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Cyborg Mathematics

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Shapes and polyforms

A polyform is a shape made by gluing copies of one tile face to face, the way Tetris pieces are made from squares. I count them for unusual cells: solids that fill space with no gaps, like the Josehedron and the cell of the sphenoid honeycomb, and hexagons, which in chemistry stand for the rings of carbon in a benzenoid molecule.

6 results · 1 article

Results

Numbered by their place in the strongest-first list

  1. 06

    A peer-reviewed census of benzenoid molecules reproduced in full, with errors found and its tables extended

    I reproduced every table of a peer-reviewed 2018 census of benzenoid molecules, shapes made of fused hexagonal rings of carbon. I found ▲ errors (in its tables, a figure caption and the text), each with a corrected value, while everything else matched exactly. I sent them to the authors, who have not yet confirmed them. I then extended the census’s breakdown tables two sizes further, to ▲ molecules at the largest size. The OEIS already had the totals at both new sizes, and mine agree with them.

    Checked outside this site

    • MatchThe totals at the two new sizes match the counts already in the OEIS. OEIS A018190
    • CreditThe errors and corrected values went to the census authors, who have not yet confirmed them,
    • ArchiveData and code archived, . Zenodo DOI
    ShapesChemistry
  2. 09

    The first OEIS counts of shapes built from Josehedron cells

    The Josehedron is a -faced solid whose copies fill space with no gaps. Its shape appeared, unnamed, in Moritz W. Schmitt’s 2016 survey of such solids, and Mathias Bernhard named it in 2026. I counted the shapes made by gluing copies of it (cells) face to face, up to cells, in the three standard ways: free (a shape, its turns and its mirror image count once), one-sided (mirror images count separately) and fixed (shapes count as the same only when one slides onto the other). The OEIS had none of these counts before; they are now A398957, A398958 and A398959.

    Checked outside this site

    • OEIS editorMichael De Vlieger approved A398958 and A398959, . OEIS A398958
    • OEIS editorA398957 published after review by further editors, . OEIS A398957
    ShapesSequences
  3. 10

    Shape counts for more space-filling cells, with an independent match

    From mid-October to early November 2026 the OEIS published sequences of mine counting the shapes built from space-filling cells, among them the cell of the Laves graph and one of Peter Engel’s -faced cells, the most faces known for a cell of this kind. For two of the cells an OEIS contributor, Alper Ferudun, computed further terms on his own, and they matched mine term for term.

    Checked outside this site

    • OEIS editorPublished by OEIS editors, the last four on this date, . OEIS A401463
    • MatchAlper Ferudun’s own terms matched mine, . OEIS A401256
    ShapesSequences
  4. 11

    Two OEIS sequences counting hexagon shapes without holes, extended

    Polyhexes are shapes made of hexagons joined edge to edge, the outlines that benzenoid molecules take. Both sequences count polyhexes without holes by perimeter, treating shapes that differ only by a turn as the same. A258204 counts them one-sided, so a shape and its mirror image count separately; A258205 counts only those with mirror symmetry. I extended each by ▲ terms, up to perimeter index ▲, where the one-sided count reaches ▲. Two separately written programs agree at every size, and an identity with published data pins down some of the values on its own.

    Checked outside this site

    • OEIS editorSean A. Irvine published A258204, . OEIS A258204
    • OEIS editorSean A. Irvine published A258205, . OEIS A258205
    ShapesSequences
  5. 15

    New terms for a shape sequence marked hard to extend

    OEIS A385028 counts the shapes built by gluing copies of an eleven-faced space-filling solid, the bisymmetric hendecahedron, face to face, and it carries the OEIS mark for sequences that are hard to extend. I computed new terms and computed one of them again with a second, separately written program.

    Checked outside this site

    • OEIS editorPublished by an OEIS editor, . OEIS A385028
    ShapesSequences
  6. 16

    Shape counts for the sphenoid hendecahedron, a space-filling solid with no earlier counts I could find

    The sphenoid hendecahedron is an eleven-faced solid whose copies fill space. The only published description of how they pack was in words, so I turned it into exact positions and showed that it is the only packing in which every copy meets its neighbors in the same way. Then I counted the shapes built by gluing copies face to face, which had no earlier counts I could find. The argument is in the published entry.

    Checked outside this site

    • OEIS editorPublished as new OEIS entries, with their full data files, . OEIS A397708
    ShapesSequences

Articles

Notes on the numbers marked ▲

  • Errors found in the published census paper while reproducing it (in its tables, a figure caption and the text), each with a corrected value: 6. I sent the errors and corrected values to the paper’s authors on August 17 and 27, 2026. They have not replied about them yet, and the corrections are not yet in a public document.
  • Benzenoid structures with 15 hexagons, two sizes past the published census (the total matches OEIS A018190): 74,207,910. The OEIS (A018190) already listed the total at this size, and my count matches it. What is new is the census’s breakdown of that total, which has not been peer reviewed yet.
  • New terms added to each of A258204 (one-sided holeless polyhexes) and A258205 (holeless polyhexes with bilateral symmetry), n = 20 through 32: 13. The new terms of both entries are published in the OEIS: A258204’s on 2026-09-08 and A258205’s on 2026-09-12.
  • Largest perimeter index n (perimeter 2n) now in A258204 (one-sided holeless polyhexes) and A258205 (holeless polyhexes with bilateral symmetry): 32. Both entries are now published in the OEIS through n = 32: A258204 on 2026-09-08 and A258205 on 2026-09-12, both approved by Sean A. Irvine.
  • One-sided holeless polyhexes of perimeter 64, counted up to rotation, mirror images counted separately (A258204 at n = 32): 750,157,868,462. A one-sided count: a shape and its mirror image count as two shapes unless they are the same.