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Counting the shapes you can build from the Josehedron

Counts of the shapes you can build by gluing copies of the Josehedron face to face; the OEIS had none before. Mathias Bernhard named this space-filling solid this spring.

Three counts, one for each standard rule about when two shapes are the same, now in the OEIS, the online database of integer sequences. The programs were written with an AI model, and the counts were checked three ways before going to the OEIS.

August 15, 2026· updated Oct 29, 2026· verified Aug 15, 2026 (3 months ago)
  • Plain version: For anyone: no notation, everyday words
  • Applied version: For practitioners: how it was done, what it took, and what carries over to other work
  • Technical version: For specialists: full notation, definitions, and the earlier best results

Switching versions changes only the wording. Numbers, dates, sources and caveats are the same in every version.

For readers in

Choose a field to open a short section, written for people who work in it, on what this result means there. The rest of the page does not change.

What was counted

The Josehedron is a solid with faces, named by Mathias Bernhard in a paper this spring. Copies of it fill space with no gaps, the way cubes stack; an arrangement like that is called a tiling, and each copy in it is called a cell. (Its shape had already appeared, without a name, in Moritz W. Schmitt's 2016 survey of such solids, which listed the solid but did not count shapes built from it.)

Squares glued edge to edge make the familiar polyominoes, such as the Tetris pieces. Cells of a space-filling solid glued face to face are the 3D version, and shapes built this way from any tile are called polyforms. I counted the Josehedron's polyforms: how many different shapes there are with two cells, with three, and so on up to . These are the first such counts in the OEIS.

To count shapes you need a rule for when two of them are the same, and there are three standard ones. The moves allowed are the symmetries of the tiling, the moves that carry the whole tiling onto itself. For shapes of cells:

  • The "fixed" count treats two shapes as the same only when one can be slid onto the other, with no turning. There are .
  • The "one-sided" count also lets a shape be turned, but not mirrored. The name comes from flat pieces that may be slid and turned on a table but never flipped onto their other side. There are .
  • The "free" count also treats a shape and its mirror image as the same, as if the piece were free to be flipped over. There are .

The last two fit together as they should. A shape that differs from its mirror image is counted twice in the one-sided count and once in the free count, while a shape that is its own mirror image is counted once in each. So twice the free count minus the one-sided count is the number of shapes that are their own mirror image, here 5. That is a consistency test on the counts; it recomputes nothing.

All checks on these counts were made inside this project, and nobody outside it has recomputed them yet. A second, separately written program confirms every count up to six cells. The counts for larger shapes come from the main program alone, which passed the internal checks described below.

In physics clusters like these are called lattice animals, and counts like these are the raw data for estimating how fast the number of clusters grows with size. How many shapes there are at each size is also one of the first things people ask about any new space-filling solid. The OEIS had no counts for the Josehedron before these, and I know of no earlier ones.

Claude agents (made by Anthropic), working under my direction, wrote and ran the counting programs and the model of the solid; I set the checks and submitted the counts. Nobody can check billions of shapes by eye, so before sending the numbers to the OEIS I ran three checks. A second program, written separately in Python, agreed with the main C++ program on every count up to six cells, which is as far as the slower Python program can go; above that, two differently set up runs of the C++ program agreed with each other. The same program, pointed at ordinary cubes, reproduced the cube counts that have been published for decades. And the solid's geometry was built a second time with exact fractions instead of decimals; both versions gave every cell the same neighbors, so no count depends on rounding.

The three counts are in the OEIS as A398957 (fixed), A398958 (one-sided) and A398959 (free).

The Josehedron is also the test case in one of my workshop papers: before an AI system was allowed to look for new space-filling solids, it had to rebuild this one from scratch.

The tiling is the Voronoi honeycomb of the Josehedron plesiohedron (Bernhard, arXiv:2604.07160): six translation classes of cells, each with face-neighbors, point group of order , and the tiling is achiral. Fixed, one-sided and free polyforms were enumerated through n = with a Redelmeier-style enumerator, with isomorph rejection by canonical hashing over the translation quotient.

Checks: an independent Python reimplementation (growth plus canonical hashing) agrees on every term through n = 6; beyond that, the fixed columns of the proper-group and full-group runs agree exactly, including at n = 10 and n = 11. The same code on the cubic honeycomb reproduces A001931, A000162 and A038119 exactly across the range. The adjacency was re-derived from an exact rational halfspace construction, replacing the floating-point Voronoi one, with identical results. One-sided = 2·free − achiral holds at every term, with small nonnegative achiral counts. All checks are internal to the project.

The same approach works for counting shapes on any unfamiliar grid. First build the grid with exact arithmetic, so the program uses the true geometry. Then confirm that the counting program reproduces known answers on a grid where they are already known, here ordinary cubes, before reporting any new number. The counting method itself is standard; the confidence in the new numbers comes from those two checks. The full run takes hours on a laptop.

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

In statistical physics, connected clusters of cells are called lattice animals, and their counts feed models of percolation and cluster sizes. Here the clusters live on the Josehedron honeycomb's network of face-adjacent cells, whose cells come in six orientations rather than one, and I could find no earlier counts of them. No physical quantity was computed from them.

For readers in geometry:how the geometry was built and checked without rounding

The solid's geometry was built twice: once in ordinary floating-point software, then again from scratch in exact fractions. Both versions give every cell the same neighbors, so no count depends on a rounding choice. Counting shapes up to rotation and reflection then uses the symmetry group of the tiling.

Limits

The counts stop at cells. That run took hours on a laptop, and the next size would take roughly ten times as long. The solid is Bernhard's work; I computed the counts.

How this was checked

  • A second program agreed on the small sizes

    A second counting program, written separately in Python with a different method, agrees with the main C++ program on every count up to six cells, as far as the Python program can reach. Above that, two runs of the C++ program, one counting up to rotation and one up to rotation and reflection, give exactly the same count of shapes that are only slid into place.

    Aug 14, 2026

  • The same program reproduced the known cube counts

    Pointed at ordinary stacked cubes, the same program reproduces the long-published cube counts (A001931, A000162, A038119) exactly. This tests the counting program, not the model of the Josehedron.

    Aug 14, 2026

  • The geometry was rebuilt with exact fractions

    The solid's geometry was first built with standard decimal software, then rebuilt from scratch in this project using exact fractions. Both versions give the same neighbors for every cell and the same counts. This rules out rounding errors; it is not a check by anyone outside the project.

    Aug 14, 2026

Revision history

  1. 2026-10-29 · revised

    Defined polyform, cell and the fixed, one-sided and free counts in plain words, explained why twice the free count minus the one-sided count gives the mirror-symmetric shapes, and said near the top which sizes a second program confirms.

  2. 2026-10-22 · revised

    Said exactly how far the second program agreed (up to six cells), relabeled the exact-fraction rebuild as an internal check, and added the third count (the fixed count, already in the OEIS since late August) to the text of this page.

  3. 2026-10-01 · revised

    Added that the solid's shape had already appeared, without a name, in Moritz Schmitt's survey of such solids.

  4. 2026-08-27 · status-change

    A398957, the count of shapes that may only be slid into place (the fixed count), published in the OEIS, completing the family.

How to cite this

Tyler Satchel Orden (2026). Counting the shapes you can build from the Josehedron. Cyborg Mathematics. https://cyborgmathematics.com/results/josehedron-polyforms (updated October 29, 2026)

This page has no DOI of its own. Cite the URL and the date; the revision history above keeps a dated citation meaningful.

BibTeX
@misc{orden2026josehedronpolyforms,
  author       = {Tyler Satchel Orden},
  title        = {Counting the shapes you can build from the Josehedron},
  year         = {2026},
  howpublished = {Cyborg Mathematics},
  url    = {https://cyborgmathematics.com/results/josehedron-polyforms}
}

See also