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 Polycube Oddities.

Dual Diagonal Mirror Symmetry

Dual diagonal mirror symmetry (Lunnon BF6) 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, Disallowing Reflection

Chiral, Allowing Reflection

Last revised 2026-10-29.

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Col. George Sicherman [ HOME | MAIL ]