This document discusses fractals and chaos theory. It begins by defining fractals as shapes made of parts that are copies of the whole, with theoretically infinite detail and self-similarity regardless of scale. Examples given include the Sierpinski gasket, Koch curve, and fractal trees. Fractals can have infinite perimeter but finite area. The concept of fractal dimensions is introduced, where fractal objects have non-integer dimensions. Chaos games and the Mandelbrot set are discussed as examples of chaos theory and fractals found in nature. Benoit Mandelbrot is credited with bringing fractals and their ability to model irregular natural forms to the mainstream.

1. Playing With Chaos
Fractals and Strange Attractors

2. Recreational Mathematics
Puzzles, games, etc.

3. Algorithmic/Generative Art
Don't get too serious about the word “art”

4. What is a fractal?

5. A shape

6. Made from other shapes

7. Each part a copy of the whole

8. Theoretically infinite

9. Sierpinski Gasket

10. Koch Curve

13. Koch Curve

14. Koch Snowflake

15. Length
4.0
5.3
7.1
+1.3
+1.8

16. 1 iteration = length 4
20 iterations = length 946

17. Infinite perimeter. Finite area.

18. Infinite triangles. Zero area.

19. Fractal tree

23. Symmetry and Regularity

24. Non-symmetry

25. Irregularity

26. Pythagorean Fractal
A
B
C

30. How long is the coast of Britain?

32. 1 iteration = length 4
20 iterations = length 946

33. How long is the coast of Britain?
It depends on the size of your measuring stick.

34. The Richardson Effect

35. Dimensions
Line:
Square:
1.0
2.0

36. Fractal Dimensions
Line:
South Coast of Africa:
West Coast of Britain:
Koch Curve:
Sierpinski Gasket:
Square:
1.0
1.02
1.25
1.2619
1.5849
2.0

37. Chaos Game

38. Chaos Game

39. Chaos Game

40. Chaos Game

41. Chaos Game

42. Chaos Game

43. Barnsley Fern

45. IFS Tree

46. Diffuse Limited Aggregation

47. The Mandelbrot Set

48. Benoit Mandelbrot

49. Imaginary Number
i = ⎷-2

50. Complex Numbers
Real Number + Imaginary Number
4 + 5i

51. x
-y
y
-x
4, 2
-2, -1

52. real
-imaginary
imaginary
-real
4, 2i
-2, -i

53. The Mandelbrot Set
z1 = z2 + c
z and c are complex numbers
z begins as 0+0i
c is a point on the complex plane
iterate for each point on the plane

54. ● z goes above a certain value
● color it based on how fast it got there
● z stays in range
● color it black
Result of iteration:

55. -r r
i
-i

56. -r r
i
-i
c = -2, -i
z = 0, 0i

58. Strange Attractors

59. Edward Lorenz

60. The Lorenz Attractor

61. Fractals in Nature

71. 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... Nature exhibits not simply a
higher degree but an altogether different
level of complexity.
- Benoit Mandelbrot

72. Thank You
http://www.bit-101.com
Twitter: @bit101

73. http://playingwithchaos.net