ACKNOWLEDGEMENT

T

HE PATH TO this project all about the Koch Snowflake has been challenging and rewarding one. My heartfelt gratitude to:

My group member (Marina, Jasmine, Farhana and Nabihah), who have been discussed together in order to make sure this project would complete before the date line.

My classmates, PPISMP Mathematics that always give some opinion and their comments about the project.

My roommate especially to Nor Faznaini binti Abu Bakar, who lend me her lap top in order to finish the project and others that encourage me to do this project although the first time I felt it is impossible to do.

My family especially my father and mother who support me from the back and my sister that give me ideas to solve the problems.

Lastly, special thanks to my lecturer, Puan Khaliza binti Mohd Khairuddin who help me a lot to solve the problem and help to makes correction when I did wrong ways.

I am truly grateful and love you all!

1

CHAPTER 1: INTRODUCTION

The Koch Curve is a mathematical curve, and one of the earliest fractal

curves to have been described. It appeared in a 1904 paper entitled "Sur une courbe

continue sans tangente, obtenue par une construction géométrique élémentaire" by

the Swedish mathematician Helge von Koch. The better known Koch Snowflake (or

Koch Star) is the same as the curve, except it starts with an equilateral triangle.

In order to create the Koch Snowflakes, von Koch began with three

development of the Koch curve. The Koch curve starts with a straight line that is

divided up into three equal parts. Using the middle segment as a base, an equilateral

triangle is created. Finally, the base of the triangle is removed, leaving us with the

first iteration of the Koch curve.

From the Koch curve comes the Koch Snowflakes. Instead one line, the

snowflakes begins with an equilateral triangle. The steps in create the Koch curve

are then repeatedly applied to each side of the equilateral triangle, creating a

‘snowflake’ shape. The Koch Snowflakes is an example of a figure that is self-similar,

meaning it looks the same on any scale.

The first four iterations of the Koch snowflake.

2

CHAPTER 2: LITERATURE REVIEW

Karl Wilhelm Theodor Weierstrass was considered one of the greatest

mathematical analysts of 19th century Europe. He was born in Ostenfeld, Germany,

on October 31, 1815. Weierstrass is well known as a cofounder of the theory of

analytic functions and their representation as power series. He made crucial

contributions to the arithematization of analysis and to the theory of

real numbers.

Weierstrass showed the importance of uniform convergence, furthered the

understanding of elliptic functions, and made contributions to the field

of differential equations. At the core of Weierstrass' mathematical

research was his work on the theory of analytic functions based on

power seriesand the process of analytic continuation. Weierstrass demonstrated that

the integral of an infinite series is equal to the sum of the integrals of the separate

terms when the series converges uniformly within a given region.

In 1861, Weierstrass demonstrated a function that is continuous over an

interval but does not possess a derivative at any point on this interval. Before this, it

had been assumed that a continuous function must have a derivative at most points.

In 1863, he provided a proof of a Gaussian theorem that complex numbers are the

only commutative algebraic extensions of the real numbers.

3

Niels Fabian Helge von Koch was born on 25 Jan 1870 in Stockholm,

Sweden. Von Koch attended a good school in Stockholm, completing his studies

there in 1887. He then entered Stockholm University and was awarded a doctorate

in mathematics by Stockholm University on 26 May 1892. In 1904, von Koch

described an interesting curiosity. He proposed a mental exercise that could be

partially carried out in visible form by anyone with pencil, paper, and patience. This is

named the Koch Snowflake.

The Koch snowflake is a continuous curve which does not have a tangent at

any point. Von Koch's 1906 paper mainly consists of a proof of this fact. He also

showed in the paper that there are two functions f and g which are both nowhere

differentiable such that the snowflake curve is x = f(t), y = g(t) where -1 ≤ t ≤ 1. At

the end of his paper, von Koch gave a geometric construction, based on the von

Koch curve, of such a function which he also expresses analytically.

Leonardo was born in Pisa, Italy in 1170 (The exact date of birth is unknown).

His father Guglielmo was nicknamed Bonaccio ("good natured" or "simple").

Leonardo's mother, Alessandra, died when he was nine years old. Leonardo was

posthumously given the nickname Fibonacci (derived from filius Bonacci, meaning

son of Bonaccio).

Guglielmo directed a trading post (by some accounts he was the consultant

for Pisa) in Bugia, a port east of Algiers in the Almohad dynasty's sultanate in North

Africa (now Bejaia, Algeria). As a young boy, Leonardo traveled there to help him.

This is where he learned about the Hindu-Arabic numeral system.

Recognizing that arithmetic with Hindu-Arabic numerals is simpler and more

efficient than with Roman numerals, Fibonacci traveled throughout the

Mediterranean world to study under the leading Arab mathematicians of the time.

Leonardo returned from his travels around 1200. In 1202, at age 32, he published

what he had learned in Liber Abaci (Book of Abacus or Book of Calculation), and

thereby introduced Hindu-Arabic numerals to Europe.

4

Johann Carl Friedrich Gauss (30 April 1777 - 23. February 1855) was a

legendary German mathematician, astronomer and physicist with a very wide range

of contributions. He is considered to be one of the greatest mathematicians of all

time.

Gauss was born in Braunschweig, Duchy of Brunswick-Luneburg (now part of

Germany) as the only son of lower-class uneducated parents. According to legend,

his genius became apparent at the age of three, when he corrected, in his head, an

error his father had made on paper while calculating finances.

Gauss made important contributions to number theory with his 1801 book

Disquisitiones Arithmeticae, which contained a clean presentation of modular

arithmetic and the first proof of the law of quadratic reciprocity. He had been

supported by a stipend from the Duke of Brunswick, but he did not appreciate the

insecurity of this arrangement and also did not believe mathematics to be important

enough to deserve support; he therefore aimed for a position in astronomy, and in

1807 he was appointed professor of astronomy and director of the astronomical

observatory in Gottingen.

5

CHAPTER 3: METHODOLOGY

In order to find the area contained inside the Koch Snowflake, I have to apply

the sequence and series method to this problem.

A set of numbers, stated in a definite order, such that each number can be

obtained from the previous number according to some rule, is a sequence. Each

number of the sequence is called a term. Consider the following:

3, 5, 7, 9, 11…

1, 4, 9, 16, 25…

1, 2, 4, 8, 16…

The ‘…’ at the end of each sequence show that each one could go on

indefinitely, i.e. the sequence is infinite. However, if we wished to restrict our

attention to a limited number of terms of a sequence, we would write 3, 5, 7, 9,

11….47. The final single full stop shows that the sequence ends when the number

47 is reached. Such as sequence is said to be finite.

An expression for the nth term (written Un) of a sequence is useful since any

specific term of the sequence can be obtained from it. The nth term of the sequence

1, 4, 9, 16, 25….is n2.

Thus, for n = 1 we obtain the first term written U1 = 12 = 1

n = 2 we obtain the second term U2 = 22= 4

n = 3 we obtain the third term U3 = 32 = 9 etc.

6

The sum of the terms of a sequence is called a series.

CHAPTER 4: ANALYZING DATA

Area contained inside the Koch Snowflake:

The area of the Koch Snowflake can be calculated by finding the area of

equilateral triangles of decreasing size and increasing number such that it follows a

geometric series.

Consider an equilateral triangle with sides s units.

The height of the triangle = √s2 − ( s2 )2

=

√32s

∴ Area of the triangle =

12

× s׿ ¿ √32s

=

√34s2

7

s units

To calculate the area of a Koch Snowflake, we start with a simple equilateral

triangle, whose area we'll just call A. So, the zeroth iteration is A0= A.

At each iteration, we add a triangle to each side. In the first iteration the

triangles are one ninth the size of the original.

N

e

x

t

,

t

o

e

a

c

h

o

f

t

8

A1 = A + 39A

= A + 13A

h

e

t

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(

19

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f

11

19)

.

A

t

e

a

c

h

n

e

w

i

t

e

r

a

t

i

12

A2 = A + 13A + 12

81A

= A + A (13 + 427 )

= A + 13A (1 + 4

9 ).

o

n

w

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a

d

d

f

o

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t

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15

w

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d

t

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r

d

s

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t

:

A3 = A + 13A (1 + 4

9 ) + 48729

A

= A + 13A (1 + 4

9+ 1681 )

= A + 13A (1 + 4

9+4

2

92 ) .A

n

d

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17

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l

l

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a

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h

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f

o

l

l

o

w

i

n

g

s

u

m

:

An = A + 13A (1 + 4

9+ 4

2

92+ 4

3

93.. . + 4n − 1

9n− 1 ).

N

18

o

w

c

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s

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a

23

l

r

e

s

u

l

t

:

A∞ = A + 13A (11 − 4

9 )= A + 1

3A × 9

5

= 85A .

Sin

ce

A

√34s2

,

hen

ce

the

are

a of

the

Koc

h

24

Sno

wfla

ke

=

85

׿ ¿

√34s2

=

2√35

s2

u

n

i

t

s

2

.

25

Developing the mathematica

26

l thinking skills by designing two

27

mathematical pattern recognition

28

applying sequence and series probl

29

em.

F

i

r

s

t

P

a

t

t

e

r

n

:

S

q

u

a

r

e

A

r

e

a

30

o

f

t

h

e

s

q

u

a

r

e

,

A

0

=

x2

L

e

n

g

t

h

o

f

t

31

h

e

s

q

u

a

r

e

s

i

d

e

s

,

l

0

=

x

32

A0 = x2

x

x

When a square is removed,

A1 = A0 - ( x3 × x

3 )

= A0 - ( 19 x2)

T

h

u

s

,

w

e

c

33

x3

When more squares are removed,

A2 = A1 - 8 ( 19 x × 1

9x)

= A1 -8 ( 181 x2)

When even more squares are removed,

A3 = A2 - 8 ( 127 x × 1

27x)

= A2 -82 ( 1729 x2)

a

n

c

o

n

c

l

u

d

e

t

h

a

t

:

A

n

=

A

n

-

1

–

8n

-

1

34

( 19n + 1 x

2)F

i

n

a

l

l

y

,

t

h

e

a

r

e

a

o

f

t

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q

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a

r

35

e

w

i

l

l

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q

u

a

l

t

o

0

.

36

Second Pattern: Triangle

37

I

f

i

t

i

s

g

i

v

e

n

n

38

=

1

,

2

,

3

,

4

.

.

.

W

h

e

n

n

=

39

1

,

12

x a20 x

√321

W

h

e

n

n

=

2

,

12

x a21

40

x

√322

W

h

e

n

n

=

3

,

12

x a22 x

√323

W

h

e

41

n

n

=

4

,

12

x a23 x

√324

T

h

e

n

w

e

c

a

n

42

c

o

n

c

l

u

d

e

t

h

a

t

:

An

= 12 x a2n−1 x

√32n

43

CHAPTER 5 : CONCLUSION

The geometry of fractals lies somewhere between dimensions. To be totally accurate

‘fractal’ is even not a ‘thing’ at all but more like a unit of measure or mathematical

characteristic. For example each fractal has a ‘fractal dimension’ which is its degree

of regularity and repetition.

The Koch Snowflake is one such fractal. The boundary of the snowflake has infinite

length by looking at the lengths at each stage of the process, which grows by 4/3

each time the process is repeated. On the other hand, the area inside the snowflake

grows like an infinite series, which is geometric and converges to a finite area.

The area of the Koch snowflake is finite, tending towards 1.6 times the size of the

original triangle, while the perimeter becomes infinite! This finite amount of area is no

larger than a credit card.

After doing this Mathematics assignment, I can conclude that sequence and series

can be applied in geometry shape. So, as a mathematics student. I should know how

to apply the concept of sequence and series.

44

REFLECTION

After received this project from lecturer on 9 September 2008,I studied the question and found that the questions were based on the Exploring Patterns And Relationship.The first thing I do is I go to the library to find a few books that contains the Sequence and Series question because I want to use it as revision and also as a guide to me to run the project. I also surf the internet to make me more confident to do the project and get some information about the Koch Snowflakes.

The difficulties that I have faced during completing this project are I found that it was very hard to find the suitable and simple pattern to be applied with the sequence and series topic..Then,to compute the area of the Koch Snowflake is quite tough because I have to know where I can apply the concept of sequence and series.The knowledges that I gained after doing this project are now I know how to use the rules of differentiation to differentiate compound expressions.Then,I know how to use the knowledge and skills of differentiation to solve problems involving applications of differentiation.

The improvement that I get after completing this project that I know how to apply the concept of sequence and series in geometric shape and maybe in the other shape.

45

REFERENCES

http: // www.unapologetic.wordpress.com/author/drmathochist

http:// www.germannotes.com/hist_carl_gauss.shtml

http :// www-history.mcs.st-andrews.ac.uk/Biographies/Oresme

http :// www.answers.com/topic/leonardo-fibonacci

http :// www..wikipedia.org/wiki/Fibonacci

http :// www.ecademy.agnesscott.edu/~lriddle/ifs/carpet/carpet.htm

http : //www.math.ubc.ca/.../m308/projects/fung/page.html

http: // www. students.umf.maine.edu/.../task.html

http://en.wikipedia.org/wiki/Koch_snowflake

46

APPENDICES

HELGE VON KOCH

JOHANN CARL FRIEDRICH

LEORNARDO OF

PISA(FIBONACCI)

KARL WILHELM

47