Glossary

Words the census uses everywhere, in the order you are likely to meet them.

Polycube

A solid made of unit cubes glued face to face, like a Tetris piece in three dimensions. A polycube with n cells is an n-cube: 1 is the monocube, 4 are tetracubes, 9 are nonacubes. Two polycubes are the same shape if one can be rotated onto the other. A shape and its mirror image count as two shapes unless they happen to be the same.

Tiling

Filling all of space with copies of one shape, no gaps and no overlaps. Copies may be rotated. A shape that admits one is a tiler. The census allows the 24 rotations of the cube and reports separately whether reflections were needed.

Periodic tiling

A tiling that repeats on a lattice, like three-dimensional wallpaper. A certificate for a periodic tiling is a lattice and a list of placements inside one repeating block. The census finds these by solving on a torus: space wrapped around so that a block’s edges meet.

Box tiling

A tiling of a rectangular box. Boxes stack, so a box tiling is a tiling of space, and the smallest box a shape fills is its box order. The box stage runs first because its certificates are the easiest to read.

Corona

One complete layer of copies wrapped around a shape so that every cell touching the shape, at a face, an edge, or a corner, is covered. A second corona wraps the first. The census uses 26-neighbour adjacency for “touching”, which is the convention from the two-dimensional literature.

Heesch number

How many complete coronas can be wrapped around a shape before it is impossible to add another. A tiler has infinitely many. A shape with Heesch number 1 can be wrapped once and provably never twice. The census’s non-tilers all have Heesch number 1 so far. Shapes known to wrap twice but not yet shown to stop are listed as Heesch number 2 or more.

Verdicts

Tiler: a certificate exists, a box or a periodic block, rechecked by plain geometry. Non-tiler: a corona depth was proven impossible by a SAT solver, and the proof was checked by an independent program. Open: the shape survived every test within its stated budgets. Open is not a shrug: the record says exactly how far each search went.

Stages

The order the census tries things: box tiling, then periodic tiling, then coronas. Each record lists which stages it went through and how each ended: certified (a tiling found), exhausted (searched to the limit, nothing), witnessed (a corona of that depth exists), refuted (none of that depth exists), or attempted (tried, not settled).

Symmetry order

How many of the 24 rotations of the cube leave the shape looking the same. A straight bar has 8. A shape with no symmetry has 1. The more symmetric a shape, the fewer distinct ways it can be placed.

Rotation number

Which of the 24 rotations a placement uses, as a number from 0 to 23. The 24 rotations page lists them all: where each new coordinate comes from, and the same turn as an axis and an angle.

Chiral

A shape is chiral if no rotation turns it into its mirror image, like a left and a right screw. Chiral shapes come in pairs, and each page links to its twin. A shape tiles space if and only if its mirror image does, so the census solves one and reflects the certificate for the other.

Certificate

Data that lets anyone recheck a claim without trusting us. A tiling certificate is a placement list, checked by counting. A refutation certificate is a proof in DRAT format, checked by drat-trim. The verify page walks through both.

SAT solver

A program that decides whether a large true-or-false formula can be satisfied. The census writes “copies of this shape fill this region” as such a formula and hands it to a solver. A satisfying assignment is a tiling. A proof of unsatisfiability is a refutation.