A343909 -id:A343909 - cp's OEIS frontend

A371062 Number of one-sided generalized polyforms on the tetrahedral-octahedral honeycomb with n cells.

1, 2, 1, 4, 11, 65, 339, 2211, 14356, 97599, 671518, 4696575, 33186688, 236773833, 1702530274, 12327097246, 89791558276

Offset: 0

Peter Kagey, Mar 09 2024

This sequence counts "one-sided" polyforms where holes are allowed. This means that two polyforms are considered the same if one is a rotation of the other. However, reflections of chiral polytopes are considered distinct.

The ratio a(n)/A343909(n) -> 2 as n -> infinity.

Peter Kagey, A photograph of the a(4)=11 polyforms 3D printed.

Cf. A000162, A343909.

a(10)-a(16) from Bert Dobbelaere, Jun 10 2025

A385028 Number of face-connected components of polyhedral cells in the bisymmetric hendecahedral honeycomb up to translation, rotation, and reflection of the honeycomb.

Original entry on oeis.org

1, 1, 4, 16, 116, 903, 8551

Offset: 0

Peter Kagey, Aug 13 2025

These are "free polyforms" because they are counted up to rotation and reflection.

A bisymmetric hendecahedron is an 11-sided polyhedron that is similar to the convex hull of (-2,1,-1), (-2,1,1), (-1,-1,0), (0,-1,-1), (0,-1,1), (0,0,-2), (0,0,2), (0,2,0), (1,-1,0), (2,1,-1), and (2,1,1).

			For n = 2, the a(2) = 4 distinct compounds of two bisymmetric hendecahedra correspond to placing the four distinct types of faces (square, kite, rhombus, and triangle) together.

Peter Kagey, Illustration of the a(3)=16 components with three polyhedra.
Wikipedia, Bisymmetric hendecahedron
Jiangmei Wu and Guy Inchbald, Folding Space-Filling Bisymmetric Hendecahedron for a Large-Scale Art Installation, Bridges 2018 Conference Proceedings.
Jiangmei Wu, Synergia (sculpture made from bisymmetric hendecahedra).

Cf. A038119 (cubic), A038173 (rhombic dodecahedral), A038181 (truncated octahedral), A343909 (tetrahedral-octahedral), A365654 (square bipyramidal), A384254 (rectified cubic), A384274 (quarter cubic), A384754 (omnitruncated cubic), A385024 (tetragonal disphenoidal), A385025 (gyrobifastigium), A385026 (hexagonal prismatic), A385027 (triakis truncated tetrahedral), A385267 (half pyramidille), A385268 (oblate cubille), A385269 (quarter cubille), A385270 (elongated dodecahedral), A385271 (hexakis cubic), A385272 (phyllic disphenoidal), A385273 (quarter oblate octahedrille), A385274 (rhombic pyramidal), A385275 (square quarter pyramidille), A385276 (trapezo-rhombic dodecahedral), A385277 (triangular prismatic), A385278 (triangular pyramidille).

cp's OEIS Frontend

A371062 Number of one-sided generalized polyforms on the tetrahedral-octahedral honeycomb with n cells.

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