Pentacube Oddities with Dual Orthogonal Symmetry


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 dual 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:

  • Pentacube Oddities with Orthogonal Mirror Symmetry
  • Pentacube Oddities with Diagonal Mirror Symmetry
  • Pentacube Oddities with Orthogonal Rotary Symmetry
  • Pentacube Oddities with Diagonal Rotary Symmetry
  • Pentacube Oddities with Inverse Symmetry
  • Pentacube Oddities with 4-Rotary Symmetry
  • Pentacube Oddities with Dual Diagonal Symmetry
  • Pentacube Oddities with Full Symmetry
  • Dual Orthogonal Mirror Symmetry

    Dual orthogonal mirror symmetry is mirror symmetry through two different coordinate axes.

    The smallest example of a polycube with dual orthogonal mirror symmetry and no stronger symmetry is the T tetracube:

    Achiral Pentacubes

    The solutions for pentacubes I, T, U, and X are trivial. Those pentacubes already have dual orthogonal mirror symmetry.

    The solutions for pentacubes F, N, P, W, and Y are minimal solutions for the corresponding pentominoes. No smaller solutions are known.

    Chiral, Disallowing Reflection

    The solution for pentacube S is its smallest known box oddity. No smaller solution for this pentacube is known.

    Chiral, Allowing Reflection

    Last revised 2024-02-10.

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    Col. George Sicherman [ HOME | MAIL ]