Fractals
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Fractals. Two important properties of a fractal F F has detail at every level. F is exactly, approximately or statistically self-similar. Koch Fractals (Snowflakes). 1/3. 1/3. 1. 1/3. 1/3. Generator. Iteration 0. Iteration 1. Iteration 2. Iteration 3. Fractal Tree. Generator.

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Fractals

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Fractals

Fractals

http://www.ugrad.cs.ubc.ca/~cs314/Vjan2010


Fractals

  • Two important properties of a fractal F

  • F has detail at every level.

  • F is exactly, approximately or statistically self-similar.


Koch fractals snowflakes

Koch Fractals (Snowflakes)

1/3

1/3

1

1/3

1/3

Generator

Iteration 0

Iteration 1

Iteration 2

Iteration 3


Fractal tree

Fractal Tree

Generator

Iteration 1 Iteration 2 Iteration 3 Iteration 4 Iteration 5


Fractal fern

Fractal Fern

Generator

Iteration 0 Iteration 1 Iteration 2 Iteration 3


Fractal landscapes

Fractal Landscapes

  • fractals: not just for “showing math”

    • triangle subdivision

    • vertex displacement

    • recursive until termination condition

http://www.fractal-landscapes.co.uk/images.html


Fractal dimension

D = log(N)/log(r) N = measure, r = subdivision scale

Hausdorff dimension: noninteger

Fractal Dimension

  • Koch snowflake

coastline of Britain

D = log(N)/log(r) D = log(4)/log(3) = 1.26

http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/Fractals.html


1d midpoint displacement

1D: Midpoint Displacement

  • divide in half

  • randomly displace

  • scale variance by half

http://www.gameprogrammer.com/fractal.html


Mandelbrot set

Mandelbrot set

Demohttp://juliamap.googlelabs.com


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