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Fractals with a Special Look at Sierpinski’s Triangle

Fractals with a Special Look at Sierpinski’s 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

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Fractals with a Special Look at Sierpinski’s Triangle

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  1. Fractals with a Special Look at Sierpinski’s Triangle By Carolyn Costello

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

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

  4. Iterated Function System • Fixed geometric replacement rule • Sierpinski’s 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

  5. Escape - Time • Formula applied to each point in space. • Mandelbrot Setstart with two complex numbers, zn and c, then follow this formula, zn+1=zn +c and keeping it bounded

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

  7. 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 • Lorenzo’s attractor use these three equations: dx / dt = 10(y - x), dy / dt = 28x – y – xz, dz / dt = xy – 8/3 y.

  8. What is the dimension? How do you know? • Line • Square • Cube

  9. What is the dimension? How do you know? • Line • Square • Cube

  10. What is the dimension? How do you know? • Line • Square • Cube

  11. Dimension • N= number of self- similar pieces • m = magnification factor • d = dimension • N = md • logN = log md • log N = d log m log N D= log m

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

  13. 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

  14. Sierpinski’s Triangle • Generated using a linear transformation • start at the origin xn+1 = 0.5xn and yn+1=0.5ynxn+1 = 0.5xn + 0.5 and yn+1=0.5yn + 0.5xn+1 = 0.5xn + 1 and yn+1=0.5yn

  15. Sierpinski’s 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.

  16. Sierpinski’s Triangle • Pascal’s Triangle

  17. Sierpinski’s Triangle • Pascal’s Triangle mod 2

  18. Sierpinski’s Triangle • Pascal’s Triangle mod 3

  19. Sierpinski’s Triangle • Pascal’s Triangle mod 6

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