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Fractals with a Special Look at Sierpinskis Triangle By Carolyn Costello

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What is a Fractal? Self-Similar Recursive definition Non-Integer Dimension Euclidean Geometry can not explain Fine structure of arbitrarily small scale

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Types of Fractals Iterated Function Systems Escape-Time Random Strange Attractor

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Iterated Function System Fixed geometric replacement rule Sierpinskis Triangle (below) by continuously removing the medial triangle Koch Curve (right) by continuously removing the middle 1/3 and replacing with two segments of equal length to the piece removed

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Escape - Time Formula applied to each point in space. Mandelbrot Set start with two complex numbers, z n and c, then follow this formula, z n+1 =z n +c and keeping it bounded

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Random created by adding randomness through probability and statistical distributions. Brownian motion the random movement of particles suspended in a fluid (liquid or gas).

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Strange Attractor start with some original point on a plane or in space, then calculate every next point using a formula and the coordinates of the current point Lorenzos attractor use these three equations: dx / dt = 10(y - x), dy / dt = 28x – y – xz, dz / dt = xy – 8/3 y.

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What is the dimension? How do you know? Line Square Cube Scale factor Magnification Factor Number of self-similar Dimension Line ½ 1 1/31/3 1 ¼ 1 Square ½ 2 1/31/3 2 ¼ 2 1/51/5 2 Cube ½ 3 1/31/3 3 ¼ 3 1/51/5 3

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What is the dimension? How do you know? Line Square Cube Scale factor Magnification Factor Number of self-similar Dimension Line ½ 21 1/31/3 31 ¼ 41 Square ½ 42 1/31/3 92 ¼ 162 1/51/5 252 Cube ½ 83 1/31/3 273 ¼ 643 1/51/5 1253

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What is the dimension? How do you know? Scale factor Magnification Factor Number of self-similar Dimension Line ½ 221 1/31/3 331 ¼ 441 Square ½ 242 1/31/3 392 ¼ 4162 1/51/5 5252 Cube ½ 283 1/31/3 3273 ¼ 4643 1/51/5 51253 Line Square Cube

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Dimension N= number of self- similar pieces m = magnification factor d = dimension N = m d log N = log m d log N = d log m log N D= log m

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Dimension of the Sierpinski Triangle Log of the number of self-similar pieces Dimension= Log of the magnification factor

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Dimension of the Sierpinski Triangle = Log 3 Log 2 1.585 Log of the number of self-similar pieces Dimension= Log of the magnification factor

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Sierpinskis Triangle Generated using a linear transformation start at the origin x n+1 = 0.5x n and y n+1 =0.5y n x n+1 = 0.5x n + 0.5 and y n+1 =0.5y n + 0.5 x n+1 = 0.5x n + 1 and y n+1 =0.5y n

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Sierpinskis Triangle Chaos Game The game starts with a triangle where each of the vertices are labeled differently, a die whose sides are marked with the labels of the vertices (two each) and a marker to be moved. Place the marker anywhere inside the triangle, then roll the die. Move the marker half the distance toward the vertex that appears on the die.

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Sierpinskis Triangle Pascals Triangle

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Sierpinskis Triangle Pascals Triangle mod 2

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Sierpinskis Triangle Pascals Triangle mod 3

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Sierpinskis Triangle Pascals Triangle mod 6

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