Calculus×Computing
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Fractals

Level SpecializationDifficulty ★★★★★Concept⌖ Open in the map

What is it?

Sets with detail at every scale and non-integer dimension. The Mandelbrot set — the cc for which z↦z2+cz \mapsto z^2 + c stays bounded from z0=0z_0 = 0 — is the most famous; strange attractors are fractal too.

Formulas

zk+1=zk2+c,z0=0,M={c∈ℂ:sup⁡k∣zk∣<∞}z_{k+1} = z_k^2 + c, \quad z_0 = 0, \qquad M = \{c \in \C : \sup_k |z_k| < \infty\}
D=lim⁡ε→0log⁡N(ε)log⁡(1/ε)D = \lim_{\varepsilon\to0}\frac{\log N(\varepsilon)}{\log(1/\varepsilon)}
box-counting dimension

Where it shows up in computing

  • Procedural generation and noise★★★★★frequentComputer graphics

    Fractal noise (fBm: octaves of Perlin noise) generates terrain, clouds and textures in games and films.

  • Media compression (JPEG, MP3, video)★★★★★historicalSignals, media and vision

    Fractal image compression (1990s) encoded images as self-similar maps; it was abandoned for transform coding.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

3D Computer graphics

∿ Signals, media and vision

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.

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