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Fractal Geometry

Fractal Geometry. IMO Mathematics Camp 20 August 2000 Leung-yiu chung S.K.H. Bishop Mok Sau Tseng Secondary School. Table of Contents. What is Fractal Examples of Fractals Properties of Fractals Two Famous Sets about Fractals -- Mandelbrot Set and Julia Set

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Fractal Geometry

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  1. 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

  3. 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,….

  4. Ferns (Leaf)

  5. Snowflakes

  6. Sierpinski Triangle • Steps for Construction • Questions

  7. 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?

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

  9. 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.

  10. 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.

  11. 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

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

  13. 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

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

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

  16. Koch Snowflake • Step One. • Start with a large equilateral triangle.

  17. 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.

  18. Amazing Phenomenon • Perimeter • Area

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

  20. Perimeter = 9 units

  21. Perimeter = ___ units

  22. Perimeter = 12 units

  23. Perimeter = ___ units

  24. Perimeter = 16 units

  25. Infinite iterations • What is the perimeter ?

  26. Area • Original Area : ? • After the 1st iteration : ?

  27. Area 2nd Iteration :

  28. Calculation

  29. Anti-Snowflake

  30. Fractal Properties • Self-Similarity • Fractional Dimension • Formation by iteration

  31. 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

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

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

  34. 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.

  35. 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.

  36. 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.

  37. 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

  38. Dimension Figure Dimension No. of copies Line 1 2=21 Square 2 4=22 Cube 3 8=23 Doubling Simliarity d n=2d

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

  40. Dimension Figure Dimension No. of copies Line 1 2=21 Sierpinski’s ? 3=2? Square 2 4=22 Cube 3 8=23 Doubling Simliarity d n=2d

  41. Dimension • Dimension for Sierpinski”s Triangle • d = log 3 / log 2 = 2.1435…...

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