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Fractals

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

  2. Two important properties of a fractal F • F has detail at every level. • F is exactly, approximately or statistically self-similar.

  3. Koch Fractals (Snowflakes) 1/3 1/3 1 1/3 1/3 Generator Iteration 0 Iteration 1 Iteration 2 Iteration 3

  4. Fractal Tree Generator Iteration 1 Iteration 2 Iteration 3 Iteration 4 Iteration 5

  5. Fractal Fern Generator Iteration 0 Iteration 1 Iteration 2 Iteration 3

  6. Fractal Landscapes • fractals: not just for “showing math” • triangle subdivision • vertex displacement • recursive until termination condition http://www.fractal-landscapes.co.uk/images.html

  7. 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

  8. 1D: Midpoint Displacement • divide in half • randomly displace • scale variance by half http://www.gameprogrammer.com/fractal.html

  9. Mandelbrot set Demohttp://juliamap.googlelabs.com

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