FIELD DAY TOK: Mathematics and Imagination

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Presentation transcript:

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 × 1 3 = 4 Perimeter 3: 48 × 1 9 = 5 1 3 Perimeter 4: 192 × 1 27 =7 1 9

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 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

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

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

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