Descartes' Circle Theorem
(k₁ + k₂ + k₃ + k₄)² = 2 (k₁² + k₂² + k₃² + k₄²)
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Four mutually tangentEvery pair touches each other at exactly one point. circles always satisfy this equation, where kCurvature — one over the radius. The outer wrapping circle gets a negative k because it bends opposite to the inside circles. = 1 / radius.
4-vector
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k₄ is the outer wrapping circle (negative). Move the sliders — the LHS and RHS above always stay equal, that's the theorem at work.
Soddy swap reveal
Click any circle on the canvas to see how its curvature is computed from its parent triple.
A Soddy swapThree mutually-tangent circles have exactly two circles tangent to all three — one in the gap, one wrapping around. Replacing one of four mutually-tangent circles with the other valid fourth is a Soddy swap. swaps one of four mutually-tangent circles for its alternate. Vieta's formulaThe algebraic relation between a quadratic's coefficients and its roots. Here it makes the swap a single subtraction: k_new = 2(k_a + k_b + k_c) − k_old. turns it into a single subtraction — that's the engine driving the recursion.
Integer curvatures
Set k₁, k₂, k₃ to integers to look for an integer gasket.
Seeds
integer
fibonacci
other
Integer-row seeds produce gaskets where every curvature is a whole number. Click any to load.