of
objects in which repetitions are
allowed and order matters.
  Hexagons
7. Has 6 sides of equal length, and this
shape is seen again and again
  Concentric circles
 Golden Ratio
NOTE:
1. Application/Integration in your module was designed only for student practice.
2. Please open your LMS and watch the documentary video about “Decoding the
Secret Patterns of Nature” (Dan McCabe & Richard Reisz, 2017;
http://youtu.be/lXyCRP871VI) and reflect on yourself the contribution of
mathematics in your life.
3. Answer Assessment 1 and Chapter Quiz 1.
Answer
key:
Using/Applying
Knowledge
(
Application/Integration
)
1.
Fractals
2.
Tessellations
3.
Concentric
Circles
4.
Fibonacci
Sequence
5.
Symmetry
6.
Sequence
7.
Hexagons

The Nature of Mathematics

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