Recursion — f(n) = f(f(n-1))

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Recursion

f(n) = f(f(n-1))

A function that calls itself — each level of depth revealing new structure, the whole containing copies of itself.

The whole containing copies of itself at every scale

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Apollonian Gasket — (k₁+k₂+k₃+k₄)² = 2(k₁²+k₂²+k₃²+k₄²)

Apollonian Gasket

(k₁+k₂+k₃+k₄)² = 2(k₁²+k₂²+k₃²+k₄²)

$Harmonograph — x = A₁sin(f₁t+φ₁)e^{-d₁t} + A₂sin(f₂t+φ₂)e^{-d₂t}$

Harmonograph

x = A₁sin(f₁t+φ₁)e^{-d₁t} + A₂sin(f₂t+φ₂)e^{-d₂t}

$Mandelbrot Orbit — z_{n+1} = z_n² + c$

Mandelbrot Orbit

z_{n+1} = z_n² + c

Strange Attractor — dx/dt = σ(y-x), dy/dt = x(ρ-z)-y, dz/dt = xy-βz

Strange Attractor

dx/dt = σ(y-x), dy/dt = x(ρ-z)-y, dz/dt = xy-βz