Mathematics in the modern world

1. PATTERNS
AND
NUMBERS
IN
NATURE
AND THE WORLD

2. WHAT DOES MATHEMATICS HAVE TO DO
WITH NATURE?
 The natural world is full of beauty and amazing
shapes and patterns. In his book, Nature by
Numbers, Ian Stewart mentioned that we live in a
universe of patterns. If you observe our nature
diligently, you can prove that our nature bounds in
mystical colors and intricate shapes and patterns
that can be described mathematically

3. PATTERN
 Patterns are things that are repetitive, which can be
found in nature as color, shape, action, or some other
sequences that are almost everywhere. Mathematics
expresses patterns. These sequences that repeat,
follows a rule or rules. A rule is a way to calculate or
solve a problem.

4. TYPES OF PATTERN IN NATURE
1. Symmetry
 According to the American Heritage Dictionary, symmetry is an exact
correspondence of form and constituent configuration on opposite
sides of a dividing line or plane or about a center or an axis. It indicates
that you can draw an imaginary line across an object and the resulting
parts are mirror images of each other.

5. TYPES OF SAYMMETRY
a. Reflection Symmetry
 It is also called mirror symmetry or line symmetry. It is made with a
line going through an object which divides it into two pieces which are
mirror images of each other. This is often termed as bilateral symmetry
as it divides the object into two (“bi “ means two) mirror images.

6. TYPES OF SAYMMETRY
b. Rotational Symmetry
 It is also called radial symmetry. In Biology, this kind of symmetry 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. Note that if you rotate the given images below by
several degrees, you can still achieve the same appearance as the
original position

7. TYPES OF SAYMMETRY
c. Translational Symmetry
 This kind of symmetry is exhibited by objects which do not change its
size and shape even if it moved to another location. Note that the
movement does not involve with reflection or rotation.
 A shape exhibits translational symmetry if displacement in some
direction-horizontal or vertical, returns the shape to its original
configuration.

8. TYPES OF PATTERN IN NATURE
2. Fractals
 These are never-ending patterns that are self-similar across different
scales. The image just reappears over and over again no matter how
many times the object is magnified.

9. TYPES OF PATTERN IN NATURE
4. Spots and stripes
Patterns are also exhibited in the external appearances of animals.

10. TYPES OF PATTERN IN NATURE
5. Flower Petals
 Flowers are easily considered as things of beauty. Their vibrant colors
and fragrant odors make them very appealing as gifts or decorations.
The flowers below have different number of petals. Flowers with 5
petals are said to be the most common. Notice that these numbers are
all Fibonacci numbers.

11. TYPES OF PATTERN IN NATURE
5. Flower Petals
FLOWERS NUMBEROF PET
ALS
Lilies 3
Buttercups 5
Delphiniums 8
Marigolds 13
Asters 21
Daisies 34, 55, 89

12. TYPES OF PATTERN IN NATURE
6. Number Patterns and Sequences
 Consider the pattern below.
 Are you familiar with this number pattern in a triangle? What do you
think will be the next layer of the numbers in the triangle? common.
Notice that these numbers are all Fibonacci numbers.

13. FIBONACCI SEQUENCE
 The Fibonacci numbers are a series of number that
often occur in nature. This 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. He is the greatest European
mathematician of the middle ages. He was born in
1170 and died in 1240. He introduced the Arabic
number system in Europe.
 Fibonacci is a short term for the latin filius bonacci,
which means “the son of Bonacci”.

14. Thinkof this!
At the beginning of a month, you are given a pair of
newborn rabbits. After a month the rabbits have produced no
offspring; however, every month thereafter, the pair of rabbits
produces another pair of rabbits. The offspring reproduce in
exactly the same manner. If none of the rabbits dies, how many
pairs of rabbits will there be at the start of each succeeding
month?
 At the start of 4th month,howmanypairs of rabbits will
there be?
 Howabout at the start of the 5th and 6th months?

19. The Golden Ratio
 The “golden ratio” is a 1.618:1 mathematical ratio,
and the number 1.618 is known as “phi.”
 Golden ratios can be found in shells, plants, flowers,
and animals, among other places. It is believed to be
one of the strongest and oldest connections between
math and creative arts.
 The Golden ratio is known by multiple names, like
Divine proportion, Golden number, or Golden mean.
Some examples of well-known works that
demonstrate this proportion are:

20. The Golden Ratio
 Some examples of well-known works that
demonstrate this proportion are:
● Khufu’s Pyramid in Egypt
● The Parthenon in Athens, Greek sculpture
● Leonardo Da Vinci’s “Mona Lisa”
● Beethoven and Mozart’s music

21.  There is a special relationship between the Golden Ratio
and Fibonacci Numbers (0, 1, 1, 2, 3, 5, 8, 13, 21, ... etc, each
number is the sum of the two numbers before it).
 When we take any two successive (one after the other)
Fibonacci Numbers, their ratio is very close to the Golden
Ratio: