# Fractals with a Special Look at Sierpinskis Triangle By Carolyn Costello.

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Fractals with a Special Look at Sierpinskis Triangle By Carolyn Costello

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

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

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

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

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

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.

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

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

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

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

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

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

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

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.

Sierpinskis Triangle Pascals Triangle

Sierpinskis Triangle Pascals Triangle mod 2

Sierpinskis Triangle Pascals Triangle mod 3

Sierpinskis Triangle Pascals Triangle mod 6

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