FIELD DAY TOK: Mathematics and Imagination
An Introduction to Fractal Geometry
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)
FIELD DAY TOK: Mathematics and Imagination
The von Koch Snowflake
FIELD DAY TOK: Mathematics and Imagination
The von Koch Snowflake
Perimeter 1: 3 × 1 = 3
Perimeter 2: 12 ×
Perimeter 3: 48 ×
1
3
1
9
Perimeter 4: 192 ×
= 4
= 513
1
27
= 719
FIELD DAY TOK: Mathematics and Imagination
The von Koch Snowflake
The AREA inside the snowflake is BOUNDED
The PERIMETER of the snowflake is UNBOUNDED
FIELD DAY TOK: Mathematics and Imagination
The von Koch Snowflake
The AREA inside the snowflake is FINITE
The PERIMETER of the snowflake is INFINITE
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)
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)?
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)?
FIELD DAY TOK: Mathematics and Imagination
Sierpinski’s Gasket
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
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
FIELD DAY TOK: Mathematics and Imagination
How long is the coastline of Britain?
In kilometres – have a guess!
FIELD DAY TOK: Mathematics and Imagination
The coastline of Britain
FIELD DAY TOK: Mathematics and Imagination
The coastline of Britain
FIELD DAY TOK: Mathematics and Imagination
The coastline of Britain
FIELD DAY TOK: Mathematics and Imagination
Self-similarity
The term "fractal" was coined by Benoit Mandelbrot in 1975. 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.
FIELD DAY TOK: Mathematics and Imagination
Self-similarity
FIELD DAY TOK: Mathematics and Imagination
Self-similarity
FIELD DAY TOK: Mathematics and Imagination
Self-similarity
1
10
11
100
101
110
111
1000
1=C
1=C
2=D
1=C
2=D
2=D
3=E
1=C
1001
1010
1011
1100
1101
1110
1111
10000
C C D C D D E C D D E D E E F C
2=D
2=D
3=E
2=D
3=E
3=E
4=F
1=C
FIELD DAY TOK: Mathematics and Imagination
Self-similarity
1
10
11
100
101
110
111
1000
1=C
1=C
2=D
1=C
2=D
2=D
3=E
1=C
1001
1010
1011
1100
1101
1110
1111
10000
C C D C D D E C D D E D E E F C
C C D C D D E C
2=D
2=D
3=E
2=D
3=E
3=E
4=F
1=C
FIELD DAY TOK: Mathematics and Imagination
Self-similarity
1
10
11
100
101
110
111
1000
1=C
1=C
2=D
1=C
2=D
2=D
3=E
1=C
1001
1010
1011
1100
1101
1110
1111
10000
C C D C D D E C D D E D E E F C
C C D C D D E C
2=D
2=D
3=E
2=D
3=E
3=E
4=F
1=C
FIELD DAY TOK: Mathematics and Imagination
Self-similarity
FIELD DAY TOK: Mathematics and Imagination
Self-similarity
FIELD DAY TOK: Mathematics and Imagination
Self-similarity
FIELD DAY TOK: Mathematics and Imagination
Self-similarity
FIELD DAY TOK: Mathematics and Imagination
Books
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”?
FIELD DAY TOK: Mathematics and Imagination
Another ToK Question
Can we trust computers?
FIELD DAY TOK: Mathematics and Imagination
A Maths Joke
Q
What is Benoit B Mandelbrot’s middle name?
FIELD DAY TOK: Mathematics and Imagination
A Maths Joke
Q
What is Benoit B Mandelbrot’s middle name?
A
Benoit B Mandelbrot
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 19 2013
FIELD DAY TOK: Mathematics and Imagination
An Introduction to Fractal Geometry