Playing With Chaos
Transcript of Playing With Chaos
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Playing With ChaosFractals and Strange Attractors
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Recreational MathematicsPuzzles, games, etc.
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Algorithmic/Generative ArtDon't get too serious about the word “art”
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What is a fractal?
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A shape
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Made from other shapes
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Each part a copy of the whole
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Theoretically infinite
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Sierpinski Gasket
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Koch Curve
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Koch Curve
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Koch Snowflake
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Length
4.0
5.3
7.1
+1.3
+1.8
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1 iteration = length 4
20 iterations = length 946
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Infinite perimeter. Finite area.
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Infinite triangles. Zero area.
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Fractal tree
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Symmetry and Regularity
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Non-symmetry
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Irregularity
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Pythagorean Fractal
A
B
C
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How long is the coast of Britain?
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1 iteration = length 4
20 iterations = length 946
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How long is the coast of Britain?
It depends on the size of your measuring stick.
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The Richardson Effect
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Dimensions
Line:
Square:
1.0
2.0
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Fractal Dimensions
Line:
South Coast of Africa:
West Coast of Britain:
Koch Curve:
Sierpinski Gasket:
Square:
1.0
1.02
1.25
1.2619
1.5849
2.0
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Chaos Game
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Chaos Game
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Chaos Game
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Chaos Game
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Chaos Game
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Chaos Game
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Barnsley Fern
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IFS Tree
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Diffuse Limited Aggregation
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The Mandelbrot Set
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Benoit Mandelbrot
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Imaginary Number
i = ⎷-2
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Complex Numbers
Real Number + Imaginary Number
4 + 5i
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x
-y
y
-x
4, 2
-2, -1
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real
-imaginary
imaginary
-real
4, 2i
-2, -i
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The Mandelbrot Set
z1 = z2 + c
z and c are complex numbers
z begins as 0+0i
c is a point on the complex plane
iterate for each point on the plane
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● z goes above a certain value ● color it based on how fast it got there
● z stays in range ● color it black
Result of iteration:
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-r r
i
-i
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-r r
i
-i
c = -2, -i z = 0, 0i
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Strange Attractors
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Edward Lorenz
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The Lorenz Attractor
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Fractals in Nature
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Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line... Nature exhibits not simply a higher degree but an altogether different level of complexity.
- Benoit Mandelbrot
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Thank You
http://www.bit-101.com Twitter: @bit101
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