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The Fractal Geometry of the Mandelbrot Set

The Fractal Geometry of the Mandelbrot Set. The Fractal Geometry of the Mandelbrot Set. How to count. The Fractal Geometry of the Mandelbrot Set. How to count. How to add. Many people know the pretty pictures. but few know the even prettier mathematics. Oh, that's nothing

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The Fractal Geometry of the Mandelbrot Set

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  1. The Fractal Geometry of the Mandelbrot Set

  2. The Fractal Geometry of the Mandelbrot Set How to count

  3. The Fractal Geometry of the Mandelbrot Set How to count How to add

  4. Many people know the pretty pictures...

  5. but few know the even prettier mathematics.

  6. Oh, that's nothing but the 3/4 bulb ....

  7. ...hanging off the period 16 M-set.....

  8. ...lying in the 1/7 antenna...

  9. ...attached to the 1/3 bulb...

  10. ...hanging off the 3/7 bulb...

  11. ...on the northwest side of the main cardioid.

  12. Oh, that's nothing but the 3/4 bulb, hanging off the period 16 M-set, lying in the 1/7 antenna of the 1/3 bulb attached to the 3/7 bulb on the northwest side of the main cardioid.

  13. Start with a function: 2 x + constant

  14. Start with a function: 2 x + constant and a seed: x 0

  15. Then iterate: 2 x = x + constant 1 0

  16. Then iterate: 2 x = x + constant 1 0 2 x = x + constant 2 1

  17. Then iterate: 2 x = x + constant 1 0 2 x = x + constant 2 1 2 x = x + constant 3 2

  18. Then iterate: 2 x = x + constant 1 0 2 x = x + constant 2 1 2 x = x + constant 3 2 2 x = x + constant 4 3 etc.

  19. Then iterate: 2 x = x + constant 1 0 2 x = x + constant 2 1 Orbit of x 2 0 x = x + constant 3 2 2 x = x + constant 4 3 etc. Goal: understand the fate of orbits.

  20. 2 Example: x + 1 Seed 0 x = 0 0 x = 1 x = 2 x = 3 x = 4 x = 5 x = 6

  21. 2 Example: x + 1 Seed 0 x = 0 0 x = 1 1 x = 2 x = 3 x = 4 x = 5 x = 6

  22. 2 Example: x + 1 Seed 0 x = 0 0 x = 1 1 x = 2 2 x = 3 x = 4 x = 5 x = 6

  23. 2 Example: x + 1 Seed 0 x = 0 0 x = 1 1 x = 2 2 x = 5 3 x = 4 x = 5 x = 6

  24. 2 Example: x + 1 Seed 0 x = 0 0 x = 1 1 x = 2 2 x = 5 3 x = 26 4 x = 5 x = 6

  25. 2 Example: x + 1 Seed 0 x = 0 0 x = 1 1 x = 2 2 x = 5 3 x = 26 4 x = big 5 x = 6

  26. 2 Example: x + 1 Seed 0 x = 0 0 x = 1 1 x = 2 2 “Orbit tends to infinity” x = 5 3 x = 26 4 x = big 5 x = BIGGER 6

  27. 2 Example: x + 0 Seed 0 x = 0 0 x = 1 x = 2 x = 3 x = 4 x = 5 x = 6

  28. 2 Example: x + 0 Seed 0 x = 0 0 x = 0 1 x = 2 x = 3 x = 4 x = 5 x = 6

  29. 2 Example: x + 0 Seed 0 x = 0 0 x = 0 1 x = 0 2 x = 3 x = 4 x = 5 x = 6

  30. 2 Example: x + 0 Seed 0 x = 0 0 x = 0 1 x = 0 2 x = 0 3 x = 4 x = 5 x = 6

  31. 2 Example: x + 0 Seed 0 x = 0 0 x = 0 1 x = 0 2 “A fixed point” x = 0 3 x = 0 4 x = 0 5 x = 0 6

  32. 2 Example: x - 1 Seed 0 x = 0 0 x = 1 x = 2 x = 3 x = 4 x = 5 x = 6

  33. 2 Example: x - 1 Seed 0 x = 0 0 x = -1 1 x = 2 x = 3 x = 4 x = 5 x = 6

  34. 2 Example: x - 1 Seed 0 x = 0 0 x = -1 1 x = 0 2 x = 3 x = 4 x = 5 x = 6

  35. 2 Example: x - 1 Seed 0 x = 0 0 x = -1 1 x = 0 2 x = -1 3 x = 4 x = 5 x = 6

  36. 2 Example: x - 1 Seed 0 x = 0 0 x = -1 1 x = 0 2 x = -1 “A two- cycle” 3 x = 0 4 x = -1 5 x = 0 6

  37. 2 Example: x - 1.1 Seed 0 x = 0 0 x = 1 x = 2 x = 3 x = 4 x = 5 x = 6

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