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

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Apollonian Gasket

(k₁+k₂+k₃+k₄)² = 2(k₁²+k₂²+k₃²+k₄²)

Descartes' Circle Theorem iterated — circles packed inside circles, the gaps filled to infinity.

Infinity nested inside a finite space

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

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

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