# Pentacube Oddities with Square 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 square 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:

## Square Symmetry

Square symmetry is 90° rotary symmetry about one orthogonal axis
with mirror symmetry through the other two orthogonal directions.
The smallest example of a polycube with square
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 square symmetry.
The solutions for pentacubes **V**
and **M**
are their smallest known oddities with full (achiral cubic/octahedral)
symmetry.
No smaller solutions are known.

The solution for pentacube
**W**
is a minimal solution for the **W** pentomino.
No smaller solution is known.

### Chiral, Disallowing Reflection

The solution for the **S** pentacube
is formed by joining three copies of the minimal known odd box.
No smaller solution is known.
No solution is known for the **G** pentacube.

### Chiral, Allowing Reflection

Last revised 2024-04-05.

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Polyform Curiosities

Col. George Sicherman
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