Hyperbolic tilings

Six equilateral triangles meet at a point and lie flat; seven make the sheet ruffle like a lettuce leaf. That extra room is the hallmark of hyperbolic geometry, where regular tilings never run out: heptagons, octagons, even squares five to a vertex. This layer fits the infinite plane into a disk or a band.

How it works

A tiling {p, q} of regular p-gons, q at each vertex, splits into identical right triangles with angles π/p at a tile’s centre, π/q at its corner and π/2 at an edge midpoint. The angles of a hyperbolic triangle add up to less than π, and the shortfall is its area; hence (p − 2)(q − 2) > 4. Equality gives the three Euclidean tilings.

In the Poincaré disk, lines are arcs perpendicular to the rim, and reflecting in one is inversion in a circle. The shader folds each pixel into the triangle by such reflections. Their parity gives the “parity” colouring; “layers” colours tiles by distance from the frame centre. “Seams” are measured in hyperbolic units and thin towards the rim with the tiles; tiles below a pixel average out rather than shimmer.

The “Wythoff point” places the pattern’s vertex in the triangle: at the tile’s corner you get {p, q}, at the edge midpoint the rectified tiling, at the centre the dual {q, p}, and truncated tilings in between. “Shift”, a translation made of two half-turns about neighbouring edge midpoints, maps the tiling onto itself, so the animation loops seamlessly.

A bit of history

Nikolai Lobachevsky (1829) and János Bolyai (1832) discovered it independently. Eugenio Beltrami devised the disk and half-plane models in 1868; they are named after Henri Poincaré, who used them in 1882. In 1954 M. C. Escher, the author of its best-known pictures, met the geometer H. S. M. Coxeter, whose drawing of triangles in a disk later, as Escher wrote, gave him quite a shock. Hence the woodcuts Circle Limit I–IV (1958–1960). Coxeter later showed that the white arcs of Circle Limit III are not hyperbolic lines but equidistant curves meeting the rim at almost 80°. The mirror-triangle trick goes back to Willem Wythoff (1918).

What to tweak

• “p — corners per tile” 7, “q — tiles per vertex” 3: its triangle, of area π/42, is the smallest possible. A “Wythoff point” of 1 gives the dual {3, 7}.
• “Colouring” “parity” with “Wythoff point” 0 splits every tile into 2p mirror triangles of alternating colour.
• “Model” “band” with animation: tiles drift along an endless ribbon. In the “half-plane” they are all congruent, just drawn smaller towards the bottom.
• A home-made “Circle Limit”: put “Truchet tiles” under this layer and set “Motif” to “picture below”; the middle of that picture fills the triangle and is mirrored across the disk.