1. Mathematics around us
Mathematics expresses itself everywhere, in almost
every facet of life - in nature all around us, and in the
technologies in our hands. Mathematics is the
language of science and engineering - describing our
understanding of all that we observe.
This presentation explores the many wonders and
uses of mathematics in daily lives. This exhibition is
divided into nine areas focusing on different aspects
of mathematics.

2. Arithmetic progression
What is
Arithmetic
progression?

Arithmetic Progression is useful in predicting
an event if the pattern of the event is known.
For Example,
If an asset costs ‘v’ when new, and is
depreciated by ‘d’ per year, its value each year can
be represented by an arithmetic progression
v, v-d, v-2d, ....

3. Arithmetic progression in nature

Old Faithful is a popular attraction at
Yellowstone National Park, because the
geyser produces long eruptions that are
fairly predictable.

When tourists visit Old Faithful, they will
see a sign that indicates an estimated
time that the geyser will next erupt.

No one controls the geyser like an
amusement park ride.

Its patterns over time have caused park
rangers to develop predictable eruption
times using an arithmetic sequence.

4. Old Faithful

The time between eruptions is based on the length of the previous
eruption

If an eruption lasts one minute, then the next eruption will occur in
approximately forty-six minutes (plus or minus ten minutes).

If an eruption lasts two minutes, then the next eruption will occur in
approximately fifty-eight minutes.

This pattern continues based on a constant difference of twelve
minutes, forming an arithmetic sequence of 46, 58, 70, 82, 94, ..

An eruption of n minutes will indicate that the next eruption, an, will
occur in an = a1 + (n î 1)d minutes, where a1is the length after a
one-minute eruption, and d is the constant difference of waiting time
among eruptions that are a one-minute difference in time.

In this particular situation, the next eruption will occur in an = 46+(n
î 1)12 minutes, if the previous eruption was n minutes long.

5. Trigonometry in daily life
What is
Trigonometr
y?And how
is it used?

Trigonometry is the
branch of mathematics
that studies triangles
and their relationships.

6. The Pyramids of Giza
Primitive forms of trigonometry were used in the construction of these
wonders of the world.

7. Architecture

In architecture, trigonometry plays a massive role in the compilation
of building plans.

For example, architects would have to calculate exact angles of
intersection for components of their structure to ensure stability and
safety.

Some instances of trigonometric use in architecture include arches,
domes, support beams, and suspension bridges.

Architecture remains one of the most important sectors of our
society as they plan the design of buildings and ensure that they are
able to withstand pressures from inside.

8. Jantar Mantar observatory
For millenia, trigonometry has played a major role in calculating distances
between stellar objects and their paths.

9. Astronomy

Astronomy has been studied for millennia by civilizations in all
regions of the world.

In our modern age, being able to apply Astronomy helps us to
calculate distances between stars and learn more about the
universe.

Astronomers use the method of parallax, or the movement of the
star against the background as we orbit the sun, to discover new
information about galaxies.

Menelaus’ Theorem helps astronomers gather information by
providing a backdrop in spherical triangle calculation.

10. Grand Canyon Skywalk
Geologists had to measure the amount of pressure that surrounding rocks
could withstand before constructing the skywalk.

11. Geology

Trigonometry is used in geology to estimate the true
dip of bedding angles. Calculating the true dip allows
geologists to determine the slope stability.

Although not often regarded as an integral profession,
geologists contribute to the safety of many building
foundations.

Any adverse bedding conditions can result in slope
failure and the entire collapse of a structure.

12. Fibonacci series around us
What are
fibonacci
series?

Were introduced in The Book of Calculating

Series begins with 0 and 1

Next number is found by adding the last two
numbers together

Number obtained is the next number in the series

Pattern is repeated over and over
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, …
Fn + 2 = Fn + 1 + Fn

13. Who Was Fibonacci?
~ Born in Pisa, Italy in 1175 AD
~ Full name was Leonardo Pisano
~ Grew up with a North African education under the Moors
~Traveled extensively around the Mediterranean coast
~ Met with many merchants and learned their systems of arithmetic
~Realized the advantages of the Hindu-Arabic system

14. Fibonacci Numbers

~ Were introduced in The Book of Calculating

~ Series begins with 0 and 1

~ Next number is found by adding the last two numbers together

~ Number obtained is the next number in the series

~ Pattern is repeated over and over
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, …
Fn + 2 = Fn + 1 + Fn

15. Fibonacci’s Rabbits
Suppose a newly-born pair of rabbits (one male, one female) are put in a field.
Rabbits are able to mate at the age of one month so that at the end of its
second month, a female can produce another pair of rabbits. Suppose that the
rabbits never die and that the female always produces one new pair (one male,
one female) every month from the second month on. How many pairs will there
be in one year?

16. Fibonacci’s Rabbits Continued

End of the first month = 1 pair

End of the second month = 2 pair

End of the third month = 3 pair

End of the fourth month = 5 pair

5 pairs of rabbits produced in one
year
1, 1, 2, 3, 5, 8, 13, 21, 34, …

17. Fibonacci Numbers in Nature

Fibonacci spiral is found in both snail and sea
shells

18. Fibonacci Numbers in Nature

Sneezewort (Achillea ptarmica) shows the
Fibonacci numbers

19. Fibonacci Numbers in Nature

Plants show the Fibonacci numbers in the arrangements of their
leaves

Three clockwise rotations, passing five
leaves

Two counter-clockwise rotations

20. Fibonacci Numbers in Nature

Pinecones clearly show the Fibonacci spiral

21. Fibonacci Numbers in Nature


A
Lilies and irises = 3
petals
Buttercups and wild roses = 5 petals
Corn marigolds = 13 petals
Black-eyed Susan’s = 21 petals

22. Fibonacci Numbers in Nature

The Fibonacci numbers are found in the arrangement of seeds on
flower heads

55 spirals spiraling outwards and 34 spirals spiraling inwards

23. Fibonacci Numbers in Nature

Fibonacci spiral can be found in cauliflower

24. Fibonacci Numbers in Nature

The Fibonacci numbers can be found in pineapples and bananas

Bananas have 3 or 5 flat sides

Pineapple scales have Fibonacci spirals in sets of 8, 13, 21

25. Golden Section
What is
golden
section ?

Represented by the Greek letter Phi

Phi equals
… and
0.6180339887 …

Ratio of Phi is 1 : 1.618 or 0.618 : 1

Mathematical definition is Phi2 = Phi + 1

Euclid showed how to find the golden
section of a line
<-------1------->
A
G
g
GB = AG or 1 – g = g
B
1-g
so that g2 = 1 – g

26. Golden Section and Fibonacci
Numbers

The Fibonacci numbers arise from the
golden section

The graph shows a line whose gradient is
Phi

First point close to the line is (0, 1)

Second point close to the line is (1, 2)

Third point close to the line is (2, 3)

Fourth point close to the line is (3, 5)

The coordinates are successive Fibonacci
numbers

27. Golden Section and Fibonacci
Numbers

The golden section arises from the Fibonacci numbers

Obtained by taking the ratio of successive terms in the Fibonacci
series

Limit is the positive root of a quadratic equation and is called the
golden section

28. Golden Section and Geometry

Is the ratio of the side of a regular pentagon to its diagonal

The diagonals cut each other with the golden ratio

Pentagram describes a star which forms parts of many flags
European Union
United States

29. Golden Section in Nature

Arrangements of leaves are the same as for seeds and petals

All are placed at 0.618 per turn

Is 0.618 of 360o which is 222.5o

One sees the smaller angle of 137.5o

Plants seem to produce their leaves, petals, and seeds based upon
the golden section

30. Golden Section in Architecture

Golden section appears in many of the
proportions of the Parthenon in Greece

Front elevation is built on the golden section
(0.618 times as wide as it is tall)

31. Golden Section in Architecture

Golden section can be found in the Great pyramid in Egypt

Perimeter of the pyramid, divided by twice its vertical height is the
value of Phi

32. Golden Section in Architecture

Golden section can be found in the design of Notre Dame in Paris

Golden section continues to be used today in modern architecture
United Nations
Headquarters
Secretariat building

33. Golden Section in Art

Golden section can be found in Leonardo da Vinci’s artwork
The Annunciation

34. Golden Section in Art

a
The Last Supper
Madonna with Child and Saints

35. Golden Section in Art

Golden section can be seen in Albrecht Durer’s paintings
Trento
Nurnberg

36. Golden Section in Music

Stradivari used the golden section to place the f-holes in his famous
violins

Baginsky used the golden section to construct the contour and arch
of violins

37. Golden Section in Music

Mozart used the golden section when composing music

Divided sonatas according to the golden section

Exposition consisted of 38 measures

Development and recapitulation consisted of 62 measures

Is a perfect division according to the golden
section

38. Golden Section in Music

Beethoven used the golden section in his famous Fifth Symphony

Opening of the piece appears at the golden section point (0.618)

Also appears at the recapitulation, which is Phi of the way through
the piece

39. Fibonacci Numbers and Golden
Section at Towson

40. Fibonacci Numbers and Golden
Section at Towson

41. Thank You
-Rachit Bhalla