# 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:

• 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 Dual Diagonal Mirror Symmetry
• Pentacube Oddities with Square Symmetry
• Pentacube Oddities with Full Symmetry
• ## 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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Col. George Sicherman [ HOME | MAIL ]