Pentacube Oddities with Inverse 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.

In 1996, Torsten Sillke reported having found a point-symmetric arrangement of 17 F pentacubes. He asked whether 17 is the least such odd number, and more generally whether an odd number of copies of a polycube can be arranged to achieve any given symmetry. This was the earliest known mention of polyform oddities.

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 inverse symmetry, or point 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 Symmetry

Inverse, or point, symmetry, is the symmetry of diametrically opposite cells. For every cell of the polycube, there is a cell of the polycube lying at the same distance from the center point, in the opposite direction.

The smallest example of a polycube with inverse symmetry and no stronger symmetry is this hexacube, found by W. F. Lunnon:

Achiral Pentacubes

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

Chiral, Disallowing Reflection

Chiral, Allowing Reflection

Last revised 2026-02-12.

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Col. George Sicherman [ HOME | MAIL ]