The whole infinite hyperbolic plane, squeezed into one finite circle. Straight lines curve into arcs, every tile in a {p, q} tiling is genuinely the same size, and the angles of a triangle add up to less than 180°. Drag anywhere to glide through a space that never runs out of room — Euclid's fifth postulate was only ever a choice.
{4,5}, {5,4}, {3,7}, {7,3}. A tiling only exists when 1/p + 1/q < 1/2; try {4,4} or {3,6} and the applet warns you — those are flat, not hyperbolic.Two thousand years of doubt. Euclid's first four postulates feel self-evident. The fifth — through a point off a line there is exactly one parallel — always felt like a theorem in disguise. For centuries geometers tried to prove it from the others and failed. In the 1820s–30s Bolyai, Lobachevsky and Gauss independently realised why: it cannot be proved, because a perfectly consistent geometry exists in which it is false. Through a point there are infinitely many parallels.
The disc as a map. Poincaré's model puts that entire infinite plane inside a unit circle. The trick is the metric: distances blow up as you approach the boundary, so the rim sits infinitely far away and you can never reach it. What your eye reads as "tiles getting smaller" is the flat paper distorting a scene in which every tile is exactly the same hyperbolic size — the same lie a flat world map tells about Greenland.
Straight = the arcs. Geodesics (shortest paths) are exactly the circular arcs that hit the boundary at 90°, plus diameters. The model is conformal: it distorts distance wildly but preserves angles perfectly — so the angle where two arcs cross on screen is the true hyperbolic angle. That is why the applet can measure a triangle's angles by reading the tilt of its arcs directly off the canvas.
The angle defect. In the plane a triangle's angles sum to π. On a sphere (positive curvature) they exceed π; in the hyperbolic plane (curvature −1) they always fall short. The shortfall δ = π − (α + β + γ) is not an error — by the Gauss–Bonnet theorem it equals the triangle's area. Big triangles are thin and pointy; the largest possible "ideal" triangle has all three vertices on the boundary, all three angles zero, and finite area π.
Building the tiling. A regular {p, q} tiling — p-sided tiles, q meeting at each corner — exists in the plane only for the three flat cases {3,6}, {4,4}, {6,3} where 1/p + 1/q = 1/2. Whenever the sum is smaller, the tiling has nowhere to live but hyperbolic space, and there are infinitely many. This applet builds one by placing a central tile, then reflecting it across its own edges (a circle inversion for each arc) over and over — the same reflection group Escher hand-drew in Circle Limit.
Where you meet it. Hyperbolic geometry is the natural home of special relativity's velocity space, of the modular group in number theory, of H² embeddings that pack tree-like data (taxonomies, networks) with almost no distortion — because a tree needs exponentially more room as it grows, and hyperbolic space is exactly the geometry with room to spare.
What the code does live. The central tile's vertices sit at Euclidean radius r = tanh(R/2) from where cosh R = cot(π/p)·cot(π/q). Each edge is the unique circle through two vertices orthogonal to the unit circle; neighbouring tiles are generated by inverting the tile in that circle. Dragging composes an SU(1,1) Möbius transform, and because the model is conformal, the triangle's on-screen arc tangents give its true hyperbolic angles — summed live in the readout.
Go deeper:
• Cannon, Floyd, Kenyon, Parry, Hyperbolic Geometry (Flavors of Geometry, MSRI) — the friendliest rigorous survey.
• Thurston, Three-Dimensional Geometry and Topology, ch. 2.
• Escher & Coxeter, Circle Limit III — the tiling this applet reconstructs.
• Sarkar (2011), "Low distortion Delaunay embedding of trees in hyperbolic plane" — why ML uses H².
Built as station #22 of the repo's 25 visual-maths ideas, following the DRIVE IT → WALKTHROUGH → AHA → EXPLANATION → RESEARCH NOTE pattern. Single self-contained HTML; no build step. Tiling by edge-reflection; angles measured conformally.