How it Works
The Poincaré disk model maps the entire hyperbolic plane into the open unit disk. We construct a central fundamental domain — a regular p-gon with interior angles 2π/q — then apply hyperbolic reflections (Möbius transformations preserving the unit disk) to propagate it outward. Each reflection maps the disk to itself and generates a new tile.
The fundamental tile is drawn using geodesic arcs. In the Poincaré disk, geodesics are circular arcs orthogonal to the boundary circle, or diameters. We draw each polygon edge as such an arc using standard Euclidean arc-drawing, after computing the center and radius of the corresponding circle in ℂ.
Möbius isometry: T_a(z) = (z - a) / (1 - āz), |a| < 1
Tile area: A = (p-2)π - p · (2π/q) = π((p-2) - 2p/q)
The Schläfli symbol {p,q} specifies a tiling where p-gons meet q per vertex. The condition for hyperbolic tiling is (p-2)(q-2) > 4 (equivalently, 1/p + 1/q < 1/2). If 1/p + 1/q = 1/2 we get a Euclidean tiling; if 1/p + 1/q > 1/2 we get a spherical tiling (Platonic solids).
Frequently Asked Questions
What is hyperbolic geometry?
Hyperbolic geometry is a non-Euclidean geometry where the parallel postulate fails: through a point not on a line, infinitely many lines pass that are parallel to the given line. It has constant negative curvature, unlike the flat Euclidean plane or the positive curvature of a sphere.
What is the Poincaré disk model?
The Poincaré disk model represents the entire hyperbolic plane inside the unit disk. Points inside are hyperbolic points; the boundary is "at infinity". Geodesics appear as circular arcs meeting the boundary at right angles, or as diameters of the disk.
What is a {p,q} tiling?
A {p,q} tiling (Schläfli symbol) places regular p-gons around each vertex with q polygons meeting there. The tiling is hyperbolic when (p-2)(q-2) > 4, Euclidean when equal to 4, and spherical when less than 4. For example {7,3} places heptagons with 3 meeting at each vertex.
What are Möbius transformations?
A Möbius transformation is a map f(z) = (az+b)/(cz+d) on the complex plane. The orientation-preserving isometries of the hyperbolic disk are exactly the Möbius transformations mapping the unit disk to itself, forming the group PSU(1,1).
Why do tiles near the boundary look smaller?
In the Poincaré disk model, the hyperbolic metric is ds = 2|dz|/(1-|z|²). Near the boundary |z|→1, the denominator approaches zero, so Euclidean distances appear compressed. All tiles have equal hyperbolic area — the distortion is an artifact of representing negative-curvature space in flat coordinates.
What is the hyperbolic area of a polygon?
The Gauss-Bonnet theorem for hyperbolic space gives the area of an n-gon as: Area = (n-2)π - Σθᵢ, where θᵢ are the interior angles. For a {p,q} tile: Area = π((p-2) - 2p/q).
Who discovered hyperbolic geometry?
Hyperbolic geometry was discovered independently by János Bolyai (1832), Nikolai Lobachevsky (1830), and Carl Friedrich Gauss (who never published). The Poincaré disk model was introduced by Henri Poincaré in the 1880s.
What is the connection to M.C. Escher's art?
M.C. Escher created his famous Circle Limit series (1958–1960) using {6,4} and {4,6} hyperbolic tilings, filling the disk with fish, angels, or devils that get smaller toward the edge. He learned the framework from geometer H.S.M. Coxeter.
What are hyperbolic geodesics?
In the Poincaré disk, geodesics are circular arcs that intersect the boundary circle at 90°, or diameters of the disk. Two geodesics can be parallel (meeting at the boundary), ultra-parallel (not meeting), or intersecting (meeting inside the disk).
What is the isometry group of hyperbolic space?
The isometry group of the hyperbolic plane is PSL(2,ℝ) (upper half-plane model) or PSU(1,1) (disk model). These Möbius transformations preserve the respective model and form a non-compact group of infinite volume, unlike the compact isometry groups of the sphere.