Pentacube Oddities with Full Symmetry

Introduction

A pentacube is a solid made of five cubes joined face to face. An oddity (or Sillke Figure) is a figure with even symmetry formed by an odd number of copies of a polyform.

Polycubes have 33 symmetry classes (including asymmetry), and 31 of them have even order. That is too many to show here. Instead I show only oddities with full cubic symmetry. For inverse (point) symmetry, see Pentacube Oddities with Inverse Symmetry.

In all pictures, the cross-sections are shown from back to front.

Thanks to Jaap Scherphuis for pointing out an error in one of my chiral tilings.

Full Symmetry

Full, or achiral octahedral, symmetry is the 48-fold symmetry of a cube.

The 5×5×5 cubes are due to Torsten Sillke.

Achiral Pentacubes

Mike Reid independently found the solution for the M pentacube.

PentacubeBoxesShapeTiles
3×12×25
3×13×25
6×8×25
7×8×25
25×25×25
cube
3125
5×5×14
5×5×19
3D cross with
arms 5×5×14
445
5×9×25
5×16×25
25×25×25
cube
3125

Chiral, Disallowing Reflection

PentacubeBoxesCubeTiles
5×9×15
6×6×15
6×9×15
15×15×15 675

Chiral, Allowing Reflection

PentacubeBoxesCubeTiles
3×5×9
5×5×6
15×15×15675

Last revised 2022-11-29.

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Col. George Sicherman [ HOME | MAIL ]