# Pentacube Oddities

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

## 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.

PentacubeBoxesCubeTiles
3×12×25
3×13×25
6×8×25
7×8×25
25×25×253125
3×15×15 15×15×15675
5×9×25
5×16×25
25×25×25 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

#### Unsolved

Last revised 2016-06-22.

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Col. George Sicherman [ HOME | MAIL ]