Fractals with a Special Look at Sierpinski

1
Fractals with a Special Look at Sierpinskis
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
• 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

5
Escape - Time

• Formula applied to each point in space.
• Mandelbrot Set
  start with two complex numbers, zn
  and c, then follow this formula, zn1zn 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
• Lorenzos 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

Scale factor Magnification Factor Number of self-similar Dimension
Line ½ 1
1/3 1
¼ 1
Square ½ 2
1/3 2
¼ 2
1/5 2
Cube ½ 3
1/3 3
¼ 3
1/5 3
9
What is the dimension? How do you know?

• Line
• Square
• Cube

Scale factor Magnification Factor Number of self-similar Dimension
Line ½ 2 1
1/3 3 1
¼ 4 1
Square ½ 4 2
1/3 9 2
¼ 16 2
1/5 25 2
Cube ½ 8 3
1/3 27 3
¼ 64 3
1/5 125 3
10
What is the dimension? How do you know?

• Line
• Square
• Cube

Scale factor Magnification Factor Number of self-similar Dimension
Line ½ 2 2 1
1/3 3 3 1
¼ 4 4 1
Square ½ 2 4 2
1/3 3 9 2
¼ 4 16 2
1/5 5 25 2
Cube ½ 2 8 3
1/3 3 27 3
¼ 4 64 3
1/5 5 125 3
11
Dimension

• N number of self- similar pieces
• m magnification factor
• d dimension
• N md
• log N log md
• log N d log m
• log N
• D log m

12

• 
  Log of the number of self-similar pieces
• Dimension Log of the magnification factor

• Dimension of the
• Sierpinski Triangle

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
Sierpinskis Triangle

• Generated using a linear transformation
• start at the origin
• xn1 0.5xn and yn10.5yn
  
  xn1 0.5xn 0.5 and
  yn10.5yn 0.5
  xn1 0.5xn
  1 and yn10.5yn

15
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.

16
Sierpinskis Triangle

• Pascals Triangle

17
Sierpinskis Triangle

• Pascals Triangle mod 2

18
Sierpinskis Triangle

• Pascals Triangle mod 3

19
Sierpinskis Triangle

• Pascals Triangle mod 6