Pentacube Oddities with Staggered Rotary/Orthogonal Mirror 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 staggered rotary/orthogonal 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.

Nomenclature

Staggered Rotary/Orthogonal Mirror Symmetry

Staggered rotary/orthogonal mirror symmetry is the symmetry of 180° rotation about a coördinate axis, and 90° rotation followed by reflection through the same axis.

The smallest example of a polycube with staggered rotary/orthogonal symmetry and no stronger symmetry is this decacube (10-cube), as found by W. F. Lunnon:

Achiral Pentacubes

The solutions for pentacubes I and X are trivial. Those pentacubes already have staggered rotary/orthogonal mirror symmetry.

The solutions for pentacubes F, N, W, X, and Z are polycube equivalents of the smallest known dual orthogonal oddities for the corresponding pentominoes. No smaller solutions are known.

Chiral, Disallowing Reflection

The solution for pentacube S is a square box formed by joining three rectangular boxes. No smaller solution is known.

Unsolved

Chiral, Allowing Reflection

Last revised 2026-11-01.

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Col. George Sicherman [ HOME | MAIL ]