fractals with a special look at sierpinski’s triangle
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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 - PowerPoint PPT PresentationTRANSCRIPT
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
• 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
1/53
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
1/5125 3
What is the dimension? How do you know?
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
1/55 125 3
• Line
• Square
• Cube
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 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
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