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

Harmonograph

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

Two damped pendulums coupled by a drawing surface — the pattern emerges from the interaction of decay rates.

Beauty emerging from the interaction of simple motions

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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)

Mandelbrot Orbit — z_{n+1} = z_n² + c

Mandelbrot Orbit

z_{n+1} = z_n² + c

Recursion — f(n) = f(f(n-1))

Recursion

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

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