Download 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
• Random
• Strange Attractor
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
Escape - Time
• Formula applied to each point in space.
• Mandelbrot Set
start with two complex numbers, zn and c, then follow this formula, zn+1=zn +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
• Lorenzo’s 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
Scale
factor
Line
• Square
Square
• Cube
Cube
Magnification
Factor
Number of
self-similar
Dimension
½
1
1/
3
1
¼
1
½
2
1/
3
2
¼
2
1/
5
2
½
3
1/
3
3
¼
1/
5
3
3
What is the dimension?
How do you know?
• Line
Scale
factor
Line
• Square
Square
• Cube
Cube
Number of
self-similar
Dimension
½
2
1
1/
3
3
1
¼
4
1
½
4
2
1/
3
9
2
¼
16
2
1/
5
25
2
½
8
3
1/
3
27
3
¼
64
3
125
3
1/
5
Magnification
Factor
What is the dimension?
How do you know?
• Line
Line
• Square
Square
• Cube
Cube
Scale
factor
Magnification
Factor
Number of
self-similar
Dimension
½
2
2
1
1/
3
3
3
1
¼
4
4
1
½
2
4
2
1/
3
3
9
2
¼
4
16
2
1/
5
5
25
2
½
2
8
3
1/
3
3
27
3
¼
4
64
3
5
125
3
1/
5
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
Dimension=
Log of the number of self-similar pieces
Log of the magnification factor
Dimension of the
Sierpinski Triangle
Dimension=
Log of the number of self-similar pieces
Log of the magnification factor
Dimension of the
Sierpinski Triangle
= Log 3
Log 2
≈ 1.585
Sierpinski’s Triangle
• Generated using a linear transformation
• start at the origin
xn+1 = 0.5xn and yn+1=0.5yn
xn+1 = 0.5xn + 0.5 and yn+1=0.5yn + 0.5
xn+1 = 0.5xn + 1 and yn+1=0.5yn
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.
Sierpinski’s Triangle
• Pascal’s Triangle
Sierpinski’s Triangle
• Pascal’s Triangle mod 2
Sierpinski’s Triangle
• Pascal’s Triangle mod 3
Sierpinski’s Triangle
• Pascal’s Triangle mod 6