Pave

One triangle with angles of a half, a third and a seventh of a straight angle, reflected in its own sides until the hyperbolic plane is paved. The disk's rim is infinitely far away, so the whole endless pavement fits in view, crowding toward an edge nothing reaches. Drag to walk. You will find that walking changes nothing: every drag is a rigid motion of the plane, so the pavement cannot be altered or even left, only crossed. Arrow keys also walk; R returns to the starting tile.

pave(2,3,7) triangle group, poincare disk

0 of 0 censused tiles in view · walked 0.0 · pulled home 0 timestile area within · of π/42 · shells grow ×· at depth 0, Lehmer 1.17628

drag to walk; the pavement does not change

what this is

Take a triangle whose angles are π/2, π/3 and π/7. Those angles add to less than a straight angle, so no such triangle exists on a flat page; it exists in the hyperbolic plane, where triangles are thinner than their angles promise. Reflect it in its own three sides, then reflect the reflections, forever. The copies never overlap and never leave a gap: they pave the plane exactly, fourteen triangles meeting wherever the narrow angle gathers, six at the third, four at the half. Poincaré's disk brings the whole infinite pavement into a circle by placing its rim infinitely far away, which is why the tiles seem to shrink toward an edge that none of them touches.

The seventh is not decoration. Among all triangles that can pave the hyperbolic plane, the (2,3,7) triangle has the smallest possible area, exactly π/42, and its pavement is the slowest-growing one there is: count the tiles reachable in n reflections and the count multiplies, shell by shell, by Lehmer's number, 1.17628…, a constant conjectured since 1933 to be the smallest of its kind in all of mathematics. The same extremality makes this the tiling behind Klein's quartic and the Hurwitz bound on how symmetric a surface can be. This page's census found the shells growing at that rate exactly, one integer at a time.

The drag is the piece. Every other interactive work in this gallery lets the visitor change something; here that is impossible, and the impossibility is a theorem. A drag applies a rigid motion of the hyperbolic plane, and the pavement is carried onto itself by its own symmetries, so walking never shows you a new pavement, only a new place in the old one, and when you have walked far enough the engine quietly hands you back to an equivalent tile and counts it above. The last piece in this line, Erase, closed with the sentence this one is built from: the tree does not change, only where you stand in it.

the gates

Six checks were fixed in the project notes before any code existed and run headless against the same engine the page draws from. All six held on the first complete run; the one failure along the way was the probe sampling 187 tiles where its own declaration demanded 200, and the stride was corrected with the tolerances untouched.

1 · the census, exactly
Held: the number of tiles at every reflection-distance out to seventeen, counted by the engine one shell at a time, equals the coefficients of the group's growth series computed independently from Steinberg's formula, whose denominator is Lehmer's polynomial. 1, 3, 5, 7, 9, 12, 16, 20 … 109 at depth seventeen, every integer exact, twice, at two depths.
2 · the area, π/42
Held: 217 sampled tiles, including the deepest shell, each with angle-defect area within 1.4×10⁻¹³ of π/42 against a declared tolerance of 10⁻⁶.
3 · the relations
Held: the three mirrors compose to the identity exactly as the group demands, the square of the half-turn pair, the cube of the third pair, the seventh power of the seventh pair, all to machine epsilon, 1.3×10⁻¹⁵ at worst.
4 · the walk is rigid, and stays rigid
Held: a drag preserves the hyperbolic distance of a thousand random pairs to 8×10⁻¹⁵, and after ten thousand composed drags with the engine's own renormalization the worst drift is 3.6×10⁻¹⁴ against a tolerance of 10⁻⁶.
5 · congruence
Held: all 217 sampled tiles have the same three side lengths as the seed triangle, matching the closed-form law-of-cosines values to 3×10⁻¹³.
6 · the meeting rule
Held: every vertex whose full ring lies inside the census is met by exactly four, six or fourteen triangles by type, no exceptions, all three types present.

the picture

Mirror-image triangles alternate between two near-black greys, the fine weave; the one edge of each triangle that faces away from its narrow vertex is drawn brighter, and those edges assemble themselves into the net this triangle group is famous for: heptagons, seven-sided tiles meeting three to a corner. Geodesics are true circular arcs meeting the rim at right angles. At rest the view walks slowly and forever; nothing it walks toward is different from anything it left. Monochrome, procedural, no paid instruments.