A Polycube Tiling Census
A polycube is a solid made of unit cubes glued face to face. This census asks, of every polycube up to nine cells, one question: can copies of it fill all of space with no gaps and no overlaps? Each answer comes with a certificate a stranger can recheck without trusting us.
In two dimensions the same census was done decades ago. In three, nobody had checked. Even the smallest polycube that cannot tile space was unknown.
What the census found
- Every polycube with one to seven cells tiles space. All 1,230 of them, each with a verified certificate.
- The smallest polycube that cannot tile space has eight cells: the square ring. A flat 3×3 ring with a hole. Its Heesch number is 1: copies can wrap it once, and provably never twice. It is the only non-tiler among the 6,922 octacubes.
- Six non-tilers through nine cells, all with Heesch number 1: the square ring, the Greek cross, the chiral pair 9/8219 and 9/8220, and the triskelion and its mirror image. All of them.
- Eight shapes that wrap themselves twice and still have no known tiling. The first shapes known to do that in three dimensions, led by the staircase. Each has a verified two-layer corona and no periodic tiling within the stated budgets. These are the census’s open questions. All of them.
How to read a page
Every shape has a page at its id, so 9/2127 is a permanent address. The page shows the shape, the tiling or corona that backs its verdict, where it went through the pipeline and how each stage ended, and the record exactly as stored. The glossary explains the words.