Hyperbolic geometry
For two thousand years, mathematicians tried to prove Euclid’s fifth postulate from the other four. They failed — because it is independent. Deny it and you get a consistent geometry where parallel lines diverge, triangles have less than 180°, and an entire infinite plane fits inside a circle. The Poincaré disk maps that plane into a finite disk, distorting distances so that the boundary represents infinity. Everything you see here is exact — the geodesics, the tessellations, the transformations. Only the scale lies.
d(z₁, z₂) = arcosh(1 + 2|z₁ − z₂|² / ((1 − |z₁|²)(1 − |z₂|²))) · Gaussian curvature K = −1
What is hyperbolic geometry?
Euclid’s Elements rests on five postulates. The fifth — the parallel postulate — says that through a point not on a line, there is exactly one parallel line. For over two millennia, mathematicians attempted to derive it from the other four. In the early 19th century, János Bolyai, Nikolai Lobachevsky, and Carl Friedrich Gauss independently realized the truth: the fifth postulate is independent. You can deny it and still get a perfectly consistent geometry.
In hyperbolic geometry, through a point not on a line, there are infinitely many lines that never intersect the original. The plane has constant negative Gaussian curvature — like a saddle at every point. Triangles have angle sums strictly less than 180°, and the area of a triangle is proportional to its angular defect: A = π − (α + β + γ). The circumference of a circle grows exponentially with radius, not linearly. There is more space here than Euclid imagined.
The Poincaré disk model
Henri Poincaré mapped the entire infinite hyperbolic plane into the interior of a finite disk. Points near the center are barely distorted; points near the edge are compressed enormously. The boundary circle represents infinity — you can never reach it. A figure moving toward the edge appears to shrink, but in hyperbolic terms it remains the same size. The model is conformal: angles are preserved exactly, so the shapes of small figures look right even as their sizes distort.
Geodesics — the “straight lines” of hyperbolic geometry — appear as circular arcs that intersect the boundary circle at right angles, or as diameters passing through the center. When you draw a geodesic between two points in this lab, the arc you see is the unique shortest path between them in hyperbolic space. It looks curved, but it is as straight as a line can be in this geometry.
Hyperbolic tessellations
In Euclidean geometry, only three regular tessellations exist: triangles {3,6}, squares {4,4}, and hexagons {6,3}. In hyperbolic geometry, infinitely many regular tessellations are possible. Any Schläfli symbol {p, q} with (p−2)(q−2) > 4 defines a valid tiling of the hyperbolic plane by regular p-gons, q meeting at each vertex.
M. C. Escher used hyperbolic tessellations in his Circle Limit series (1958–1960), working from correspondence with the geometer H.S.M. Coxeter. Escher’s Circle Limit III is based on the {6,4} tessellation — a pattern of interlocking fish that tile the hyperbolic plane, shrinking toward the boundary in exact conformal symmetry. The tiles look smaller near the edge, but they are all the same hyperbolic size.
Curvature and parallel lines
The key invariant is Gaussian curvature. A flat plane has curvature K = 0. A sphere of radius r has K = 1/r². The hyperbolic plane has K = −1 (for the standard model). This single number controls everything: the angle sum of triangles, the growth rate of circles, the number of parallel lines through a point.
On a sphere, there are no parallel lines — all great circles intersect. On the Euclidean plane, there is exactly one parallel through any external point. On the hyperbolic plane, there are infinitely many. Two of them are special — the limiting parallels that asymptotically approach the original line at infinity. All lines between them also fail to intersect, giving a continuous fan of non-intersecting lines through a single point.
Applications
Hyperbolic geometry is not merely a mathematical curiosity. In special relativity, the velocity addition of collinear boosts follows hyperbolic trigonometry — rapidities add linearly in hyperbolic space while velocities do not. The Minkowski hyperboloid model of the hyperbolic plane is the natural setting for Lorentz transformations.
In computer science, hyperbolic space is used to embed hierarchical data — trees and taxonomies — because hyperbolic space grows exponentially, matching the exponential branching of trees. Hyperbolic neural network embeddings outperform Euclidean ones for data with latent hierarchical structure.
In topology, Thurston’s geometrization conjecture (proved by Perelman in 2003) shows that most 3-manifolds decompose into pieces carrying one of eight model geometries, and hyperbolic geometry is by far the most common. Most of the universe of possible 3-dimensional shapes is hyperbolic.