Chapter 1
Mathematics in Our World
Learning Outcomes
At the end of the chapter, the students shall be able to:
1. Identify patterns in nature and regularities in the world.
2. Solve problems involving sequences.
3. Illustrate how mathematics in nature is used in inventions.
4. Articulate the importance of mathematics in one’s life.
Mathematics is an old, broad, and deep discipline (field of study). People working to
improve math education need to understand "What is Mathematics?"
We define mathematics as a way of thinking about nature and the world in general. It
is a system of knowing or understanding our surroundings. It provides glimpses into the
nature of mathematics and how it is used to understand our world. This understanding, in
conjunction with other disciplines, contributes to a more complete portrait of the world. Its
central purpose is to explore those facets of mathematics that will strengthen your
quantitative understandings of our environs. Thus, patterns and numbers that are useful in
the world like Fibonacci sequence and other arrays of number will be discussed and
understand how they were used to predict and control the behavior of nature and the
phenomena in this world. In the future, we think about numerous applications of mathematics
as tools in decision-making.
Several uses of Mathematics in different fields are to calculate the results of different
activities, predict the behavior of a variable when the other variables are known, identify well
the requirements of a particular dosage of medicine to cure a certain illness, and to verify
whether a particular solution is applicable to the problem set. The chronology of events in
the past can identify patterns of situation.
1.1 Patterns and Numbers in Nature and the World
Patterns are repetitive, which can be found in nature as color, shape, action, or some
other sequence that are almost everywhere.
Mathematics is the science of patterns and relationships, and nature exploits just about
every pattern that there is. As a theoretical discipline, mathematics explores the possible
relationships among abstractions without concern for whether those abstractions have
counterparts in the real world. Mathematics uncovers these patterns and the rules governing
it.
The simplest mathematical objects are numbers and the simplest of nature's patterns are
numerical. These can be observed in the things and the events in our surrounding as shown
in the following examples.
1
2
➢ We live in the universe of patterns.
➢ Nature abounds in spectral colors and intricate shapes.
These miraculous creations not only delight the imagination; they also challenge our
understanding.
3
Types of Patterns
1. Fractals. It is a never-ending pattern that are self-similar across different scales, i.e., the
image reappears over and over again no matter how many times the object is
magnified. It is an infinite iteration of itself. Some examples are shown in the pictures.
2. Spirals. It is curved patterns made by series of circular shapes revolving around a central
point. They are patterns that occur in plants and natural systems and has been the
inspiration for architectural forms and ancient symbols. They were studied by
mathematicians including Leonardo Fibonacci, who tried to understand order in
nature. Below are some examples.
4
3. Chaos. Chaos is a simple pattern created from complicated underlying behavior. A
chaotic pattern is used to describe a kind of order which lacks predictability. It is a
kind of apparent randomness whose origins are entirely deterministic as depicted in
the pictures below.
4. Tessellations. These are patterns formed by repeating tiles all over a flat surface. Some
examples are shown below.
Dragonfly Eye
5. Symmetry. It is an exact correspondence of form on opposite sides of a dividing line or
plane or about a center or an axis. It has sense of harmonious and beautiful balance and
proportion and remains unchanged after transformations, such as rotations and scaling.
Two Common Types of Symmetry
i. Reflection Symmetry - It is also known as mirror symmetry or line symmetry or
bilateral symmetry. Exactly the same image is formed when divided in half.
5
ii. Rotational Symmetry - It is also known as radial symmetry in biology. It is exhibited
by objects when their similar parts are regularly arranged around a central axis
and the pattern looks the same after a certain amount of rotation. This includes
three-fold, four-fold to n-fold symmetry. The following are examples of objects in
nature exhibiting rotational symmetry. Notice that the two pictures on right have
three-folds and six-fold symmetrical patterns.
Shapes in Nature
Mathematical shapes can always be reduced to numbers, which is how computers handle
graphics. Each tiny dot in the picture is stored and manipulated as a pair of numbers: how
far the dot is along the screen from right to left, and how far up it is from the bottom,
(coordinates of the dot). A general shape is a collection of dots and can be represented as
a list of pairs of numbers. Simple shapes in nature are triangles, squares, pentagons,
hexagons, circles, ellipses, spirals, cubes, spheres, cones, and so on.
6
➢ The rainbow is a collection of circular arcs, one for each color.
➢ Circles also form in the ripples on a pond, in the human eye, on butterflies' wings,
etc.
 Crystals - composed of solid materials having a composition enclosed and arranged
in symmetrical plane surfaces, intersecting at definite angles.
7
➢ Rock
rocks are of
various shapes.
Formations
irregular
and
➢ Animal Kingdom - various patterns and shapes in animal skin
8
Exercise 1-A
A. Identify the following picture as to its pattern (i.e fractals, tessellation, spots and stripes,
symmetry, chaos, etc.) If symmetric, identify whether it is rotational, bilateral, or n-fold
symmetry. You can answer more than one pattern for each picture (if applicable).
9
What is a Mathematical pattern?
Mathematics is a tool of science to described patterns in nature. In our environment, as
portrayed from the pictures above, we can observe different shapes and geometrical figures
which form a pattern. Like butterfly’s wings, the repetition of designs, forms a pattern. Having
been observable, what pattern of the design can you see in the surroundings? Everything
contains a pattern. The shell of the Snell’s has their own pattern.
Patterns are repetitive which can be found in nature, shape, action, or some other
sequences that are almost everywhere. Mathematics as a tool illustrate this patterns. These
repetitive patterns forms sequences which could be traced following a rule of
correspondence.
A mathematical pattern is a repetitive array of numbers or a geometrical figure.
Examples of mathematical patterns include the following:
1. Symmetry. Observe a staircase or a brick wall; you’ll notice that a pattern repeats
again and again. There are also patterns on floor tiling, wallpaper, or decorative
vases. These patterns repeat themselves and see all examples of symmetries.
2. Number Patterns and Sequences. Consider the figure below.
You will notice that the figure has an increasing number of rectangles from
left to right. It suggests that a certain pattern occurs, that is, the number of rectangles
is being doubled, from 1 to 2, from 2 to 4, and from 4 to 8. Hence, the rectangles can
be expressed into an array of numbers such as {1, 2, 4, 8}, where the numbers inside
the parenthesis are called terms. Establishing the domain as the counting numbers
and calculating a mathematical formula suited to this array of numbers yields a
sequence. This sequence, i.e. {1, 2, 4, 8}, is finite. By inspection, it suggests an
infinite number of terms defined by the formula {2𝑛−1 }, where 𝑛 = 1, 2, 3, … . Thus, a
new sequence may be written as {1, 2, 4, 8, … , 2𝑛−1 , … } = {2𝑛−1 } , an infinite
1
sequence. Taking the reciprocal of {2𝑛−1 } yields a new infinite sequence { 𝑛−1} =
2
1 1 1
2 4 8
{1, , , , … }.
3. Continued fractions.
Fractions written in continued form, i.e. numerator and denominator are
fractions having continuous addition or subtraction of simple fractions, are known as
continued fractions.
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Example: a) 1 +
1
1+
b) 1 +
1
1+
1+
1
1
1
1+
1+⋯
2
2
2
1+
2
1+
2
1+1+⋯
1+
Simplify Examples a) and b).
Solution a) Let 𝑥 = 1 +
1
1
1+
1+
1
𝑥
. Thus, 𝑥 = 1 + .
1
1
1+
1
1+
1+⋯
𝑥 2 = 𝑥 + 1, multiplying the equation by 𝑥.
𝑥 2 − 𝑥 − 1 = 0, rewriting the equation in quadratic form.
1+√5
,
𝑎𝑛𝑠𝑤𝑒𝑟
1
},
1−√5
=
, 𝑒𝑥𝑡𝑟𝑎𝑛𝑒𝑜𝑢𝑠 𝑟𝑜𝑜𝑡
1
𝑥=
𝑥
by quadratic formula.
1+√5
The number 𝑥 =
is known as the golden number or golden
1
ratio due to its aesthetic beauty. It could be traced on a regular five sided
star, the golden spiral, the ratio of two consecutive terms in a Fibonacci
𝐹
sequence (i.e. 𝑛+1 , 𝑎𝑠 𝑛 𝑎𝑝𝑝𝑟𝑜𝑎𝑐ℎ𝑒𝑠 𝑎 𝑙𝑎𝑟𝑔𝑒 𝑛𝑢𝑚𝑏𝑒𝑟), the golden rectangle,
𝐹𝑛
etc.
Solution b) Let 𝑦 = 1 +
2
1+
2
2
𝑦
. Thus 𝑦 = 1 + .
2
1+
2
1+
2
1+1+⋯
𝑦 2 = 𝑦 + 2, multiplying the equation by 𝑥.
𝑦 2 − 𝑦 − 2 = 0, rewriting the equation in quadratic form.
𝑦 = 2,
𝑎𝑛𝑠𝑤𝑒𝑟
}, by factoring or quadratic formula.
𝑦 = −1, 𝑒𝑥𝑡𝑟𝑎𝑛𝑒𝑜𝑢𝑠 𝑟𝑜𝑜𝑡
4. Continued radical.
Radicals written in continued form, i.e. the expressions written inside the
radicals are also radicals having continuous addition or subtraction of simple radicals,
are known as continued radicals.
Example: a) √1 + √1 + √1 + √1 + ⋯
b) √2 + √2 + √2 + √2 + ⋯
Simplify Examples a) and b).
Solution a) Let 𝑥 = √1 + √1 + √1 + √1 + ⋯. Thus, 𝑥 = √1 + 𝑥
𝑥 2 = 𝑥 + 1, squaring both sides the equation.
𝑥 2 − 𝑥 − 1 = 0, rewriting the equation in quadratic form.
11
1+√5
,
𝑎𝑛𝑠𝑤𝑒𝑟
1
},
1−√5
=
, 𝑒𝑥𝑡𝑟𝑎𝑛𝑒𝑜𝑢𝑠 𝑟𝑜𝑜𝑡
1
𝑥=
𝑥
by quadratic formula.
This is another form of the golden number.
Solution b) Let 𝑥 = √2 + √2 + √2 + √2 + ⋯. Thus, 𝑥 = √2 + 𝑥
𝑥 2 = 𝑥 + 2, squaring both sides the equation.
𝑥 2 − 𝑥 − 2 = 0, rewriting the equation in quadratic form.
𝑥 = 2,
𝑎𝑛𝑠𝑤𝑒𝑟
}, by quadratic formula.
𝑥 = −1, 𝑒𝑥𝑡𝑟𝑎𝑛𝑒𝑜𝑢𝑠 𝑟𝑜𝑜𝑡
5. The Golden Line Segment.
A line segment divided into two, i.e. the longer length is represented by 𝑘 and
the shorter one is one unit measure, where the ratio of the original length over the
longer one is equal to the ratio of the longer segment over the shorter one is known
as the golden line segment.
𝒌
𝑘
1
𝑘+1
In symbols,
= 𝑘 . Thus, 𝑘 2 = 𝑘 + 1, 𝑜𝑟 𝑘 2 − 𝑘 − 1 = 0. Another
1
quadratic form, similar to the quadratic equation whose root is the golden ratio above.
6. The golden rectangle.
A rectangle divided into a square and a smaller rectangle where the ratio of
the sides of the original rectangle is equal to the ratio of the corresponding sides of
the smaller rectangle is known as the golden rectangle.
𝒌
𝟏
Drawing an arc from the largest square and connecting it to the next smaller
square up to infinity yields golden spiral as shown in the figure below.
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7. Number Patterns within the Pascal’s Triangle
The Pascal’s Triangle which comes from the coefficients of a binomial
expansion also exhibits number patterns. The sum of the coefficients in each row is
represented by the formula {2𝑛−1 }.
As you build the layers of the triangle, you will notice an interesting pattern.
The inside number in each row can be found by adding the previous numbers and
the number above it. Thus, the middle number of the third line of the Pascal’s Triangle
is 1 + 1 = 2.
20
21
22
23
24
25
26
27
= 1
= 2
= 4
= 8
= 16
= 32
= 64
= 128
=
=
=
=
=
=
=
=
1
1
1
1
1
1
1
1
1
1
1
2
3
4
5
6
7
8
1
3
1
6
4
1
10 10 5
1
15 20 15 6
21 35 35 21
1
1
2
3
5
8
13
21
34
55
.
.
.
a Fibonacci sequence
1
7
1
This Fibonacci sequence could also be traced on a golden spiral by drawing
arcs from an innermost unit square up to a desired length of a golden rectangle.
To further understand mathematical patterns, consider this suggested pattern below
pattern and find out the product of the 7th equation.
12345679 x 9 x 1 = 111111111
12345679 x 9 x 2 = 222222222
12345679 x 9 x 3 = 333333333
12345679 x 9 x 4 = 444444444
12345679 x 9 x 5 = 555555555
12345679 x 9 x 6 = 666666666
12345679 x 9 x 7 = _________
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Exercise 1-B
A. Directions: Look for the pattern and write the missing number.
1. 3, 6, 9, 12, ______, ______, ______, ______
2. 3, 10, 17, 24, ______, ______, ______, ______
3. 2 ½, 3, 3½, 4, ______, ______, ______, ______
4. 11, 8, 5, 2, -1, ______, ______, ______, ______
5.
3, 6, 11, 18, _______, ______, ______, ______
B. Directions. Find the missing numbers or shapes in each of the following patterns.
1.
2.
3.
4.
5.
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1.2 Fibonacci Sequence
Fibonacci Numbers are the series of numbers that often occur in nature. The number
sequence was developed in the Middle Ages, and it was named after Leonardo Pisano
Bigollo, a famous Italian mathematician who also happened to discover Fibonacci. The term
“Fibonacci” is a short term for the Latin filius bonascci, which means “the son of Bonacci”.
In 1202, Leonardo Pisano Bigollo published his most prominent work the Liber Abaci (The
Book of Calculating). He introduced his famous rabbit problem:
If a pair of rabbits is put into a walled enclosure (room) to breed, how many pairs of rabbits
will there be after a year if it is assumed that every month each pair produces one new
pair, which begins to bear young two months after its own birth?
Construction of Table 1.1 will be helpful to find the number of pairs of rabbits there will be
after a year. First, represent the adult pair of rabbits as A, and the baby pair of rabbits as B.
For the fourth month, replace A by AB and B by A.
Table 1.1 Breeding of Rabbits
Number of
Months
Pair
No. of No. of
A’s
B’s
No. of
pairs
1st month
A
1
0
1
2nd month
AB
1
1
2
3rd month
ABA
2
1
3
4th month
ABAAB
3
2
5
5th month
ABAABABA
5
3
8
6th month
ABAABABAABAAB
8
5
13
7th month
ABAABABAABAABABA
13
8
21
8th month
ABAABABAABAABABAABAAB
21
13
34
9th month
ABAABABAABAABABAABAABABAABAAB
34
21
55
10th month
55
34
89
11th month
---
---
---
12th month
__
__
__
Can you see the pattern? The number of A’s in 8th month is the sum of the number of A’s
in the 7th month and the number of B’s that became A’s. And the number of B’s in the 8th
month is the same as the number of A’s in the 7th month. Therefore, the number od A’s in
the 9th month will be the sum of the number of A’s in the 8th month, which is the number of
B’s that will be changed to A, (21+13=34). The number of B’s is the same as the number of
A’s in 8th month, which is 21.
Referring to table 1.1, it reveals that each entry in the columns of numbers may be found
through a pattern. It means that the set of numbers in each column of the table forms a
sequence, known as “Fibonacci Sequence”.
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Many applications and occurrence of Fibonacci numbers can be observed everywhere
in the world around us. Many plants and animals show the Fibonacci numbers in the
arrangement of the leaves around the stem, the number of petals on a flower tend to be a
Fibonacci number like sunflowers. A sunflower can contain the number 89, or even 144.
Palm trees, on the other hand, show the numbers in the rings on their trunks. Hus, Fibonacci
numbers are associated with the growth of every living thing, including a single cell, a grain
of wheat, a hive of bees, and even all of mankind.
What is a Fibonacci sequence?
Now, let 𝐹𝑛 be a term in the Fibonacci sequence and 𝑛 = 0, 1, 2, 3, 4, … . The function
defined by 𝐹𝑛 = 𝑛, when 𝑛 = 0 𝑜𝑟 𝑛 = 1, and 𝐹𝑛 = 𝐹𝑛−2 + 𝐹𝑛−1 , when 𝑛 > 1 is known as
Fibonacci sequence, a recursively defined function.
The sequence begins with zero or one. Each subsequent number is the sum of the
two preceding numbers.
In particular,
2 is found by adding the two numbers before it (1+1)
3 is found by adding the two numbers before it (1+2);
5 is found by adding the two numbers before it (2+3);
and 8 is from (3+5), and so on…
Example: The next number in the sequence above is 34 + 55 = 89
A sequence (noun) is an ordered set of numbers, shapes, or any other mathematical
objects arranged into a rule.
Consider all the Fibonacci sequences, and number each term (n) from 0 to onwards.
Table 1.2 Fibonacci Sequences
n
0
1
2
3
4
5
6
7
8
9
…
F(n)
0
1
1
2
3
5
8
13
21
34
…
So the term number (n) 6 is called F6 (which equals 8).
Example: If we want to find the 8th term, that is 7th term plus the 6th term:
F8 = F7 + F6
Means
So,
F8 = F8-7 + F8-2
F8 = 13 + 8
Therefore, F8 = 21
A Recurrence Relation makes the rule of Fibonacci sequence that the next number is the
sum of the two previous numbers. Mathematically, this is written into a functional notation
that is:
Fn = Fn-1 + Fn-2 using f0 = 0 and f1 = 1
where: Fn - is term number n
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Fn-1 - is the previous term ( n – 1 )
Fn-2 - is the term before that ( n – 2 )
Example: Calculate the value of F9.
Solution:
F9 = F8 + F7
= 21 + 13
F9 = 34
In Fibonacci sequence the interesting thing to be considered is that it shows multiples
of a number. Look at the table below.
Table 1.3 Fibonacci Sequence
n
0
1
2
3
4
5
6
7
8
9
10
11 …
Fn
0
1
1
2
3
5
8
13
21
34
55
89 …
X3 = 2. Every 3rd number is a multiple of 2(2,8,34,144,610,…)
X4 = 3, every 4th number is a multiple of 3 ( 3, 21, 144, …)
X5 = 5, so every 5th number is a multiple of 5 ( 5, 55, 610, …)
X6 = 8, and every 6th number is a multiple of 8.
In conclusion, every nth number is a multiple of xn.
Fibonacci Spiral
Fibonacci spiral is the application of geometric design in using Fibonacci numbers in
our nature. The following instructions may be followed to construct the spiral.
Drawing the Fibonacci
Spiral
The well-known sunflowers have a Golden Spiral seed arrangement. This provides a
biological advantage because it maximizes the number of seeds that can be packed into a
seed head.
17
Similarly, we can see a double set of spirals in the pinecones or pineapples- one going
in a clockwise direction and one in the opposite direction. When these spirals are counted,
the two sets are found to be adjacent Fibonacci numbers.
Exercise 1-C
A. True or False: Write T if the statement or equation is TRUE. Otherwise, write F.
_______ 1. 13 is a prime number; thus, F13 is a prime number.
_______ 2. F19 = 144, the index number is equal to its digit sum.
_______ 3. Every 4th Fibonacci is a multiple of 3.
_______ 4. The digit sum of 89 is 17; therefore, F17 = 89
_______ 5. F6 = 8 is the first Fibonacci number with 2 as a factor.
_______ 6. 7 is a factor of F8 .
_______ 7. F3 and F4 are prime factors of F12.
_______ 8. Every 3rd Fibonacci number is a multiple of 2.
_______ 9. Every 5th Fibonacci number is an even number.
_______ 10. F12 = 144, the index number 12 is a factor of 144; therefore 15 is a factor of
F15 .
B. Compute for the ratio of the following and discuss how it compares to the golden ratio:
1. length and width of your face
2. shoulder circumference and your waist
3. distance between the lips and where the eyebrows meet to the length of the nose
4. distance between the navel and knee to the distance between the knee and the
end of the foot.
C. Write synthesis paper (not less than 200 words) focusing the importance of
mathematics in one’s life.
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The Golden Ratio
Observe the results if we take any two successive (one after the other) Fibonacci numbers,
as shown below.
a
b
b/a
2
3
1.5
3
5
1.6666…
5
8
1.6
8
13
1.625
Their ratio is very close to the Golden Ratio “φ”, which is approximately 1.618034….
Golden Ratio is an irrational number and is typically represented by the Greek letter Phi “φ”.
Golden Ratio is a special number also known as the Golden Section, Golden Mean,
Divine Proportion, or Greek letter Phi, which exists when a line is divided into two parts, and
the longer part (a) divided by the smaller part (b) is equal to the sum of (a) + (b) divided by
(a), which both equal 1.618.
If point M divides line AB into two pieces, one of length a and the other of length b
where a is larger than b, the total length is a + b. (See figure below.)
a+b
a
b
Figure 1.8
Observe that the ratio of the longest part to the shorter part is the same as the ratio of
the whole line AB to the longer part. In other words, as the longer part is to be shorter part,
so is the whole line to the longer part. In equation: a:b = (a+b) : a then the special ratio
between the two numbers is called the Golden Ratio.
It means, a/b is Golden Ratio if the following equation holds true:
a/b = (a + b)/ a
Using the Golden Ratio, we can calculate any Fibonacci number. In Table 1.4, the
notation φn/ fn ≈ 2.236… = √5.
Table 1.4 Calculation of Fibonacci Numbers
N
0
1
2
3
4
5
6
7
….
Φn/fn
0
1.618
2.618
2.118
2.285
2.218
2.243
2.236
….
However, the following theorem gives the exact formula for computing the nth term of
the Fibonacci sequence.
Theorem: The nth term, fn of the Fibonacci sequence is given by:
𝑓𝑛 =
1 (1+√5)𝑛 −(1−√5)𝑛
[
]
2𝑛
√5
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Golden Ratio Around Us
The golden ratio, a simple and insignificant number as we might think it is, has
intrigued and fascinated mathematicians and even thinkers in other disciplines. While it is
unknown when exactly this number was discovered, it could be seen in history that this was
used by Greeks in designing and constructing the renowned Parthenon (see bottom left
photo). Da Vinci’s famous artwork, Mona Liza (bottom, 2nd from left photo) also followed this
ratio. Amazingly, many things that are pleasing to the eyes follow the golden ratio. (Consider
the face in the photo below).
A lot of investigations have been done on this mysterious number as it occurs in
almost everything around us: from the seeds of a sunflower, flower petals, DNA, snail shells,
buildings, to our very own bodies!
1.3 Patterns and regularities in the world
Patterns that we observe in nature are the regularities of form in our natural world.
These patterns may recur in different circumstances and sometimes can be mathematically
modeled. These are illustrated in the following examples.
The flow of fluids provides an inexhaustible supply of nature's patterns.
waves radiating from
an underwater
earthquake.
waves surging towards a
beach in parallel ranks
waves spreading in a Vshape behind a moving boat
- waves swirling spiral whirlpools and
tiny vortices.
- apparent structureless, random
frothing of turbulent flow of waves
20
Similar patterns can be found in the atmosphere including the vast spiral of hurricane
as seen by orbiting astronaut.
Noctilucent Clouds
Wave patterns on land includes the strikingly mathematical landscapes on Earth in the
great ergs or sand oceans of the Arabian and Sahara deserts
The simplest pattern is of transverse dunes (upper leftmost photo below) which, like ocean
waves, line up in parallel straight rows at right angles to the prevailing wind direction which
are called barchanoid ridges when the rows are wavy. Sometimes they break up into
innumerable shield-shaped barchan dunes. On the other hand, a slightly moist sand with a
little vegetation to bind it together forms parabolic dunes, shaped like a U, with the rounded
end pointing in the direction of the wind which when in clusters, resembles the teeth of a
rake. If the wind direction is variable, clusters of star-shaped dunes form with several
irregular arms radiating from a central peak, arranged in a random pattern of spots.
21
Nature's love of stripes, spots and other patterns extend in the animal and plant kingdoms
Spots, stripes and several other patterns in plants and animals are almost always present
in our surroundings. Few examples are shown in the following photos.
‣ stripes and spots in plants and animals
viruses assume regular geometric shapes, like icosahedron-a regualr solid formes
from twenty equilateral triangles
‣ many animals are bilaterally symmetric
‣ Pinecones, sunflowers, and daisies (among other flora) have spiral patterns
associated with the well-known Fibonacci sequence.
22
In addition to patterns of form, there are also patterns of movement.
In human, the feet strike the ground in a regular rhythm. When a four-legged creature,
like horse walks, there is more complex but equally rhythmic pattern. The prevalence of
pattern in locomotion extends to the scuttling of insects, the flight of birds, the pulsation of
jellyfish, and the wavelike movements of fish, worms, and snakes. The sidewinder, a desert
snake, moves like a single coil of a helical spring, thrusting its body forward in a series of Sshaped curves in an attempt to minimize its contact with the hot sand. Tiny bacteria propel
themselves along using microscopic helical tails which rotates rigidly like a ship's screw.
Each of nature's patterns is a puzzle, nearly always a deep one. Mathematics is brilliant
at helping us to solve puzzles. It is a more or less systematic way of digging out the rules
and structures that lie behind some observed pattern or regularity, and then using those rules
and structures to explain what's going on. Indeed, mathematics has developed alongside
our understanding of nature, each reinforcing the other.
1.4 Use of mathematics to control nature and occurrences in the world for
human benefits
Mathematics is useful in everyday life, because everyone needs mathematics in
doing something, like the baker or cook, which uses mathematics to measure the quantity
of ingredients; a dressmaker uses mathematics to measures the figure and length of cloths;
a farmer can use mathematics to plan how and when to sow seeds or count the number of
plants; an artist can use mathematics to paint, design collages, dance, and also to measures
the size of the canvas, size of the stage, space required to for an art; a motorist uses
mathematics to estimate the distance travel where the equivalent proportion of fuel expense
of the vehicle should be estimated and the distance of the destination should be accounted
for, and finally, everything in the world requires numbers in nature.
Mathematics can make our life orderly and systematic, and it prevent chaos. It guides
us to see patterns to generalize a solution to a problem. Mathematics is also used to express,
solve, and interpret the puzzles observed in nature. It can expound our power of reasoning,
creativity, abstract or spatial thinking, problem-solving ability, and even effective
communication skills.
Just as any invention of man, mathematics is a product of necessity. Its invention,
with all its branches starting from the simplest to the most complex, has a long story of
continuously catering human needs and problems. It was never a human endeavor intended
to complicate things, but rather, to make complicated phenomena within human
understanding and, if possible, control for the benefit of humanity.
Organize the underlying patterns and regularities in the most satisfying way.
Nature's patterns are not just there to be admired, they are vital clues that govern natural
processes.
‣ Four hundred years ago, German astronomer, Johannes Kepler, argued that
snowflakes must be made by packing tiny identical units together based on the
sixfold symmetry of snowflakes, which is natural consequence of regular packing.
(That is, if you place a large number of identical coins on a table and try to pack
them as closely as possible, the arrangement would be that of a honeycomb in
which every coin, except those at the edges, is surrounded by six others, arranged
in a perfect hexagon.)
‣ By performing a mathematical analysis of astronomical observations made by the
contemporary Danish astronomer Tycho Brahe, Kepler was eventually driven to
the conclusion that planets move in ellipses.
‣ The regular nightly motion of stars is a clue that Earth rotates
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‣ Waves and dunes are clues to the rules that govern the flow of water, sand, and
air
‣ The tiger's stripes and hyena's spots attest to mathematical regularities in
biological growth and form,
‣ Rainbows tell us about the scattering of light, and indirectly confirm that raindrops
are spheres.
‣ Lunar haloes are clues to the shape of the ice crystals.
Patterns possess utility as well as beauty. Once we have learned to recognize a
background pattern, exceptions suddenly stand out. The following are some illustrations:
‣ The desert stands still, but the lion moves.
‣ Against the circling background of stars, a small number of stars that move quite
differently beg to be singled out for special attention. The Greeks called them
planetes, meaning "wanderer," a term retained in our word "planet." (It took a lot
longer to understand the patterns of planetary motion than it did to work out why
stars seem to move in nightly circles. One difficulty is that we are inside the Solar
System, moving along with it, and things that look simple from outside often look
much more complicated from inside.)
‣ The planets were clues to the rules behind gravity and motion.
Predict. One powerful use of mathematics is in the aspect of predicting phenomena.
With these, we are able to prepare and limit the negative impact of these phenomenon to
humanity. With a well-anticipated and beneficial phenomenon that rarely happens, its
prediction makes us not miss its happening. Here are just some instances that uses
mathematics for prediction.
‣ Understanding the motion of heavenly bodies, astronomers could predict
lunar and solar eclipses and the return of comets. They knew where to
point their telescopes to find asteroids that had passed behind the Sun,
out of observational contact.
‣ Because the tides are controlled mainly by the position of the Sun and
Moon relative to the Earth, they could predict tides many years ahead.
(The chief complicating factor in making such predictions is not
astronomy: it is the shape of the continents and the profile of the ocean
depths, which can delay or advance a high tide. However, these stay
pretty much the same from one century to the next, so that once their
effects have been understood it is a routine task to compensate for them.)
‣ It is much harder to predict the weather. We know just as much about the
mathematics of weather as we do about the mathematics of tides, but
weather has an inherent unpredictability. Despite this, meteorologists can
make effective short-term predictions of weather patterns— say, three or
four days in advance.
Control. Many human undertakings and events happen in controlled environment or
within controlled processes. Standards are set in order for a process to attain the desired
results. It may not be visible to us but most electronic inventions of man that we use from the
kitchen, to the streets and even to the outer space operate under controlled conditions.
These control mechanisms are possible because of mathematics. Examples of control
systems range from the thermostat on a boiler, which keeps it at a fixed temperature, to the
medieval practice of coppicing woodland. Without a sophisticated mathematical control
system, the space shuttle would fly like the brick, for no human pilot can respond quickly
enough to correct its inherent instabilities. The use of electronic pacemakers to help people
with heart disease is another example of control.
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1.5 Applications of math as a tool thus making math indispensable
Our world rests on mathematical foundations, and mathematics is unavoidably
embedded in our global culture. The only reason we don't always realize just how strongly
our lives are affected by mathematics is that, for sensible reasons, it is kept as far as possible
behind the scenes.
When you go to the travel agent and book a vacation, you don't need to understand
the intricate mathematical and physical theories that make it possible to design computers
and telephone lines, the optimization routines that schedule as many flights as possible
around any particular airport, or the signal-processing methods used to provide accurate
radar images for the pilots.
When you watch a television program, you don't need to understand the threedimensional geometry used to produce special effects on the screen, the coding methods
used to transmit TV signals by satellite, the mathematical methods used to solve the
equations for the orbital motion of the satellite, the thousands of different applications of
mathematics during every step of the manufacture of every component of the spacecraft that
launched the satellite into position.
When a farmer plants a new strain of potatoes, he does not need to know the
statistical theories of genetics that identified which genes made that particular type of plant
resistant to disease.
Understanding the nature's secret regularities has many uses in our physical
environment. The following are some instances that use such.
‣ steer artificial satellites to new destinations with far less fuel than anybody had
thought possible,
‣ help avoid wear on the wheels of locomotives and other rolling stock,
‣ improve the effectiveness of heart pacemakers
‣ manage forests and fisheries,
‣ make efficient dishwashers, and most importantly;
‣ give a deeper vision of the universe in which we live and of our own place in it.
In addition, using mathematics to understand the structure of some plants and animals and
how they behave empowered us to develop useful mechanism that help us perform jobs
efficiently. Few examples are:
‣ Helicopter is designed from studying the built and movement of a dragonfly
‣ Climbing pads capable of supporting human weight are a mimic of the
biomechanics of gecko feet.
‣ The aerodynamics of the famous Japanese Bullet train was inspired by the shape
of a bird’s beak.
‣ The first flying machine heavier than the air from the Wright brothers, in 1903, was
inspired by flying pigeons.
‣ Architecture is inspired by termite mounds to design passive cooling structures.
‣ Velcro is born from the observation of the hooks implemented by some plants for
the propagation of their seeds via animal’s coat.
‣ The study of shark skin is at the origin of particularly effective swimming suits, as
well as a varnish for plane’s fuselage.
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In sum, mathematics definitely is a way to think about nature as it helps to understand
how things happen, to understand why things happen, to organize the underlying patterns
and regularities in the most satisfying way, to predict how nature will behave, and to control
nature for our own ends, and to make practical use of what we have learned about our world
REFERENCES
Tolentino, A., et. al. (2018). Mathematics in the Modern World. Mutya Publishing House,
Manila
Stewart, I. (1995). Nature’s Numbers. USA: BasicBooks
Adam, J. (2003). Mathematics in Nature: Modelling Patterns in the Natural World. New
Jersey: Princeton University Press
Adam, J. (2009). Mathematical Nature Walk. New Jersey: Princeton University Press
Akiyama and Ruiz (2008). A Day’s Adventure in Math Wonderland. Singapore: World
Scientific Publishing Co. Pte. Ltd.
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