Apollonian gasket

Apollonius of Perga; Descartes, 1643

Three circles touching each other leave curved gaps; fill each gap with the largest circle that fits, and keep going. Descartes found the rule linking the bends of four touching circles. Start from bends −1, 2, 2, 3 and every circle in the gasket has a whole-number bend, written on the larger ones.

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