1. S.K.H. Bishop Mok Sau Tseng Secondary School
Fractal Geometry
IMO Mathematics Camp
20 August 2000
Leung-yiu chung
S.K.H. Bishop Mok Sau Tseng Secondary School

2. Table of Contents What is Fractal Examples of Fractals
Properties of Fractals
Two Famous Sets about Fractals -- Mandelbrot Set and Julia Set
Applications and Recent Developments

4. Coastline Measure with a mile-long ruler
Measure with a foot-long ruler
Measure with a inch-long ruler
Any difference?
The measurement will be longer,longer,….

5. Ferns (Leaf)

6. Snowflakes

10. Sierpinski Triangle
Steps for Construction
Questions

11. Step One
Draw an equilateral triangle with sides of 2 triangle lengths each.
Connect the midpoints of each side.
How many equilateral triangles do you now have?

12. Shade out the triangle in the center
Shade out the triangle in the center. Think of this as cutting a hole in the triangle.

13. Step Two
Draw another equilateral triangle with sides of 4 triangle lengths each. Connect the midpoints of the sides and shade the triangle in the center as before.

14. Step Three
Draw an equilateral triangle with sides of 8 triangle lengths each. Follow the same procedure as before, making sure to follow the shading pattern. You will have 1 large, 3 medium, and 9 small triangles shaded.

15. Step Four
How about doing this one on a poster board? Follow the above pattern and complete the Sierpinski Triangle.
Use your artistic creativity and shade the triangles in interesting color patterns. Does your figure look like this one?
Back

16. Question 1
in Step One. What fraction of the triangle did you NOT shade?
Back to step One

17. Question 2 What fraction of the triangle in Step Two is NOT shaded?
What fraction did you NOT shade in the Step Three triangle?
Back to Step Two

18. Question 3
Use the pattern to predict the fraction of the triangle you would NOT shade in the Step Four Triangle.

19. Question 4
CHALLENGE:
Develop a formula so that you could calculate the fraction of the area which is NOT shaded for any step

20. Koch Snowflake
Step One.
Start with a large equilateral triangle.

21. Koch Snowflake Step Two
1.Divide one side of the triangle into three parts and remove the middle section.
2.Replace it with two lines the same length as the section you removed.
3.Do this to all three sides of the triangle.
Do it again and again.

27. Amazing Phenomenon
Perimeter
Area

28. Perimeter
In Step One, the original triangle is an equilateral triangle with sides of 3 units each.

29. Perimeter = 9 units

30. Perimeter = ___ units

31. Perimeter = units

32. Perimeter = ___ units

33. Perimeter = 16 units

34. Infinite iterations
What is the perimeter ?

35. Area
Original Area : ?
After the 1st iteration : ?

36. Area
2nd Iteration :

37. Calculation

38. Anti-Snowflake

39. Fractal Properties Self-Similarity Fractional Dimension
Formation by iteration

40. Self-Similarity
To the right is the Sierpinski Triangle that we make in this unit. Notice that the outline of the figure is an equilateral triangle. Now look inside at all the equilateral triangles. Remember that there are infinitely many smaller and smaller triangles inside. How many different sized triangles can you find? All of these are similar to each other and to the original triangle - self similarity

41. If the red image is the original figure, how many similar copies of it are contained in the blue figure?

42. Fractional Dimension Point --- No dimension Line --- One dimension
Plane --- Two dimensions
Space --- Three dimensions

43. A definition for Dimension
Take a self-similar figure like a line segment, and double its length. Doubling the length gives two copies of the original segment.

44. Take another self-similar figure, this time a square 1 unit by 1 unit.
Now multiply the length and width by 2. How many copies of the original size square do you get? Doubling the sides gives four copies.

45. Take a 1 by 1 by 1 cube and double its length, width, and height.
How many copies of the original size cube do you get? Doubling the side gives eight copies.

46. Table for comparison
when we double the sides and get a similar figure, we write the number of copies as a power of 2 and the exponent will be the dimension
Table

47. Dimension Figure Dimension No. of copies Line 1 2=21 Square 2 4=22
Cube =23
Doubling Simliarity d n=2d

48. Sierpinski Triangle
How many copies shall we get after doubling the side?

49. Dimension Figure Dimension No. of copies Line 1 2=21
Sierpinski’s ? =2?
Square =22
Cube =23
Doubling Simliarity d n=2d

50. Dimension Dimension for Sierpinski”s Triangle
d = log 3 / log 2 = …...

51. Iterative Process on a Function
Mandelbrot Set
Julia Set

54. Mandelbrot Set Iterative Function : f(z) = z2 +C
For each complex number C, start z from 0,
f(0)=C, f(f(0)), f(f(f(0))),….
C belongs to Mandelbrot set if and only if
the sequence of iteration converges

55. Calculation and Coloring
For each point in a designated region in Complex Plane, construct the sequence of iteration

56. Calculation and Coloring
Determination of Convergence:
One can prove that
if the distance of the point from the origin become greater than 2, it will grow to infinity
For certain iteration, the value of the term >2, then the sequence grows to infinity--> out of Mandelbrot Set

57. Calculating and Coloring
Within certain iterations (e.g. 200), if all the terms have the magnitudes not greater than 2, then the sequence is assumed to be convergent.
The corresponding value of C should be regarded as an element of Mandelbrot Set and colored in black.

58. Calculating and Coloring
For other value of C, the point will be colored depending on the number of iteration needed to have a term with magnitude greater than 2.
If the colors used are chosen in such a way that the no. of iterations follows a certain spectrum of color, we may get fantastic pictures.

59. Mandelbrot Set
Animation

62. Julia Set
The Process to get Julia Set is similar to that for Mandelbrot Set except
In Mandelbrot Set, C corresponds to an element and we start from z= 0
In Julia Set, C is pre-fixed and the varies z for starting value, the z for convergence corresponds to an element.

63. Julia Set
Animation

65. Applications Fractal Gallery -- Simulate Natural Behaviors
Fractal Music
Fractal Landscape
Application on Chaotic System
Damped pendulum problems
Image Compression -- highly jaggly texture

66. Useful Resources Websites: http://math.rice.edu/~lanius/frac/
Search on Yahoo.com
key word : Fractal Geometry, Fractal, Mandelbrot

67. Softwares Java Applets from Websites
Freeware/Shareware downloaded from Internet:
Fractal Browers
Fantastic Fractals 98