# Pentacube Oddities with Square Box 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 on one page.
Instead I show only pentacube oddities with square box 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 Box Symmetry

Square box symmetry is 90° rotary symmetry about one orthogonal axis
with mirror symmetry in all three orthogonal directions.
The smallest example of a polycube with square box
symmetry and no stronger symmetry
is the dicube:

### Achiral Pentacubes

The solutions for pentacubes
**I** and
**X**
are trivial.
Those pentacubes already have square box symmetry.
The solutions for pentacubes
**A**,
**K**,
**M**,
**N**, and
**V**
are their smallest known oddities with full (achiral cubic/octahedral)
symmetry.
No smaller solutions are known.

The solution for pentacube **T**
is formed by joining two prime boxes.
No smaller solution is known.

The solution for pentacubes
**F**,
**W**,
and
**Z**
are minimal solutions for the
corresponding pentominoes.
No smaller solutions are 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.
The solutions for the **E**
and **H** pentacubes have full (achiral cubic/octahedral)
symmetry. No smaller solutions are known.

No solution is known for the **G** pentacube.

### Chiral, Allowing Reflection

Last revised 2024-04-20.

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Polyform Curiosities

Col. George Sicherman
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