Hyperbolic Gardens (Digital Edition)

Kleinian groups are discrete groups of Möbius transformations of the Riemann sphere — and the orbits of points under these groups produce some of the most intricate geometric objects in mathematics. Apollonian circle packings, the limit case where the group fills space with nested tangent circles, were studied by Apollonius of Perga around 200 BCE and rediscovered as a deep topic in hyperbolic geometry in the 20th century. The limit sets of more complex Kleinian groups produce dendritic, foam-like, and lace-like structures that tile the hyperbolic plane. Every image and animation in this collection is computed by iterating actual Möbius transformations and inversions, with the resulting circle packings rendered at full mathematical precision.