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. Here I show pentacube oddities with full cubic symmetry. For other classes of symmetry, see Polycube Oddities.

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 or a regular octahedron.

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

Achiral Pentacubes

Mike Reid independently found the solution for the M pentacube.

The oddity for the B pentacube can also be tiled by the Q pentacube.

Pentacube Boxes Shape Tiles

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

Boxes	Shape	Tiles
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

Unsolved

Chiral, Disallowing Reflection

Pentacube Boxes Cube Tiles

5×9×15
6×6×15
6×9×15 15×15×15 675

Pentacube	Boxes	Cube	Tiles
	5×9×15 6×6×15 6×9×15	15×15×15	675

Unsolved

Chiral, Allowing Reflection

Pentacube Boxes Cube Tiles

3×5×9
5×5×6 15×15×15 675

Pentacube	Boxes	Cube	Tiles
	3×5×9 5×5×6	15×15×15	675

Last revised 2024-03-07.

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Col. George Sicherman [ HOME | MAIL ]