GEED 10053:
Mathematics
in the Modern
World
2
Cynthia P. Equiza
Professor
Patterns and
Numbers in Nature
3
TOPIC
OUTLINE
Fibonacci
Sequence
Mathematics for
Our World
4
What is
Mathematics?
Mathematics is a branch of science, which deals
with numbers and their operations.
5
It involves calculations, computation, solving of
problems, etc.
Mathematics helps us to organize and systemize
our ideas about patterns, in so doing, not only
can we admire and enjoy these patterns, we can
also use them to infer some of the underlying
principles that govern the world of nature.
6
Patterns and
Numbers in
Nature
“Mathematics is a study of patterns and relationship, a
way of thinking, an art, a language, and a tool. It is
about patterns and relationships. Numbers are just a
way to express those patterns and relationships.”
— National Council of Teachers of
Mathematics (1991)
7
PATTERN
A pattern is an arrangement which helps
observers anticipate what they might see
or what happens next.
A pattern also shows what may have
come before.
8
Natural patterns include symmetries, fractals, spirals, meanders, waves, foams,
tessellations, cracks, and spots & stripes. Studying patterns allows one to watch,
guess, create, and discover. The present mathematics is considerably more
than arithmetic, algebra, and geometry.
9
A. SYMMETRY
Symmetry can be found
everywhere. It can be seen
from different viewpoints
namely; nature, the arts and
architecture, mathematics;
especially geometry and
science.
Symmetry occurs when there
is congruence in dimensions,
due proportions and
arrangement. It provides a
sense of harmony and
balance.
10
TYPES OF SYMMETRY
1. Bilateral or reflection symmetry is the simplest kind of symmetry. It can also be
called mirror symmetry because an object with this symmetry looks unchanged if
a mirror passes through its middle.
11
TYPES OF SYMMETRY
2. Radial symmetry is rotational symmetry around a fixed point known as the
center. Images with more than one lines of symmetry meeting at a common point
exhibits a radial symmetry.
12
OTHER CLASSIFICATIONS OF SYMMETRIC
PATTERNS
Rosette patterns consist of taking motif or an element and rotating and/or
reflecting that element. There are two types of rosette patterns namely cyclic and
dihedral.
13
OTHER CLASSIFICATIONS OF SYMMETRIC PATTERNS
Frieze pattern is a pattern in which a basic motif repeats itself over and over in one
direction. It extends to the left and right in a way that the pattern can be mapped
onto itself by a horizontal translation.
7 TYPES:
1. Hop - only admits a translational symmetry.
2. Step - only admits a translational and glide symmetries.
14
OTHER CLASSIFICATIONS OF SYMMETRIC PATTERNS
3. Sidle - only admits translations and vertical reflections.
4. Spinning Hop - only admits translations and 180◦ rotations
(half-turns).
5.Spinning Sidle - only admits translations, vertical reflections,
rotations, and glide reflections.
15
OTHER CLASSIFICATIONS OF SYMMETRIC PATTERNS
6. Jump - only admits translations, a horizontal reflection, and
glide reflection.
7. Spinning Jump - admits translations, vertical reflections,
horizontal reflections, rotations, and glide reflections.
16
OTHER CLASSIFICATIONS OF SYMMETRIC PATTERNS
Wallpaper pattern is a pattern with translation symmetry in two directions. It is,
therefore, essentially an arrangement of friezes stacked upon one another to fill
the entire plane.
17
18
Symmetry in everyday language refers to a sense of harmonious and beautiful
proportion and balance. In mathematics, "symmetry" has a more precise definition,
and is usually used to refer to an object that is invariant under some
transformations; including translation, reflection, rotation or scaling. Although these
two meanings of "symmetry" can sometimes be told apart, they are intricately
related.
B. TESSELLATION
A tessellation is a pattern of one or more shapes where the shapes do not overlap
or have no space between them.
19
C. WAVES
As waves in water or wind pass over sand, they create patterns of ripples. When
winds blow over large bodies of sand, they create dunes, sometimes in
extensive ...
20
D. FRACTALS
Fractals are never-ending patterns. The beauty of fractals is that their infinite complexity is formed through the
repetition of simple equations. These repeating patterns are displayed at every scale.
A fractal is a kind of pattern that we observe often in nature and in art. As Ben Weiss explains, “whenever you
observe a series of patterns repeating over and over again, at many different scales, and where any small
part resembles the whole, that’s a fractal.
21
E. SPIRAL
A spiral is a curved pattern that focuses on a center point and a series of circular shapes that revolve around
it. Examples of spirals are pine cones, pineapples, hurricanes. The reason for why plants use a spiral form like
the leaf picture below is because they are constantly trying to grow but stay secure.
22
F. MEANDERS,FLOW,CHAOS
The relationship between chaos and fractals is that strange attractors in chaotic systems have a fractal
dimension. ... Meanders are bends in a sinuous form that appears as rivers or other channels, which form as a
fluid, most often water, flows around bends. Chaos is the study of how simple patterns can be generated from
complicated underlying behavior.
Many events were considered to be chaotic, unpredictable and random. The dripping of a tap, the weather,
the formation of clouds, the fibrillation of the human heart, the turbulence of fluid flows or the movement of a
simple pendulum under the influence of a number of magnets are a few examples.
23
G. SPOTS, STRIPES
Leopards and ladybirds are spotted; angelfish and zebras are striped.
These patterns have an evolutionary explanation: they have functions which increase the chances that the
offspring of the patterned animal will survive to reproduce.
One function of animal patterns is camouflage; for instance, a leopard that is harder to see catches more
prey.
24
H. CRACKS
Cracks are linear openings that form in materials to relieve stress. When a material fails in all directions it results
in cracks. The patterns created reveal if the material is elastic or not.
Cracks are overlooked because they are so common. It is often a pattern engineers want to avoid, for
example a crack in a bridge or a road or a glass. Engineers spend a lot of time trying to determine when a
crack can become a catastrophe.
25
I. FOAM & BUBBLES
Foam is a mass of bubbles; foams of different materials occur in nature
- A foam is a substance made by trapping air or gas bubbles inside a solid or liquid. Typically, the volume of
gas is much larger than that of the liquid or solid, with thin films separating gas pockets.
- bubble is a spherically contained volume of air or other gas, especially one made from soapy liquid while
foam is a substance composed of a large collection of bubbles or their solidified remains.
26
27
Fibonacci
Sequence
FIBONACCI SEQUENCE
The Fibonacci sequence was invented by the Italian Leonardo Pisano
Bigollo (1180-1250), who is known in mathematical history by several
names: Leonardo of Pisa (Pisano means “from Pisa”) and Fibonacci
(which means “son of Bonacci”).
To formally, define the Fibonacci sequence, we start by defining F 1 = 1
and F2 = 1. For n > 2, we define
Fn = Fn−1 + Fn−2
The sequence F1, F2, F3,… is then the Fibonacci sequence. Such a
definition is called a recursive definition because it starts by defining
some initial values and defines the next term as a function of the
previous terms.
28
One of the exercises in Fibonacci’s book :
“A man put a pair of rabbits in a place surrounded on all sides by a wall. How
many pairs of rabbits are produced from that pair in a year, if it is supposed that
every month each pair produces a new pair, which from the second month
onwards becomes productive?”
RABBIT HABIT
29
GROWTH OF RABBIT COLONY
The Fibonacci sequence is the sequence f1, f2,
f3, f4, … which has its first two terms f1 and f2
both equal to 1 and satisfies thereafter the
recursion formula fn = fn–1 + fn–2.
The sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,
144, 233, 377, … is called the Fibonacci
sequence and its terms the Fibonacci numbers.
30
Fibonacci numbers appear in nature in various places.
Pinecones, Speed Heads,
Vegetables and Fruits Spiral
patterns curving from left and
right can be seen at the array
of seeds in the center of a
sunflower.
Flowers and Branches
Honeybees
Most flowers express the
Fibonacci sequence if you
count the number of
petals on these flowers.
Some plants also exhibit
the Fibonacci sequence in
their growth points, on
the places where tree
branches form or split.
The family tree of a
honeybee perfectly
resembles the Fibonacci
sequence. A honeybee
colony consists of a
queen, a few drones and
lots of workers.
31
Luca Pacioli found the relationship between Fibonacci
sequence and the golden ratio.
The golden ratio was first called as the Divine Proportion
in the early 1500s in Leonardo da Vinci’s work was
explored by Luca Pacioli (Italian mathematician) entitled
“De Devina Proportione” in 1509.
Luca Pacioli
D a V in c i ’s dra w ings of t he f i ve pl ato n i c
solids and it was probably da Vinci who first
called it the “section aurea” Latin for Golden
Section
32
33
Two quantities are in the Golden ratio if their ratio is the same of their sum to the larger of
the two quantities.
The Golden Ratio is the relationship between numbers on the Fibonacci sequence where
plotting the relationships on scales results in a spiral shape
34
The Fibonacci numbers can be applied to the proportions of a rectangle, called the Golden
rectangle.
Golden Rectangle is known as one of the most visually satisfying of all geometric forms – hence, the
appearance of the Golden ratio in art.
The Golden rectangle is also related to the Golden spiral, which is created by making adjacent
squares of Fibonacci dimensions.
A Fibonacci spiral which approximates the golden spiral, using Fibonacci sequence square sizes up to
34.
35
GOLDEN RECTANGLE
A golden rectangle can be broken into squares the size of the next
Fibonacci number down and below.
Fibonacci spiral – Take a golden rectangle, break it down into smaller
squares based from Fibonacci sequence and divide each with an arc.
36
Human body has many elements
numbers and the golden ratio.
follow the Fibonacci sequence
measurements of the human body
terms of the golden ratio.
that show the Fibonacci
Most of your body parts
and the proportions and
can also be divided up in
Geography, Weather and Galaxies Fibonacci numbers and
the relationships between these numbers are evident in
spiral galaxies, sea wave curves and in the patterns of
stream and drainages.
37
The Golden Ratio and/or the Golden Spiral can also be observed in music, art, and designs. Appearing in many
architectural structures, the presence of the golden ratio provided a sense of balance and equilibrium.
Architecture
The Great Pyramid of Giza: The Great
Pyramid of Giza built around 2560 BC
is
one of the earliest examples of
the use of the golden ratio.
The Greek sculptor Phidias sculpted
many things including the bands of
sculpture that run above the columns
of the Parthenon.
38
Arts
Mona-Lisa by Leonardo Da Vinci: It
i s b el i e v ed th a t L e o n a r d o ,
as a
mathematician tried
to incorporate
of mathematics into art.
39
MATHEMATICS FOR OUR WORLD
“Neglect of mathematics works injury to all knowledge,
since he who is ignorant of it cannot know the other
sciences or the things of the world..”
— Roger Bacon
(1214-1294)
40
Mathematics is everywhere; whether it is on
land, sea or air, online or on the front line,
mathematics underpins every nook and
cranny of modern life.
41
Math helps us understand or make sense of
the world - and we use the world to
understand math. It is therefore important that
we learn math contents needed to solve
complex problems in a complex world
42
Applications of Mathematics in our
world
Mathematics helps
organize patterns
and regularities in the
world;
Mathematics helps
predict the behavior
of nature and many
phenomena;
Mathematics helps
control nature and
occurrences in the
world for our own
good;
Mathematics has
applications in many
human endeavors.
Arithmetic Sequences
Sequence is a list of numbers typically with a
pattern.
Each number in the list called a term.
43
ADDITIONAL
TOPICS:
Arithmetic
Sequences
and Series
2, 4, 6, 8, …
a1, a2, a3, a4, …
The first term in a sequence is denoted as a1, the
second term is a2, and so on up to the nth term an.
Sequence – a set of numbers in a specific order.
Terms – the numbers in the sequence
Arithmetic sequence – if the difference between successive terms is constant.
Common difference – the difference between the terms
Identify Arithmetic Sequences:
Ex. 1) 7, 12, 17, 22, 27
+5
+5
+5
+5
Since this sequence has a common
difference it is an arithmetic
sequence.
Ex.2)
+1
1,
+2
2,
+4
4,
8, . . .
This is not an arithmetic sequence
because the difference between
terms is not constant
44
An arithmetic sequence can be found as follows
a1, a1+d, a2+d, a3+d,…
Ex. 3) 74
-7
67
-7
60
-7
53
?
-7
?
?
-7 -7
The common difference is -7
Add -7 to the last term of the sequence to find the next three terms.
(a4+d)=53+(-7)= 46
(a5+d)=46+(-7)= 39
(a6+d)=39+(-7)= 32
Ans:
46, 39, 32
45
How do you find any term in a sequence?
To find any term in an
arithmetic sequence,
use the formula
an = a1 + (n – 1)d
where d is the
common difference.
46
Ex. 4) Find the 14th term in the arithmetic
sequence
9, 17, 25, 33,…
Write an equation/formula for a sequence
Ex. 5) Write an equation for the nth term of the
sequence, 12, 23, 34, 45, …
Sol.
First step: get the common difference (d)
d= (a2 – a1 )= 17-9 = 8
an = a1 + (n – 1)d
a1 = 12, d = 11
an = 12 + (n -1)11
(a3 – a2 )= 25-17= 8
an = 12 + 11n – 11
(a4 – a3 )= 33-25= 8
an = 11n + 1
Distributive property
The common difference is +8
Given: a1 = 9, n = 14, d = 8
Use the equation to solve for the 10th term
an = 11n + 1
Use the formula for the nth term
a10 = 11(10) + 1
an = a1 + (n – 1)d
a14 = 9 + (14– 1)8
n = 10
�� replace n with 10
a10 = 111 ans.
a14 = 9 + 104
a14 = 113 ans.
47
Writing Terms of Sequences
Ex. 6) Write the first five terms of the sequence an = 2n + 3.
SOLUTION
a1 = 2(1) + 3 = 5
1st
term
a2 = 2(2) + 3 = 7
2nd
term
a3 = 2(3) + 3 = 9
3rd
term
a4 = 2(4) + 3 = 11
4th
term
a5 = 2(5) + 3 = 13
5th
term
Ans: 5, 7, 9, 11, 13
48
Writing Terms of Sequences
Ex. 7) Write the first five terms of the sequence f(n) = (–2)n – 1 .
SOLUTION
f(1) = (–2)1–1 = 1
1st
term
f(2) = (–2)2–1 = –2
2nd
term
f(3) = (–2)3–1 = 4
3rd
term
f(4) = (–2)4–1 = – 8
f(5) = (–2)5–1 = 16
Answers:
4th
5th
term
term
1, -2, 4, -8, 16
49
ADDITIONAL TOPICS: Geometric Sequence
A sequence is geometric if the ratios of consecutive terms are the same.
r - is the common ratio.
Finding the nth term of a Geometric Sequence
Formula:
an = a1r(n – 1)
r ➤ is the common ratio
an ➤ nth term
50
Are these geometric?
Ex 1)
2, 4, 8, 16, …
Ex 3) 1, 4, 9, 16, …
4, 2.25, 1.78
This is not a geometric sequence because
4/2, 8/4, 16/8,
Yes, r =2
Ex 2)
the ratio between terms is not constant
12, 36, 108, 324
36/12, 108/36, 324/108,
Yes r =3
51
Ex. 9) Find the 15th term of the geometric sequence whose first term is 20 and whose
common ratio is 1.05
r = 1.05, a1=20,
a15=?
an = a1r(n – 1)
a15 = (20)(1.05)(15 – 1)
a15 = (20)(1.05)14
a15 = (20)(1.979931...)
a15 = 39.599 or 39.60 ans.
52
Ex. 10) Find a formula for the nth term 5, 15, 45, …
Sol. compute r=?
and What is the 9th term?
15/5=3
a9 = 5(3)9–1
45/15=3
a9 = 5(3)8
r=3
a9 = 5(6,561)
a9 = 32,805 ans.
Given: a1 = 5, r=3
an = a1rn – 1
an = 5(3)n – 1
formula
53
Ex. 11) Find the common ratio and the seventh term
of the following sequence: 2/9, 2/3, 2, 6, 18,..
To find the common ratio, divide a successive pair
of terms.
Sol.
(2/3)/(2/9)=(2/3) x ( 9/2)= 3/1 or 3
2/(2/3)=(2/1) x (3/2)=6/2 or 3
6/2 = 3
Given the five terms, so the sixth
term is the very next term, the
seventh will be the term after that.
a6 = (18)(3)=54
(54/18=3,
a7 = (54)(3)=162
(162/54=3, r=3)
18/6 =3
Answers:
The ratio is, r = 3.
common ratio: r = 3
r=3)
seventh term: 162
Note: A geometric sequence goes from one term
to the next by always multiplying (or dividing) by
the same value.
54
Additional topic: Difference table
A difference table shows the differences between successive terms of the sequence. The differences in rows
maybe first, second and third differences. Each number in the first row of the table is the differences between
the closest numbers just above it. If the first differences are not the same, compute the successive differences
of the first differences .
The following examples will show how to predict the next term of a sequence and we look for a pattern in a
row differences.
Ex. : Construct the difference table to predict the next term of each sequence.
a. 3, 7, 11, 15, 19, ?
b. 2, 4, 9, 17, 28, ?
c. 6, 9, 14, 26, 50, 91, ?
55
Solutions
a. 3, 7, 11, 15, 19, ?
Sequence
V V
3 7 11 15 19 ?
V V
First differences
4
4
4
Add 4+19= 23
The next term is 23 ans.
4
b. 2, 4, 9, 17, 28,
?
Sequence
9
2
4
V
First differences
V
17
V
Second differences
V
3
5
8
11
V
3
add all the last digits,
The next term is 42
?
V
2
V
28
3
3+11+28= 42
ans.
56
Solutions
c. 6, 9, 14, 26, 50, 91, ?
Sequence
6 9 14 26 50 91
V V
V
First differences
V V
V V
3 5 12 24 41
V
Second differences
V
V
Third differences
?
V
2
7 12 17
V
5
5
5
add all the last digits, 5+17+41+91=154
The next term is 154 ans.
57
ASSESSMENT 1:
Part 1.
1. Look for patterns Inside or outside of your house then take pictures of the patterns explored using smart
phones or digital camera. Explore, take photos,
make list and identify what patterns can be seen in
nature inside your house, at the garden or park nearby or any part of the neighborhood.
Answer the ff. questions
2.
How do you find the golden ratio of your face?
3. In relation to golden ratio, give 3 examples of Celebrities with almost
why?
perfect faces, explain
58
ASSESSMENT 1:
Part 2:
A) Construct a difference table to predict the next term of each sequence.
1) 6, 9, 14, 26, 50, 91, ?
2) 4, 8, 14, 22, 32, 44, ?
B) Use the given nth-term formula to compute the first three terms of the given sequence.
1) an= 2n3-n2
2) an= 5n2-3n
C) Geometric sequence, using an=a1r(n-1) compute the next term.
1) 1, 2, 4, 8, 16, 32, 64, ?
2) 6, 12, 24, 48, 96, ?
D) Essay
a) Will the universe exist without mathematics or vice versa?
59
“The essence of mathematics is not to make simple
things complicated, but to make complicated things
simple.”
— S. Gudder
60