Playing With Chaos

Playing With ChaosFractals and Strange Attractors

Recreational MathematicsPuzzles, games, etc.

Algorithmic/Generative ArtDon't get too serious about the word “art”

What is a fractal?

A shape

Made from other shapes

Each part a copy of the whole

Theoretically infinite

Sierpinski Gasket

Koch Curve

Koch Curve

Koch Snowflake

Length

4.0

5.3

7.1

+1.3

+1.8

1 iteration = length 4

20 iterations = length 946

Infinite perimeter. Finite area.

Infinite triangles. Zero area.

Fractal tree

Symmetry and Regularity

Non-symmetry

Irregularity

Pythagorean Fractal

A

B

C

How long is the coast of Britain?

1 iteration = length 4

20 iterations = length 946

How long is the coast of Britain?

It depends on the size of your measuring stick.

The Richardson Effect

Dimensions

Line:

Square:

1.0

2.0

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

Chaos Game

Chaos Game

Chaos Game

Chaos Game

Chaos Game

Chaos Game

Barnsley Fern

IFS Tree

Diffuse Limited Aggregation

The Mandelbrot Set

Benoit Mandelbrot

Imaginary Number

i = ⎷-2

Complex Numbers

Real Number + Imaginary Number

4 + 5i

x

-y

y

-x

4, 2

-2, -1

real

-imaginary

imaginary

-real

4, 2i

-2, -i

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

● 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:

-r r

i

-i

-r r

i

-i

c = -2, -i z = 0, 0i

Strange Attractors

Edward Lorenz

The Lorenz Attractor

Fractals in Nature

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

Thank You

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