A385269 Number of face-connected components of quarter cubille cells in the quarter cubille up to translation, rotation, and reflection of the honeycomb.

1, 1, 2, 4, 18, 67, 374, 2063, 12482, 76835, 486375, 3119695, 20275051, 133031450, 880300617, 5866722906

Offset: 0

Author

Peter Kagey and Bert Dobbelaere, Jun 23 2025

Keywords

• nonn
• hard
• more

Comments

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

The quarter cubille is the dual to the runcic cubic honeycomb (equivalently, the cantellated tetrahedral-octahedral honeycomb) which consists of rhombicuboctahedra, cubes, and tetrahedra.

Each cell is similar to the convex hull of (0,2,2), (2,0,2), (2,2,0), (2,2,2), and (1,1,1).

Links

• Peter Kagey, Animation illustrating the a(4)=18 connected 4-cell components.
• Wikipedia, Architectonic and catoptric tessellation
• Wikipedia, Tetrahedral-octahedral honeycomb (see subsection "Quarter cubille.")

Crossrefs

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).