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Explore the Fractal Geometry of the Mandelbrot Set

Dive into the intriguing mathematics behind the Mandelbrot Set, understand orbits, seed iteration, and infinity tendencies. Start learning now!

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Explore the Fractal Geometry of the Mandelbrot Set

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  1. Excel quad iteration M-set iterator Movie maker 75

  2. The Fractal Geometry of the Mandelbrot Set How the computer has revolutionized mathematics

  3. The Fractal Geometry of the Mandelbrot Set You need to know:

  4. The Fractal Geometry of the Mandelbrot Set You need to know: How to count

  5. The Fractal Geometry of the Mandelbrot Set You need to know: How to count How to add

  6. Many people know the pretty pictures...

  7. but few know the even prettier mathematics.

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

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

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

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

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

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

  14. 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.

  15. Start with a function: 2 x + constant

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

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

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

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

  20. 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

  21. 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.

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

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

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

  26. 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

  27. 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

  28. 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 = BIGGER 6

  29. 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

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

  31. 2 Example: x + 0 Seed 0 x = 0 0 x = 0 1 x = 2 x = 3 x = 4 x = 5 x = 6

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

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

  34. 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

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

  36. 2 Example: x - 1 Seed 0 x = 0 0 x = -1 1 x = 2 x = 3 x = 4 x = 5 x = 6

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

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