# 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:

• Pentacube Oddities with Orthogonal Mirror Symmetry
• Pentacube Oddities with Diagonal Mirror Symmetry
• Pentacube Oddities with Orthogonal Rotary Symmetry
• Pentacube Oddities with Diagonal Rotary Symmetry
• Pentacube Oddities with Inverse Symmetry
• Pentacube Oddities with 4-Rotary Symmetry
• Pentacube Oddities with Dual Orthogonal Mirror Symmetry
• Pentacube Oddities with Square Symmetry
• Pentacube Oddities with Square Box Symmetry
• Pentacube Oddities with Full Symmetry
• ## 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, Allowing Reflection

Last revised 2024-02-15.

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Col. George Sicherman [ HOME | MAIL ]