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

Mandelbrot Orbit

z_{n+1} = z_n² + c

Individual orbits of the Mandelbrot iteration — not the set boundary, but the trajectories themselves.

The invisible paths that define the boundary of chaos

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

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(k₁+k₂+k₃+k₄)² = 2(k₁²+k₂²+k₃²+k₄²)

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

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

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

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