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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 .”

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FIELD DAY TOK: Mathematics and Imagination

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

  2. 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

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

  4. The von Koch Snowflake Perimeter 1: Perimeter 2: Perimeter 3: Perimeter 4: FIELD DAY TOK: Mathematics and Imagination

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

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

  7. 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

  8. 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

  9. 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

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

  11. 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

  12. 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

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

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

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

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

  17. 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

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

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

  20. 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 2 = D 2 = D 3 = E 2 = D 3 = E 3 = E 4 = F 1 = C FIELD DAY TOK: Mathematics and Imagination C C D C D D E C D D E D E E F C

  21. 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 2 = D 2 = D 3 = E 2 = D 3 = E 3 = E 4 = F 1 = C FIELD DAY TOK: Mathematics and Imagination 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. 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 2 = D 2 = D 3 = E 2 = D 3 = E 3 = E 4 = F 1 = C FIELD DAY TOK: Mathematics and Imagination 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. Self-similarity FIELD DAY TOK: Mathematics and Imagination

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

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

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

  27. Books FIELD DAY TOK: Mathematics and Imagination

  28. 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

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

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

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

  32. 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

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

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