# Pentacube Oddities with Plane Diagonal Rotary 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 plane diagonal rotary 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:

## Plane Diagonal Rotary Symmetry

Plane diagonal rotary symmetry is the symmetry of 180° rotation
around an axis lying diagonally in a plane parallel to some faces of the cells.
The smallest example of a polycube with plane diagonal rotational
symmetry and no stronger symmetry
is this tetracube, found by W. F. Lunnon:

### Achiral Pentacubes

The solutions for pentacubes
**I**,
**M**,
**V**,
**W**,
and
**X**
are trivial.
Those pentacubes already have plane diagonal rotary symmetry.
The solution for pentacube **N**
is the smallest diagonal mirror oddity for the N pentomino.

### Chiral, Disallowing Reflection

The solutions for pentacubes
**G**
and
**S**
are trivial.
Those pentacubes already have plane diagonal rotary symmetry.

### Chiral, Allowing Reflection

The solutions for pentacubes
**G**
and
**S**
are trivial.
Those pentacubes already have plane diagonal rotary symmetry.

Last revised 2023-02-19.

Back to Polyform Oddities
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Polyform Curiosities

Col. George Sicherman
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