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Playing With Chaos

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Playing With Chaos Fractals and Strange Attractors
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Page 1: Playing With Chaos

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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http://playingwithchaos.net


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