Pentacube Oddities with Orthogonal-Diagonal Rotary 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. See Polycube Symmetries. Here I show oddities with orthogonal and diagonal rotary symmetry (BC10). 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.

Orthogonal and Diagonal Rotary Symmetry

Orthogonal and diagonal rotary symmetry (Lunnon BC10) is rotary symmetry about a coordinate axis and a plane diagonal axis.

The smallest example of a polycube with orthogonal-diagonal rotary symmetry and no stronger symmetry is this decacube, as found by W. F. Lunnon:

Achiral Pentacubes

The solutions for pentacubes I, X, and Z, are trivial. Those pentacubes already have orthogonal and diagonal rotary symmetry.

The solution for pentacube W is the polycube equivalent of the minimal rotary oddity for the W pentomino. No smaller solutions are known.

Chiral, Disallowing Reflection

No solution is known for the G pentacube without reflection.

Chiral, Allowing Reflection

Last revised 2026-11-01.

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Col. George Sicherman [ HOME | MAIL ]