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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

Fractals

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

slide2

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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