From networks to nature — hyperbolic geometry in the real world
Far from a mathematical curiosity, hyperbolic geometry appears throughout science and technology. Hierarchical networks — the internet, social graphs, biological phylogenies — embed naturally in hyperbolic space because its exponential area growth matches exponential branching. Nature builds hyperbolic surfaces in coral reefs, lettuce leaves, and sea slugs. Einstein's special relativity uses the Klein model for velocity addition. And immersive VR flythroughs of tessellated H3 turn abstract math into visceral experience.
This page showcases four application domains: network embedding, natural forms, immersive exploration, and the deep connection to special relativity.
A binary tree of depth 4 is embedded in the Poincare disk using a radial layout where each deeper level is placed closer to the boundary. Edges are drawn as geodesic arcs. Compare with the Euclidean layout on the right, where the same tree becomes crowded and overlapping — hyperbolic space naturally has room for exponentially many nodes.
Try it: Compare the two layouts. Notice how the hyperbolic embedding keeps every node visible while the Euclidean one collapses leaf nodes into a dense band.
Organisms across the tree of life have independently discovered hyperbolic geometry. Coral reefs maximize surface area with ruffled forms, leaf edges curl into saddle shapes, and sea slugs sport elegant hyperbolic frills. Each card below describes an example and visualizes the characteristic ruffled edge.
Reef-building corals maximize surface area for nutrient absorption by growing in hyperbolic forms. The ruffled, folded shapes of brain coral and lettuce coral are physical realizations of negative Gaussian curvature.
The curly edges of lettuce, kale, and other leafy greens arise because the leaf margin grows faster than the interior. This differential growth rate forces the edge into a hyperbolic surface, creating the characteristic ruffled shape.
Many nudibranch species sport elaborate, ruffled cerata and mantles that follow hyperbolic geometry. These structures increase surface area for gas exchange while creating stunning visual displays for camouflage and warning.
Fly through a tessellated region of hyperbolic 3-space (H3) using WASD keys and mouse drag. A raymarching fragment shader traces geodesics through the hyperboloid model, rendering the cell structure in real time. The exponential volume growth of H3 means the corridor ahead seems to stretch on forever.
Click to Activate
WASD to move, drag to look around
Try it: Click the viewport to activate. Use W/S for forward/backward, A/D for strafe, and drag the mouse to look around. Adjust the speed slider to navigate faster or slower.
Einstein's velocity addition formula is hyperbolic geometry in disguise. In the Klein disk model, a velocity v corresponds to a point at distance |v|/c from the center. Composing two velocities is a hyperbolic translation, and the result always stays inside the disk (|v| < c). Rapidity — the hyperbolic angle — is additive, while velocity is not.
The Klein disk represents velocity space where the boundary is the speed of light c. Rapidity φ = atanh(|v|/c) is the hyperbolic distance from the origin. For collinear velocities, rapidity is additive (φ1 + φ2 = φrel) while velocity is not (— it follows the formula vrel = (v1 + v2) / (1 + v1v2/c2)).