field day tok: mathematics and imagination

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)



The von Koch Snowflake



Perimeter 1: Perimeter 2: Perimeter 3: Perimeter 4:



The AREA inside the snowflake is BOUNDED

The PERIMETER of the snowflake is UNBOUNDED



The AREA inside the snowflake is FINITE

The PERIMETER of the snowflake is INFINITE



We are claiming that a FINITE area (2-D) can have an INFINITELY long boundary (1-D)



So can a FINITE volume (3-D) have an INFINITELY large surface area (2-D)?


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 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 FINITEThe PERIMETER is INFINITE


How long is the coastline of Britain?

In kilometres – have a guess!


The coastline of Britain


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.


Self-similarity


Self-similarity

1 = C1 = C2 = D1 = C2 = D2 = D3 = E1 = C

11011

100101110111

1000

1001101010111100110111101111

10000

2 = D2 = D3 = E2 = D3 = E3 = E4 = F1 = C

C C D C D D E C D D E D E E F C


Self-similarity

1 = C1 = C2 = D1 = C2 = D2 = D3 = E1 = C

11011

100101110111

1000

1001101010111100110111101111

10000

2 = D2 = D3 = E2 = D3 = E3 = E4 = F1 = 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


Self-similarity


Self-similarity

http://www.fourmilab.ch/images/Romanesco/Lcr1.html

http://www.fourmilab.ch/images/Romanesco/Lcr4.html


Books


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”?


Another ToK Question

Can we trust computers?


A Maths Joke

Q What is Benoit B Mandelbrot’s middle name?


A Maths Joke


A Benoit B Mandelbrot


A Maths Joke


A Benoit B Mandelbrot1

Reference:1 Wearden WP, private conversation, November 19 2013

field day tok: mathematics and imagination

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