Pentacube Oddities with Rotary/Orthogonal Mirror 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 rotary/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
Polycube Oddities.
Rotary/Orthogonal Mirror Symmetry
Rotary/orthogonal mirror symmetry is mirror symmetry across a
coordinate axis
with 2-rotary symmetry about the same axis.
The smallest example of a polycube with rotary/orthogonal mirror
symmetry and no stronger symmetry
is the N tetracube:
Achiral Pentacubes
The solutions for pentacubes
I,
X, and
Z,
are trivial.
Those pentacubes already have rotary/orthogonal mirror symmetry.
The solutions for pentacubes
F,
N,
P, and
W
are polycube equivalents of the minimal rotary oddities
for the corresponding pentominoes.
No smaller solutions are known.
Chiral, Disallowing Reflection
No solution is known for the G pentacube
without reflection.
Chiral, Allowing Reflection
Last revised 2026-09-01.
Back to Polycube Oddities
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Polyform Oddities
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Polyform Curiosities
Col. George Sicherman
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