A polycube can have any of 33 forms of symmetry, including asymmetry.
Here are examples of them.
Polycube symmetries (conjugacy classes of subgroups of the achiral
octahedral group) were first identified by W. F. Lunnon in
Symmetry of Cubical and General Polyominoes,
in Graph Theory and Computing,
Ronald C. Read, editor, New York, Academic Press, 1972.
Lunnon's codes are given below.
An asterisk means that Lunnon's example differs from mine.
1-Fold Symmetry (Asymmetry)
This pentacube has no symmetry.
So do three other pentacubes.
Polycubes may have 5 forms of binary symmetry,
generated by orthogonal rotation, plane diagonal rotation,
orthogonal reflection, plane diagonal reflection, or inversion.
(Lunnon: B6*, C4, E4, F5, CF6)
Polycubes may have one form of ternary symmetry,
generated by rotations on a solid diagonal axis.
Polycubes have 9 types of quaternary symmetry.
The second type shown is unusual: the transform that generates it
a 90° orthogonal rotation followed by reflection through
the plane perpendicular to the axis of rotation.
(Lunnon: A12, J10, BC10, BB10, CK6, BE4, CE3, BF6, EE4)
Polycubes have 3 forms of senary symmetry:
rotation around the solid diagonal combined with (a) plane diagonal rotations,
(b) plane diagonal reflections, or (c) inversion.
(Lunnon: CD10, FF4, H12)
Polycubes have 7 forms of octonary symmetry.
(Lunnon: AB16, EF6, BFF8, CJ6, AE8, EFF7, EEE6)
Polycubes have two forms of duodenary symmetry.
The first shown is called chiral tetrahedral symmetry, or T,
because it is the group of proper motions of a platonic tetrahedron.
The second is generated by rotation around a solid diagonal and reflections
through the three conjugate plane diagonals.
(Lunnon: BD34, BF6)
Polycubes have one form of sedenary symmetry, the symmetry group
of a square prism.
Polycubes have three forms of 24-fold symmetry.
The first is called the chiral octahedral group O,
because it is the group of proper motions of a platonic octahedron.
The other two are derived
from the tetrahedral group by adding diagonal reflection or inversion.
The first of these is called the achiral tetrahedral group,
The second is called the pyritohedral group, or Th.
(Lunnon: R56, CCC20, DEE25)
The maximum symmetry for a polycube is the full symmetric
group of a cube, with 48 elements.
Crystallographers call this group the achiral octahedral group,
Last revised 2013-03-12.
Back to Polyform Curiosities.
Col. George Sicherman