# 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 other classes of symmetry, see:

• Pentacube Oddities with Orthogonal Mirror Symmetry
• Pentacube Oddities with Diagonal Mirror Symmetry
• Pentacube Oddities with Orthogonal Rotary Symmetry
• Pentacube Oddities with Plane Diagonal Rotary Symmetry
• Pentacube Oddities with Inverse Symmetry
• Pentacube Oddities with 4-Rotary Symmetry
• Pentacube Oddities with Dual Orthogonal Mirror Symmetry
• Pentacube Oddities with Dual Diagonal Mirror 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 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.

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 2024-01-23.

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Col. George Sicherman [ HOME | MAIL ]