Pentacube Oddities with Full 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. That is too many to show here. Instead I show only oddities with full cubic symmetry.

For other classes of symmetry, see:

  • Pentacube Oddities with Orthogonal Mirror Symmetry
  • Pentacube Oddities with Diagonal Mirror Symmetry
  • Pentacube Oddities with Orthogonal Rotary Symmetry
  • Pentacube Oddities with Plane Diagonal Rotary Symmetry
  • Pentacube Oddities with Inverse Symmetry
  • Pentacube Oddities with 4-Rotary Symmetry
  • Pentacube Oddities with Dual Orthogonal Mirror Symmetry
  • Pentacube Oddities with Dual Diagonal Mirror Symmetry
  • In all pictures, the cross-sections are shown from back to front.

    Thanks to Jaap Scherphuis for pointing out an error in one of my chiral tilings.

    Full Symmetry

    Full, or achiral octahedral, symmetry is the 48-fold symmetry of a cube or a regular octahedron.

    The 5×5×5 cubes are due to Torsten Sillke.

    Achiral Pentacubes

    Mike Reid independently found the solution for the M pentacube.

    The oddity for the B pentacube can also be tiled by the Q pentacube.

    PentacubeBoxesShapeTiles
    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

    Unsolved

    Chiral, Disallowing Reflection

    PentacubeBoxesCubeTiles
    5×9×15
    6×6×15
    6×9×15
    15×15×15 675

    Unsolved

    Chiral, Allowing Reflection

    PentacubeBoxesCubeTiles
    3×5×9
    5×5×6
    15×15×15675

    Last revised 2024-01-23.


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    Col. George Sicherman [ HOME | MAIL ]