About the basic knowledge of patterns and sequence

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Mathematics in the Modern
World

2. Chapter 1 - MATHEMATICS in our WORLD
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Chapter 1 – Mathematics in our World
Objectives:
To cite examples of some applications of
mathematics in our everyday lives.
To define the meanings of the different
mathematical patterns applied to our daily lives.
To prove that mathematics has importance not
only in science but in our surroundings as well.

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This lesson presents the nature of mathematics. It
reveals hidden beautiful patterns found in nature
and introduce the famous mathematical number
sequence that is related to nature.
1.1 Patterns and Numbers in Nature and the World

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• The word mathematics comes from the Greek word
“máthēma” (noun) which means "learning or
knowledge”.
• According to Webster’s dictionary, “Mathematics is the
science of number and their operations, interrelations,
combinations, generalizations and abstractions and of
space configurations and generalizations.
1.1.1 Nature of Mathematics

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1. Mathematics is the science of quantity. – Aristotle
2. Mathematics is the language in which god has written in the universe. - Galileo
3. The science of indirect measurement. - Auguste Comte
4. Mathematics is the classification and study of all possible patterns. - Walter
Warwick Sawyer
5. Mathematics is our one and only strategy for understanding the complexity of
nature. – Ralph Abraham
6. Mathematics is a formal system of thought for recognizing, classifying, and
exploiting patterns and relationships. – Ian Stewart
1.1.1 Nature of Mathematics

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It can be seen that the definition of
mathematics changes according to specific lights. Thus,
mathematics maybe defined as the study of patterns
which may be numerical, logical or geometric.
Mathematics as the study of patterns will be the focus
of the lesson.
1.1.1 Nature of Mathematics

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1. The snowflake
2. The honeycomb
3. The sunflower
4. The snail’s shell
5. Flower’s petals
6. Weather
1.1.2 Patterns in Nature

9. Snowflakes
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10. Honeycomb
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11. The
Sunflower
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All the sunflowers in
the world show a
number of spirals
that are within the
Fibonacci sequence

12. The Snail’s
Shell
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Petals of flowers
FLOWERS NUMBER OF PETALS
Lilies 3
Buttercups 5
Delphiniums 8
Marigolds 13
Asters 21
Daisies 34, 55, 89

14. Cycle of
Seasons
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A pattern is an organized arrangement of objects in space
or time. It must have something that is repeated either
exactly or according to recognizable transformations. It is
the opposite of chaos.
example:
0, 5, 10, 15, 20, 25, ?
1.1.2 Patterns in Nature

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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.
1.1.2 Patterns in Nature

17. Symmetry
is when a shape looks
identical to its original shape
after being flipped or turned.
Reflective symmetry, or line
symmetry means that one half of an
image is the mirror image of the other
half.
Rotational symmetry means that the
object or image can be turned around a
center point and match itself some
number of times.
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18. Cracks
are linear openings that
form in materials to
relieve stress. The pattern of
cracks indicates whether the
material is elastic or not.
Some examples are old
pottery surface, drying
inelastic mud, and palm
trunk with branching
vertical cracks.
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19. Tessellation or Tiling
Distinct shapes are formed
from several geometric units
(tiles) that all fit together
with no gaps or overlaps to
form an interesting and
united pattern.
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20. Fractal Pattern
is when an object exhibits self-
similar shape or form at any
scale and repeat itself overtime.
Trees are natural fractals,
patterns that repeat smaller and
smaller copies of themselves to
create the biodiversity of a
forest.
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The natural world is full of sets of numbers. The
Fibonacci sequence (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 and so
on), that is, it follows a trivial logic in which the sum of
the later two numbers gives rise to the next number in the
sequence.
It is a simple pattern, but it appears to be a kind of built-in
numbering system to the universe.
1.1.3 Numbers in Nature

22. Number in Flowers
An interesting fact
is that the number
of petals on a
flower always turns
out to be a
fibonacci number.
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23. Another example is if you look at the bottom of pinecone,
and count clockwise and anti-clockwise number of spirals,
they turn out to be adjacent fibonacci numbers.
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24. The head of a flower is also subject to Fibonaccian
processes. Typically, seeds are produced at the center, and
then migrate towards the outside to fill all the space.
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This lesson presents a general view of
mathematical sequence, Fibonacci sequence and
Golden Ratio. It explains the existence of
Fibonacci sequence and the golden ratio in nature.
1.2 Fibonnaci Sequence and the Golden Ratio

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A sequence is an ordered list of numbers; the numbers in this
ordered list are called the elements or the terms of the
sequence. The arrangement of these terms is set by a definite
rule. If a1, a2, a3, a4,……… etc. Where “a” denote the terms of a
sequence, then 1,2,3,4,…..denotes the position of the term.
A sequence can be defined based upon the number of terms i.e.
either finite sequence or infinite sequence. If a1, a2, a3, a4, ……. is
a sequence, then the corresponding series is given by
Sn = a1+a2+a3 + ... + an
1.2.1 Sequence
Emerson Jay Bellon, DMNS

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Example 1. Describe the following sequences.
{1, 2, 3, 4,…} is a very simple sequence (and it is an infinite
sequence)
{20, 25, 30, 35,…} is also an infinite sequence.
{1, 3, 5, 7} is the sequence of the first 4 odd numbers (and is a
finite sequence)
{1, 2, 4, 8, 16, 32, …} is an infinite sequence where every term
doubles
{a, b, c, d, e} is the sequence of the first 5 letters alphabetically.
1.2.1 Sequence

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An arithmetic sequence is a list of numbers with a definite
pattern. If you take any number in the sequence then subtract it by
the previous one, and the result is always the same or constant then
it is an arithmetic sequence.
The constant difference in all pairs
of consecutive or successive numbers in a sequence is called
the common difference, denoted by the letter d.
1.2.1 Sequence

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Example 2. Find the next term in the sequence below.
2, 5, 8,11,14, ___
1.2.1 Sequence

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A geometric sequence is a sequence of numbers where each term
after the first is found by multiplying the previous one by a fixed,
non-zero number called the common ratio.
A geometric sequence goes from one term to the next by
always multiplying (or dividing) by the same value. So 1, 2, 4, 8,
16,... is geometric, because each step multiplies by two;
The number multiplied (or divided) at each stage of a
geometric sequence is called the "common ratio" r, because
dividing (that is, if you find the ratio of) successive terms, the value
are the same.
1.2.1 Sequence

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Example 4. Find the common ratio and the seventh term of the
following sequence:
To find the common ratio, divide a successive pair of terms.
1.2.1 Sequence
The ratio is always 3, so r = 3.

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1.2.1 Sequence
Since the problem has given the five terms, the sixth term is the very
next term; the seventh will be the term after that. To find the value
of the seventh term, multiply the fifth term by the common ratio
twice:
a6 = (18)(3) = 54
a7 = (54)(3) = 162
Thus, the common ratio: r = 3 and the seventh term is 162.

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1.2.2 The Fibonacci Sequence
The Fibonacci sequence is a series of numbers where a number
is found by adding up the two numbers before it. Starting with 0
and 1, the sequence goes 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so forth.
Written as a rule, the expression is

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1.2.2 The Fibonacci Sequence
The Fibonacci spiral also known as golden spiral has an association
with the golden mean, and it is based on the Fibonacci sequence.
Fibonacci spiral is also reffered to as golden spiral.

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Binet’s Formula
1.2.2 The Fibonacci Sequence

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FIBONACCI
IN
NATURE

37. Fibonacci
in
Pineapple
Pineapples have spirals formed by their
hexagonal nubs. The nubs on many
pineapples form 8 spirals that rotate
diagonally upward to the left and 13
spirals that rotate diagonally upward to
the right. The numbers 8 and 13 are
consecutive Fibonacci numbers.
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38. Fibonacci in Pinecone
The number of spirals going
from the center of the cone (where it
attached to the tree) to the outside
edge. Count the spirals in both
directions. The resulting numbers are
usually two consecutive Fibonacci
numbers.
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Fibonacci in Plants

40. Fibonacci in
Fruits
Inside the fruit of many plants
we can observe the presence of
Fibonacci order.
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1.2.3 The Golden Ratio
Golden ratio, also known as the golden section, golden
mean, or divine proportion, in mathematics, the
irrational number (1 + √5)/2, often denoted by the Greek
letter or τ, which is approximately equal to 1.61803 39887
ϕ
49894 84820. In the world of mathematics, the numeric
value is called "phi", named for the Greek sculptor Phidias.

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1.2.3 The Golden Ratio
It is the ratio of a line segment cut into two pieces of different lengths such
that the ratio of the whole segment to that of the longer segment is equal to
the ratio of the longer segment to the shorter segment. The origin of this
number can be traced back to Euclid, who mentions it as the “extreme and
mean ratio” in the Elements.
𝒂
𝒃
=
𝒂+ 𝒃
𝒂

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1.2.3 The Golden Ratio
The relationship of the Fibonacci sequence to the Golden Ratio lies not
in the actual numbers of the sequence, but in the ratio of the consecutive
numbers.
2/1 = 2.0
3/2 = 1.5
5/3 = 1.67
8/5 = 1.6
13/8 = 1.625
21/13 = 1.615
34/21 = 1.619
55/34 = 1.618
89/55 = 1.618

44. Golden Ratio in Nature
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45. Golden Ratio in Architecture and Engineering
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46. Golden Ratio in Arts
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47. Golden Ratio in Humans
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Thank you!