Counting the shapes built from two space-filling solids, including one with no earlier counts

Some solid shapes fill space completely with copies of themselves, fitting face to face with no gaps, the way cubes stack into a block. A packing like that is called a honeycomb, and each copy in it a cell. A standard question about any honeycomb is how many different clusters you can build by gluing a given number of cells together face to face. For cubes these clusters are the polycubes, the three-dimensional cousins of Tetris pieces, and they have been counted to large sizes. For most other shapes, very little has been counted.

I worked on two shapes, each with eleven faces. The second had never been counted because of a step that comes before counting. The only published description of how its copies pack was in words, and a counting program needs exact positions. So the first job was to turn that description into an exact one, and to show it was the only arrangement of its type. That missing step is why I picked this shape as the follow-up to the first. Claude, the AI model I work with, wrote and ran the code for both shapes; I reviewed the checks below and submitted the results.

The counts are now in the OEIS (the Online Encyclopedia of Integer Sequences), a public database of number sequences that mathematicians use and cite.

What counts as a different cluster#

Two clusters count as the same if one can be moved onto the other by a symmetry of the honeycomb: sliding it, turning it, or mirroring it. Both honeycombs here include mirror symmetries, so a cluster and its mirror image count once.

The same idea sorts the faces of a cell into kinds. Two faces are the same kind if a symmetry of the honeycomb carries one onto the other. The smallest clusters show why this matters: two cells glued together make one cluster for each kind of face they can share.

The first shape: the bisymmetric hendecahedron#

A hendecahedron is any solid with eleven faces. This one has four triangles, four kites, two large rhombi (diamond shapes) and one square. It is one of five space-filling solids that Guy Inchbald described in 1996. All four kites are the same kind in the sense above, and so are the triangles and the two rhombi, so there are four kinds of face and 4 two-cell clusters.

Its counts were already in the OEIS as A385028, up to clusters of 6 cells. I extended them to 11 cells. Here is the full list, starting from clusters of 1 cell:

So there are 1,170,906,897 different clusters of 11 cells. The value for 7 cells, 85,365, had been computed by the authors of an earlier paper on these counts with their own software and printed in that paper, but it had never been entered in the database. The program arrived at the same number on its own.

The second shape: the sphenoid hendecahedron#

The second shape is another of Inchbald's five. It also has eleven faces: six kites, four triangles and one rhombus. Inchbald described in words how its copies pack: six of them meet around a shared edge like the petals of a flower, and these flower clusters stack to fill space. I could find no description in print that gave the exact position of each neighboring cell, which is what a counting program needs.

So the code worked it out from the shape itself. For each face, it found every position in which a second copy could sit flush against it, matching face to face. Ten of the faces allow two positions each and the rhombus allows four. Choosing one position for every face gives 2 multiplied by itself ten times (1,024), times 4, which is 4,096 possible rules for how a cell meets its neighbors.

For each rule, the code checked whether the cells meeting around every edge of the cell fit exactly, with their angles adding up to a full turn and no gap or overlap. Exactly one rule passes. The code then built that honeycomb out to 714 cells and confirmed that no two cells overlap, and it confirmed exactly that the pattern repeats in blocks of twelve cells that fill space with nothing left over.

So among honeycombs in which every cell is attached to its neighbors by the same rule, this shape has exactly one. The search did not look at arrangements where different cells follow different rules. The one it found matches Inchbald's description: six cells around each such edge, and a repeating block of twelve cells, which is two flower clusters.

With the honeycomb fixed, counting could start. These are the first counts I could find for this shape, in the OEIS as A397708:

The 6 for two cells comes from the kinds of face. The six kites come in three mirror-image pairs, and a mirror symmetry of the honeycomb carries each kite onto its partner, so there are three kinds of kite. The four triangles make two kinds in the same way, and the rhombus is one more, which makes six kinds and 6 two-cell clusters.

How the counts were checked#

The counting program was first run on cubes, where it reproduces the known polycube counts. On the first shape it reproduces every count that was already published, plus the 7-cell value from the earlier paper.

A second program, written on a different method, counted the same clusters for both shapes. It agrees with the first program on every count through clusters of 8 cells, and beyond that it becomes too slow. The larger values (9 to 11 cells for the first shape, 9 and 10 for the second) come from the first program alone, which had passed the checks above.

For specialists#

The same method was applied to two cell-transitive honeycombs: the bisymmetric hendecahedron honeycomb (A385028), which already had published terms, and the sphenoid hendecahedron honeycomb (A397708), which had none. Inchbald's account of the sphenoid packing is qualitative, and I found no explicit description of the tiling in print, so the honeycomb had to be constructed before anything could be counted.

Starting from the vertices of one cell (Inchbald's coordinates, as corrected on his own site), the faces were computed in exact rational arithmetic, and for each face the face-to-face placements of a neighboring cell were enumerated: two per face except the rhombus, which admits four, each realized by two isometries differing by the cell's mirror. All 4,096 resulting assignments are locally overlap-free. An exact edge-closure test (around each of the 20 edges, the induced walk must return to the base cell with dihedral angles summing to a full turn) leaves exactly one. Grown to a 714-cell patch it shows no overlapping pairs, and the exact volume identity (12 cells times the cell volume equals the lattice determinant) shows the 12 cells of one primitive cell of the translation lattice fill it. The lattice is simple hexagonal and the point group has order 24, acting transitively on the 12 cell positions. The search is the uniqueness argument; it covers isohedral face-to-face tilings, in which every cell uses the same gluing rule, and does not exclude non-isohedral ones. The match with Inchbald's florets is a consistency check, not part of the argument.

Counting: free polyforms are counted with a canonical-minimum test over the point-group operations, using a Redelmeier-style enumerator generalized to any honeycomb whose cells are all copies of one cell. The independent check is a Python breadth-first growth program with canonical-hash deduplication, sharing only the adjacency and symmetry tables. The main enumerator reproduces the cubic sequences A001931 and A038119 through n = 8 and the reference program reproduces A038119 through n = 6; the two programs agree through n = 8 on both honeycombs. On the bisymmetric hendecahedron the enumerator reproduces every published term and matches a(7) = 85365 from Table 1 of Dobbelaere, Kagey, Thomas and Vindas-Meléndez, which had not been entered in the database. On both honeycombs the ratio of fixed counts (up to translation only; A397707 and A397709) to free counts climbs toward the point-group order, 16 for the bisymmetric hendecahedron and 24 for the sphenoid.

For readers in physics:where these counts sit in statistical physics, and where the comparison stops

A cluster of cells glued face to face is what statistical physicists call a lattice animal, here on the cell-adjacency graph of each honeycomb, where every cell has 11 neighbors. For series work the relevant counts are the fixed ones, counted up to translation only: A397707 and A397709. The ratio between successive free counts at the last terms is about 11.1 for the bisymmetric hendecahedron and 10.5 for the sphenoid, a rough indication of the growth constant. This work counts clusters only. It gives no perimeter counts, so it is not enough on its own for percolation series, and it computes no physical quantity.

@misc{orden2026honeycombconstruction,
  author       = {Tyler Satchel Orden},
  title        = {Counting the shapes built from two space-filling solids, including one with no earlier counts},
  year         = {2026},
  howpublished = {Cyborg Mathematics},
  url    = {https://cyborgmathematics.com/writing/honeycomb-construction}
}