# Pentacube Oddities with Dual Diagonal 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 dual diagonal mirror symmetry.
In all pictures, the cross-sections are shown from top to bottom.
If you find a smaller solution, please write.

For other classes of symmetry, see:

## Dual Diagonal Mirror Symmetry

Dual diagonal mirror symmetry is mirror symmetry through two different
plane diagonal axes.
The smallest example of a polycube with dual diagonal mirror
symmetry and no stronger symmetry
is this hexacube, found by W. F. Lunnon:

### Achiral Pentacubes

The solutions for pentacubes
**I** and
**X**
are trivial.
Those pentacubes already have dual diagonal mirror symmetry.
The solution for pentacube
**W**
is a minimal solution for the **W** pentomino.
No smaller solution is known.

### Chiral, Disallowing Reflection

### Chiral, Allowing Reflection

Last revised 2024-02-15.

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Polyform Curiosities

Col. George Sicherman
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