Pentacube Oddities with Inverse/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. Here I show oddities with inverse/diagonal 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.

Inverse/Diagonal Symmetry

Inverse/diagonal symmetry is rotary symmetry through a plane diagonal axis plus inverse (point) symmetry. It necessarily also entails plane diagonal mirror symmetry.

The smallest example of a polycube with inverse/diagonal 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 inverse/diagonal symmetry.

Chiral, Disallowing Reflection

Chiral, Allowing Reflection

Last revised 2026-09-03.

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Col. George Sicherman [ HOME | MAIL ]