# Pentacube Oddities with Dual Orthogonal 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.
In 1996, Torsten Sillke reported
having found a point-symmetric arrangement of 17 F pentacubes.
He asked whether 17 is the least such odd number,
and more generally whether an odd number of copies of a polycube can be arranged
to achieve any given symmetry.
This was the earliest known mention of polyform oddities.

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 orthogonal 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 Orthogonal Mirror Symmetry

Dual orthogonal mirror symmetry is mirror symmetry through two different
coordinate axes.
The smallest example of a polycube with dual orthogonal mirror
symmetry and no stronger symmetry
is the T tetracube:

### Achiral Pentacubes

The solutions for pentacubes
**I**,
**T**,
**U**, and
**X**
are trivial.
Those pentacubes already have dual orthogonal mirror symmetry.
The solutions for pentacubes
**F**,
**N**,
**P**,
**W**,
and
**Y**
are minimal solutions for the corresponding pentominoes.
No smaller solutions are known.

### Chiral, Disallowing Reflection

The solution for pentacube **S**
is its smallest known box oddity.
No smaller solution for this pentacube is known.

### Chiral, Allowing Reflection

Last revised 2024-02-10.

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Polyform Curiosities

Col. George Sicherman
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