Mathematics in the Modern World Lecture 1

Chapter 1: Nature of Mathematics
Section 1.1 Patterns and Numbers in Nature and the World
Anna Clarice M. Yanday
Pangasinan State University
August, 2018

PATTERNS
In this discussion, we will be looking at patterns and regularities in
the world, and how MATHEMATICS comes into play, both in
nature and in human endeavor.
Definition
Patterns are regular, repeated or recurring forms or designs.
Example
layout of floor tiles
designs of buildings
the way we tie our shoelaces
Studying patterns helps us in identifying relationships and finding
local connections to form generalizations and make predictions.
Anna Clarice M. Yanday Pangasinan State University

PATTERNS
1. Which of the ﬁgures can be used to continue the series given
below?

PATTERNS
2. Which of the ﬁgures, you think best ﬁts the series below?

PATTERNS
3. Which of the ﬁgures can be used to continue the series given
below?

PATTERNS
4. Which number should come next in this series?
10, 17, 26, 37, ?
A. 46
B. 52
C. 50
D. 56

PATTERNS
5. Which number should replace the question mark "?"
A. 4
B. 5
C. 6
D. 7

PATTERNS
1 The base figure rotates at an angle of 45◦ in the
counterclockwise direction. Hence choice C is the perfect
match.
2 The base figure rotates at an angle of 90◦ in the clockwise
direction. Hence choice A is the best fit.
3 In the given series, a figure is followed by the combination of
itself and its vertical inversion. Thus D. is the right choice.
4 Beginning with 3, each number in the series is a square of the
succeeding no. plus 1.
5 For each row the sum of the first two columns is equal to the
multiple of the last two columns.

SYMMETRY
Definition
Symmetry indicates that you can draw an imaginary line across an
object and the resulting parts are mirror images of each other.
Example
butterfly
Leonardo da Vinci’s Vitruvian Man
starfish

SYMMETRY
Figure 1
The butterﬂy is symmetric about the axis indicated by the black line.
Note that the left and right portions are exactly the same. This type
of symmetry is called bilateral symmetry.

SYMMETRY
Figure 2
Leonardo da Vinci’s Vitruvian Man shows the proportion and
symmetry of the human body.

SYMMETRY
There are other types of symmetry depending on the number of
sides or faces that are symmetrical.
Figure 3

SYMMETRY
Note that if you rotate the starﬁsh in Figure 3 by 72◦ , you can still
achieve the same appearance as the original position. This is known
as the rotational symmetry. The smallest measure of angle that
a ﬁgure can be rotated while still preserving the original position is
called the angle of rotation. A more common way of describing
rotational symmetry is by order of rotation.

ORDER OF ROTATION
A ﬁgure has a rotational symmetry of order n (n- fold rotational
symmetry) if
1
n
of a complete turn leaves the ﬁgure unchanged. To
compute for the angle of rotation, we use the formula
Angle of rotation =
360◦
n

ORDER OF ROTATION
Example
Figure 4

ORDER OF ROTATION
As seen in Figure 4, the pattern on the snowﬂake repeat six times,
indicating that there is a 6-fold symmetry. Using the formula, the
angle of rotation is 60◦.

HONEYCOMB
Figure 5
Do you wonder why bees used hexagon in making honeycomb and
not any other polygons? The reason is that more area will be
covered using hexagon compared to other polygons. The following
computation proves this.

PACKING PROBLEM
Definition
Packing problem involve finding the optimum method of filling up
a given space such as a cubic or spherical container.
Claim: If hexagonal structure is used, then more area will be
covered.

PACKING PROBLEM
Proof.
Suppose you have circles of radius 1 cm, each of which will then
have an area of π cm2. We are then going to ﬁll a plane with these
circles using square packing and hexagonal packing.
Figure 6

PACKING PROBLEM
Proof (Cont.)
For square packing, each square will have an area of 4 cm2. Note
from Figure 6 that for each square, it can only ﬁt one circle. The
percentage of square’s area covered by circles will be
area of the circles
area of the square
× 100% =
π cm2
4 cm2
× 100% ≈ 78.54%
For hexagonal packing, we can think of each hexagon as composed
of six equilateral triangles with side equal to 2 cm.

PACKING PROBLEM
Proof (Cont.)
Figure 7
The area of each triangle is given by
A =
√
3
4
× side2
=
√
3
4
× (2 cm)2
=
√
3 cm2

PACKING PROBLEM
Proof (Cont.)
Thus, the area of the hexagon is 6
√
3 cm2. Looking at Figure 7,
there are 3 circles that could ﬁt inside one hexagon (the whole
circle in the middle and 6 one-thirds of a circle), which gives total
area as 3 π cm2. The percentage of the hexagon’s area covered by
circles will be
area of the circles
area of the hexagon
× 100% =
3 π cm2
6
√
3 cm2
× 100% ≈ 90.69%
Comparing the two percentages, we can clearly see that using the
hexagons will cover a larger area than when using squares.

OTHER MATHEMATICS IN NATURE AND THE WORLD
Examples
Figure 8 Hyena’s Spot and Tiger Stripes
According to Alan Turing, a British Mathematician, the for-
mation of hyena’s spot and tiger stripes is governed by a set of
equations. What Turing proposed was that there are two chemi-
cals interacting inside the embryo of an animal. He did not know
what these chemicals were, so he named them morphogens and
proposed that they reacted with each other and

Examples
diffused through the embryo according to a system of
"reaction-diffusion equations."
Fibonacci numbers on flowers and nautilus shell (play videos)
Mathematics used to model population growth with the
formula
A = Pert
where A is the size of the population after it grows, P is the
initial number of people, r is the rate of growth, t is time and
e is the Euler’s constant with an approximately value of 2.718.

Example
The exponential growth model A = 30e0.02t describes the
population of a city in the Philippines in thousands, t years after
1995.
1 What was the population of the city in 1995?
Solution: Since our exponential growth model describes the
population t years after 1995, we consider 1995 as t = 0 and
then solve for A. Thus, A = 30 and the city population in
1995 is 30,000.
2 What will be the population in 2017?
Solution: We need to ﬁnd A for the year 2017. To ﬁnd t, we
subtract 2017 and 1995 to get t = 22. Hence, A = 46.5813
and the city population in 2017 is approximately 46,581.

REFERENCES
Mathematics in the World book from RBSI
https://www.iqtestexperts.com/pattern-recognition-
sample.php
https://www.iqtestexperts.com/maths-sample.php
http://www.mathscareers.org.uk/article/how-the-tiger-got-its-
stripes/

Mathematics in the Modern World Lecture 1

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