1. FIELD DAY TOK: Mathematics and Imagination
An Introduction to Fractal Geometry

2. FIELD DAY TOK: Mathematics and Imagination
An Introduction to Fractal Geometry
“Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.”
Benoit B Mandelbrot (1924 – 2010)

3. FIELD DAY TOK: Mathematics and Imagination
The von Koch Snowflake

4. FIELD DAY TOK: Mathematics and Imagination
The von Koch Snowflake
Perimeter 1: 3 × 1 = 3
Perimeter 2: 12 × = 4
Perimeter 3: 48 × 1 9 =
Perimeter 4: 192 × =7 1 9

5. FIELD DAY TOK: Mathematics and Imagination
The von Koch Snowflake
The AREA inside the snowflake is BOUNDED
The PERIMETER of the snowflake is UNBOUNDED

6. FIELD DAY TOK: Mathematics and Imagination
The von Koch Snowflake
The AREA inside the snowflake is FINITE
The PERIMETER of the snowflake is INFINITE

7. FIELD DAY TOK: Mathematics and Imagination
The von Koch Snowflake
We are claiming that a FINITE area (2-D) can have an INFINITELY long boundary (1-D)

8. FIELD DAY TOK: Mathematics and Imagination
The von Koch Snowflake
So can a FINITE volume (3-D) have an INFINITELY large surface area (2-D)?

9. FIELD DAY TOK: Mathematics and Imagination
The von Koch Snowflake
So can a FINITE volume (3-D) have an INFINITELY large surface area (2-D)?

10. FIELD DAY TOK: Mathematics and Imagination
Sierpinski’s Gasket

11. FIELD DAY TOK: Mathematics and Imagination
Sierpinski’s Gasket
The sum of all the white areas is equal to the original area of the black triangle
This means the black parts ultimately form a 1-D boundary enclosing a 2-D area

12. FIELD DAY TOK: Mathematics and Imagination
Sierpinski’s Gasket
The sum of all the white areas is equal to the original area of the black triangle
This means the black parts ultimately form a 1-D boundary enclosing a 2-D area
The AREA is FINITE
The PERIMETER is INFINITE

13. FIELD DAY TOK: Mathematics and Imagination
How long is the coastline of Britain?
In kilometres – have a guess!

14. FIELD DAY TOK: Mathematics and Imagination
The coastline of Britain

15. FIELD DAY TOK: Mathematics and Imagination
The coastline of Britain

16. FIELD DAY TOK: Mathematics and Imagination
The coastline of Britain

17. FIELD DAY TOK: Mathematics and Imagination
Self-similarity
The term "fractal" was coined by Benoit Mandelbrot in It comes from the Latin fractus, meaning an irregular surface like that of a broken stone. Fractals are non-regular geometric shapes that have the same degree of non-regularity on all scales. Just as a stone at the base of a foothill can resemble in miniature the mountain from which it originally tumbled down, so are fractals self-similar whether you view them from close up or very far away.

18. FIELD DAY TOK: Mathematics and Imagination
Self-similarity

19. FIELD DAY TOK: Mathematics and Imagination
Self-similarity

20. FIELD DAY TOK: Mathematics and Imagination
Self-similarity 1 10 11
100
101
110
111
1000
1 = C
2 = D
3 = E
1001
1010
1011
1100
1101
1110
1111
10000
2 = D
3 = E
4 = F
1 = C
C C D C D D E C D D E D E E F C

21. FIELD DAY TOK: Mathematics and Imagination
Self-similarity 1 10 11
100
101
110
111
1000
1 = C
2 = D
3 = E
1001
1010
1011
1100
1101
1110
1111
10000
2 = D
3 = E
4 = F
1 = C
C C D C D D E C D D E D E E F C
C C D C D D E C

22. FIELD DAY TOK: Mathematics and Imagination
Self-similarity 1 10 11
100
101
110
111
1000
1 = C
2 = D
3 = E
1001
1010
1011
1100
1101
1110
1111
10000
2 = D
3 = E
4 = F
1 = C
C C D C D D E C D D E D E E F C
C C D C D D E C

23. FIELD DAY TOK: Mathematics and Imagination
Self-similarity

24. FIELD DAY TOK: Mathematics and Imagination
Self-similarity

25. FIELD DAY TOK: Mathematics and Imagination
Self-similarity

26. FIELD DAY TOK: Mathematics and Imagination
Self-similarity

27. FIELD DAY TOK: Mathematics and Imagination
Books

28. FIELD DAY TOK: Mathematics and Imagination
A ToK Question
The von Koch snowflake exists only in the mind of a mathematician or a computer ROM; you can never actually make one – so – to what extent does it “exist”?

29. FIELD DAY TOK: Mathematics and Imagination
Another ToK Question
Can we trust computers?

30. FIELD DAY TOK: Mathematics and Imagination
A Maths Joke
Q What is Benoit B Mandelbrot’s middle name?

31. FIELD DAY TOK: Mathematics and Imagination
A Maths Joke
Q What is Benoit B Mandelbrot’s middle name?
A Benoit B Mandelbrot

32. FIELD DAY TOK: Mathematics and Imagination
A Maths Joke
Q What is Benoit B Mandelbrot’s middle name?
A Benoit B Mandelbrot1
Reference:
1 Wearden WP, private conversation, November

33. FIELD DAY TOK: Mathematics and Imagination
An Introduction to Fractal Geometry