Pattern · geometry · a floor for every curvature

The Floor That Won't Lie Flat

Lay regular seven-sided tiles on a floor, three to a corner. The corners don't fit — three heptagons crowd 385.7° into a spot that has only 360° to give. Something has to bend, and it's the floor.

Pick a regular tile with p sides and a rule that q of them meet at every corner — a {p, q} floor. There is exactly one number that decides whether such a floor can exist, and what it must be shaped like. Turn the two knobs and watch it happen.

The {p, q} floor

{7,3} · hyperbolic

p — sides per tile: 7

q — tiles per corner: 3

each corner of a {p}-gon: 128.571°
q of them at a point: 385.714°
a flat floor allows: 360.000°
1/p + 1/q: 0.4762 (< 0.5)

Hyperbolic Three heptagons overshoot a flat corner by 25.7°. The floor can't stay flat — it ruffles into the hyperbolic plane, and the same tile repeats forever, shrinking toward a horizon it never reaches.

Look at one corner. q wedges of the tile's angle, laid around a point. They overshoot 360° — there isn't room, so the floor must warp to fit them.

Drag the sliders, or pick a floor. The disk is a real Poincaré-disk map of the hyperbolic plane: every tile is genuinely the same size and shape — they only look smaller as they near the edge, the way a map of Earth stretches near its rim.

The check

Nothing here is decorative. Every tiling in the disk is grown by reflecting the central tile across its own edges — the genuine reflection group — and every number is recomputed in your browser from the geometry. The same maths is re-derived from scratch, offline, in a deterministic verifier (research/hyperbolic-tiling/verify.mjs, 495/495 green), which also extracts the canonical function from this page and checks it hasn't drifted:

The verdict is sign(1/p + 1/q − 1/2), proven to always match sign(4 − (p−2)(q−2)) across the grid.
The five spherical floors are the Platonic solids — checked by Euler's V − E + F = 2 and by their faces covering exactly 4π of sphere.
There are exactly three flat regular tilings — {3,6}, {4,4}, {6,3} — and no others; the corner defect is exactly 0° there.
In the disk, each edge is a geodesic: a circular arc meeting the boundary at 90° (|O|² = 1 + s², to 1e-7). Reflections preserve hyperbolic distance (to 1e-7), so every tile is congruent.
Exactly q tiles close up at every interior corner — counted directly in the generated tiling — and every tile has the same hyperbolic area, π·(p − 2 − 2p/q) (Gauss–Bonnet), even the ones that look vanishingly small.
The slow drift is a rotation about the disk's centre — a hyperbolic isometry — so the motion never distorts a single tile.

Run it yourself: node research/hyperbolic-tiling/verify.mjs — the count and any failure print in the terminal.

Where the tiles can go

A regular tiling is written {p, q}: regular p-sided tiles, q of them meeting at each corner (the notation is Ludwig Schläfli's). A regular p-gon has corners of (p−2)·180°/p — 60° for a triangle, 90° for a square, 128.571° for a heptagon. Lay q of them around a point and the corners must add up to exactly 360° for the floor to lie flat. That single requirement forks three ways.

Fall short of 360° and the floor puckers inward at every corner and closes on itself: you don't get a floor, you get a solid. There are only five ways to do it, and they are precisely the five Platonic solids — the tetrahedron, cube, octahedron, dodecahedron, icosahedron. A dodecahedron is nothing but a {5,3} floor that ran out of flatness: three pentagons at a corner make 324°, 36° short, so it curls into a die.

Hit 360° exactly and you get the flat, endless tilings we actually put on floors. There are only three — squares {4,4}, triangles {3,6}, hexagons {6,3} — and no others, because only those corner-angles divide 360° with regular tiles. This is the knife-edge, and almost nothing lands on it.

Overshoot 360° — as three heptagons do, by 25.7° — and there is too much tile for a flat corner and too much for a sphere's. The excess has nowhere to go but into the floor itself, which buckles into a saddle at every point: the hyperbolic plane, where a surface curves away from itself in all directions at once. Now there is room for infinitely many regular tilings — one for every {p,q} with 1/p + 1/q < 1/2 — and each fits infinitely many ever-shrinking copies of one tile into a bounded picture. The whole trichotomy is one inequality: the sign of 1/p + 1/q − 1/2 is the sign of the curvature the floor is forced to have.

The map, not the territory

The hyperbolic plane doesn't fit in a flat picture without distortion — that's a theorem, not a shortcoming (Gauss, Theorema Egregium, presented 1827). So the disk you're looking at is the Poincaré disk model: the entire infinite hyperbolic plane, drawn conformally (angles honest, sizes not) inside a circle you can never reach. Straight lines become circular arcs that hit the rim at right angles. Every heptagon in there is the same size as the one in the middle — the shrinking is the map's, exactly as every flat map of the round Earth has to stretch something. (The model is usually named for Poincaré, ~1882, but Beltrami drew it first, in 1868.)

Escher reached it with a compass

In 1954, at the International Congress of Mathematicians in Amsterdam, the printmaker M. C. Escher met the geometer H. S. M. Coxeter. Three years later Coxeter sent him a reprint of a paper with a figure in it — a hyperbolic tessellation, triangles shrinking toward a circular rim. Escher wrote back that it had given him "quite a shock." He had been looking for exactly this: a way to capture infinity inside a finite frame. Between 1958 and 1960 he cut the four Circle Limit woodcuts, the third — interlocking fish in four colours, printed from five blocks — the most famous picture of the hyperbolic plane ever made.

He did it without the mathematics. Escher had no training in hyperbolic geometry; Coxeter's explanatory letters were, he complained, "hocus-pocus" to him. He reconstructed the diminishing scale by compass, straightedge, and a craftsman's stubborn eye. Years later Coxeter analysed Circle Limit III and found something startling: the white arcs along the fishes' spines aren't the hyperbolic straight lines you'd expect — they meet the boundary not at 90° but at almost exactly 80° (79.97°, to be exact — the arcs are hypercycles, curves that hold a constant distance from a true geodesic). Escher had drawn that angle right, by hand, to a precision Coxeter called "marvellous."¹

You can hold one

If a flat picture must lie about sizes, a physical model must ruffle — and it does. In 1997 the mathematician Daina Taimiņa, at Cornell, worked out that you could crochet the hyperbolic plane: add stitches at a fixed exponential rate and the fabric has no choice but to frill, because there is more and more area to fit into each step outward. Hers was the first durable model anyone could pick up and fold.

Nature got there first. The ruffled edge of a kale or lettuce leaf, the frill of certain corals and sea slugs, the crenellations of a brain coral — these are everyday approximations of constant negative curvature, the same "too much area near the edge" that overshoots a flat corner. The artists Margaret and Christine Wertheim built an entire touring Crochet Coral Reef out of the idea. The heptagons in the disk above and the ruffle on a leaf are the same fact, seen two ways: when a surface carries more angle than flatness allows, it curls — into a die if it has too little, into a frill if it has too much. Only the exact ration lies flat, and the floor, given the choice, almost never does.