# 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:

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.

#### Unsolved

### Chiral, Disallowing Reflection

#### Unsolved

### Chiral, Allowing Reflection

Last revised 2024-03-07.

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Polyform Curiosities

Col. George Sicherman
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