Scientific Calculation and Visualization

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William McKenzie
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1 Scientific Calculation and Visualization Topic Iteration Method for Fractal 2 Classical Electrodynamics

2 Contents A First Look at Quantum Physics. Fractals.2 History of Fractal.3 Iteration Method for Fractal.4 Reference

3 . Fractal Fractal: A complex geometric pattern presents self-similarity in that small details of its structure at every scale.

4 Fractal in Nature

5 Fractal Art

.2 History of Fractal In the 7th century: Gottfried Leibniz

Weierstrass proposed a function which exhibited self-similarity

6 .2 History of Fractal In the 7th century: Gottfried Leibniz researched the problem of recursive self-similarity In 872: Karl Weierstrass proposed a function which exhibited self-similarity Weierstrass function: f x a b x n where n n cos a b is a positive odd integer n n f x a cos b x witha 5. andb 5 In 883: Georg Cantor published the so-called Cantor set which also showed self-similarity n

$earliest fractal curves In 95:$ Waclaw Sierpinski construct the

7 In 94: Koch curve one of the earliest fractal curves In 95: Waclaw Sierpinski construct the famous triangle structure unit basis In 98: Gaston Julia constructed the patterns with self-similarity by associating with mapping complex numbers and iterative functions

8 In 967: Benoît Mandelbrot discussed the self-similar curves in the paper of How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension In 975: Benoît Mandelbrot illustrated the mathematical model for selfsimilar patterns and named the phenomenon of self-similarity as fractal

9 .3 Iteration Method for Fractal Representation of points in complex plane ŷ ŷ A x y C x, y, ŷ 3 3 B x, y 2 2 ˆx x3 x cos sinx2 x y3 y sin cos y2 y x3 x cos sinx2 x y3 y sin cos y2 y complex number: z x iy xy, x, y ˆx ˆx x y x y x y i i i i z x iy re re e ze 2 2 i z xiy x y i r cos isin re The vector z rotate counterclockwise through an angle z i ze 2

10 Im z z z x, y x, y z x, y Re z z x iy z2 x2 iy2 z3 x3 iy3 i z z z z e from eq. and eq i z3 z z2 z e 3 Arbitrary point z 3 on complex plane can related to other points z and z 2 according to eq.(3). For the system composed of many points which obey some regular rules, it is possible to represent any point z k by some other points z k-,z k-2, that is Fractal is the systems with highly regularity use iteration mathematical model to exhibit fractal structure z f z, z k k k2

11 Iteration Method for Fractal tree First iteration z 9 z z 7 z3 z5 z6 z8 z z2 z4 Second iteration i 2 z : initial point z, z9 e z9 : final point z z9 z, z2 z, z4 z 3 z 2 z 3 z, z z, z z i i , z z z z e z z z z e 5 segments in the "first" iteration of fractal tree: z z, k=,, 4 2k 2k+ initial point of each segment: z++k z2k final point of each segment: z9++k z2k+ z z z, z z, z z 3 2 z z z, z z, z z 3 k 9k k 2k k 4k k 5k 9k k 6k 5k 8k 5k 3k 4k 5k 4k 7 k 8k 9k 8k z z z z e z i 6 i 6 z z z e

12 Third iteration 25 segments in the "second" iteration of fractal tree: z z, k=,, 24 initial point of each segment: z final point of each segment: z ++5+k +2k 9++5+k z z +2k+ 2k 2k+ z z 3 z, z z, z z z 2 z 3 z, z z, z z z 5 k 95k 5k 25k 5k 45k 5k 55k 95k 5k 65k 55k 85k 55k 35k 45k 55k 45k 75k 85k 95k 85k z z z e z z z z e i 6 i 6

13 Im( z) Im( z) Re( z) Re( z) Im( z) Im( z) Re( z) Re( z).5

14 Extended examples: rice ears trees

15 Iteration Method for Koch Snowflake 2 i k Iteration equation: z z z z e, k, 2, k k k k ,, ,, ,, A A A A A,,

16 Im( z).6.4 Im( z).6.4 Im( z) Re( z) Re( z) Re( z) Im( z) Re( z).866 Click to play the animation

17 Koch curve snowflake in square snowflake in hexagon

18 .4 Reference [] [2] [3] [4] [5] [6] [7] [8] [9] []

Fractals: Self-Similarity and Fractal Dimension Math 198, Spring 2013

Fractals: Self-Similarity and Fractal Dimension Math 198, Spring 2013 Background Fractal geometry is one of the most important developments in mathematics in the second half of the 20th century. Fractals