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First polyform counts on the Josehedron honeycomb

The polyforms of the Josehedron honeycomb are enumerated for the first time, in three symmetry classes, all published in the OEIS: A398957 (fixed), A398958 (one-sided) and A398959 (free).

A recent preprint introduced a new space-filling solid. This entry records the first enumeration of the shapes buildable from its honeycomb, verified several independent ways before submission.

August 15, 2026· verified Aug 15, 2026 (27 days ago)
  • Plain: Plain version: no notation, everyday words
  • Applied: Applied version: the method, the cost, and what transfers
  • Technical: Technical version: full notation, definitions, and prior bounds

Figures, dates, citations and caveats are identical in every reading; only the words around them change.

The result

The Josehedron is a solid with

Value
12
Basis
observed
Source
Mathias Bernhard, The Josehedron, arXiv:2604.07160 · the solid's defining face list; independently re-derived as the convex hull of the paper's twelve vertices
Checked
Hull of the published vertex coordinates recomputed during prep; exact rational halfspace construction reproduces the same facet signature on every honeycomb cell.
As of
2026-08-15
faces, introduced this spring, that fills space with copies of itself. This entry records the first counts of the shapes you can build by gluing its cells together face-to-face: at cells there are one-sided shapes and shapes when mirror images are also identified.

Think of a jigsaw made from one oddly-shaped block that happens to pack perfectly with copies of itself (a new one, described for the first time this spring). A natural question for any such block: how many different shapes can you make by sticking 2, 3, 4… of them together? (For flat squares this is the classic pentominoes puzzle.) Nobody had counted these for the new block, so we did, up to

Value
11
Basis
observed
Source
OEIS A398958 / A398959 DATA (published 15 Aug 2026) · both sequences carry terms a(1)..a(11)
Checked
Term count read from the published DATA lines of both entries.
As of
2026-08-15
blocks, where there are already distinct shapes. Before publishing, the counts were checked several separate ways: completely different programs had to agree on every number; the same program had to reproduce the long-known answers for ordinary cubes; and the block's geometry itself was rebuilt with exact fraction arithmetic so no rounding error could hide anywhere.

Polyform enumeration on the Voronoi honeycomb of the Josehedron plesiohedron (Bernhard, arXiv:2604.07160): six cell types, each with

Value
12
Basis
observed
Source
Mathias Bernhard, The Josehedron, arXiv:2604.07160 · the solid's defining face list; independently re-derived as the convex hull of the paper's twelve vertices
Checked
Hull of the published vertex coordinates recomputed during prep; exact rational halfspace construction reproduces the same facet signature on every honeycomb cell.
As of
2026-08-15
face-neighbours, point group of order , achiral tiling. Fixed, one-sided and free counts through n = (free class at the top order: ). Isomorph rejection by canonical hashing over the translation quotient; the identity one-sided = 2·free − achiral holds at every term with small nonnegative achiral counts. Verification: independent reimplementation, cubic-control anchors (A001931/A000162/A038119, exact across the range), and an exact rational halfspace re-derivation of the adjacency replacing the floating Voronoi construction. OEIS: A398957 (fixed), A398958 (one-sided) and A398959 (free) are all published, the fixed class on 2026-08-27.

The method is the transferable part: to count objects on a novel substrate, first pin the substrate with exact arithmetic (the adjacency here was rebuilt from integer cell centres with rational halfspace clipping, no library Voronoi in the trust chain), then anchor the counter on a solved neighbouring problem (the cubic lattice, known for decades) before believing any new number. The enumeration itself is a standard Redelmeier walk; every guarantee comes from the anchors around it. Cost: the full run to n =

Value
11
Basis
observed
Source
OEIS A398958 / A398959 DATA (published 15 Aug 2026) · both sequences carry terms a(1)..a(11)
Checked
Term count read from the published DATA lines of both entries.
As of
2026-08-15
is hours on a laptop; the verification scaffolding is what took the care.

Limitations

Counts end at n =

Value
11
Basis
observed
Source
OEIS A398958 / A398959 DATA (published 15 Aug 2026) · both sequences carry terms a(1)..a(11)
Checked
Term count read from the published DATA lines of both entries.
As of
2026-08-15
(the next order costs roughly an order of magnitude more). The underlying solid is Bernhard's; only the enumeration is ours.

How this was checked

  • cross implementation

    Independently written enumerators (a C++ Redelmeier-style counter and a Python growth-and-canonical-hash counter) agree on every term of every class in the shared range.

    Aug 14, 2026

  • known value anchor

    Run on the simple cubic honeycomb, the same enumerator reproduces the published cubic sequences A001931, A000162 and A038119 exactly across the anchor range.

    Aug 14, 2026

  • independent reproduction

    The honeycomb cell adjacency, originally built with a floating-point Voronoi library, was re-derived by an exact rational-arithmetic halfspace construction with no floating point: the facet signature, symmetric adjacency, lattice volume, and an independent fixed-count enumeration all match the banked values across the anchor range.

    Aug 14, 2026

Revision history

  1. 2026-08-27 · status-change

    A398957, the fixed class, published after an editor round (the process comment came out and the verified self-contained enumerator went up as an a-file). The family is now complete and live; source: PROGRAM_LEDGER.md, entry dated 2026-08-27.

  2. 2026-08-15 · created

    Written the day A398958 and A398959 published; A398957 (fixed class) still in review.

How to cite this

Tyler Satchel Orden (2026). First polyform counts on the Josehedron honeycomb. Checkable. https://cyborgmathematics.com /results/josehedron-polyforms (updated August 15, 2026)

No DOI yet. Cite the URL and the date; this page carries a revision history so a dated citation stays meaningful.

BibTeX
@misc{orden2026josehedronpolyforms,
  author       = {Tyler Satchel Orden},
  title        = {First polyform counts on the Josehedron honeycomb},
  year         = {2026},
  howpublished = {Checkable},
  url    = {https://cyborgmathematics.com
/results/josehedron-polyforms}
}