Pentacube Oddities with Pyritohedral 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 pentacube oddities with pyritohedral 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.

Pyritohedral Symmetry

Pyritohedral symmetry (Lunnon DEE25) is a 24-fold symmetry with every kind of symmetry except plane diagonal.

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

Achiral Pentacubes

Most of these minimal oddities have full (achiral cubic/octahedral) symmetry. No smaller solutions are known.

No solution is known for pentacube X.

Pentacube Boxes Shape Tiles

3×12×25
3×13×25
6×8×25
7×8×25 25×25×25
cube 3125

5×5×14
5×5×19 3D cross with
arms 5×5×14 445

5×9×25
5×16×25 25×25×25
cube 3125

Boxes	Shape	Tiles
3×12×25 3×13×25 6×8×25 7×8×25	25×25×25 cube	3125
5×5×14 5×5×19	3D cross with arms 5×5×14	445
5×9×25 5×16×25	25×25×25 cube	3125

Chiral, Disallowing Reflection

The solution for pentacube E has full (achiral cubic/octahedral) symmetry. No smaller solution is known.

No solution is known for the G pentacube.

Pentacube Boxes Cube Tiles

5×9×15
6×6×15
6×9×15 15×15×15 675

Pentacube	Boxes	Cube	Tiles
	5×9×15 6×6×15 6×9×15	15×15×15	675

Chiral, Allowing Reflection

Pentacube Boxes Cube Tiles

3×5×9
5×5×6 15×15×15 675

Pentacube	Boxes	Cube	Tiles
	3×5×9 5×5×6	15×15×15	675

Last revised 2026-11-04.

Back to Polycube Oddities < Polyform Oddities < Polyform Curiosities

Col. George Sicherman [ HOME | MAIL ]