1 / 351

Exploring the Intriguing Mathematics of the Mandelbrot Set

Discover the beauty and complexity of the Mandelbrot Set through fractal geometry and orbit calculations; learn to interpret the fate of orbits and fascinating mathematical patterns.

djanice
Download Presentation

Exploring the Intriguing Mathematics of the Mandelbrot Set

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related