Polycube Symmetries

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. (Lunnon: I5*)

2-Fold Symmetry

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)

3-Fold Symmetry

Polycubes may have one form of ternary symmetry, generated by rotations on a solid diagonal axis. (Lunnon: D7)

4-Fold Symmetry

Polycubes have 9 types of quaternary symmetry. The second type shown is unusual: the transform that generates it consists of 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)

6-Fold Symmetry

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)

8-Fold Symmetry

Polycubes have 7 forms of octonary symmetry. (Lunnon: AB16, EF6, BFF8, CJ6, AE8, EFF7, EEE6)

12-Fold Symmetry

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, reflections through the three conjugate plane diagonals, and inversion. (Lunnon: BD34, DF6)

16-Fold Symmetry

Polycubes have one form of sedenary symmetry, the symmetry group of a square prism. (Lunnon: BBC2)

24-Fold Symmetry

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, or Td. The second is called the pyritohedral group, or Th. (Lunnon: R56, CCC20, DEE25)

48-Fold Symmetry

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, or Oh. (Lunnon: G1)

Relations Among Symmetry Classes

This table shows the relations among the symmetry classes:

12346812162448
I5
E		C4
V		B6
I		CF6
E~		F5
E/		E4
E|		D7
Y		A12
L		J10
I$		BC10
K		BB10
*		CK6
V/~		BE4
I|~		CE3
V|/		BF6
I/		EE4
I|		CD10
X		FF4
Y/		H12
Y~		AB16
#		EF6
L|/		BFF8
*/$		CJ6
K|$		AE8
L|~$		EFF7
K|/~		EEE6
*|~		BD34
T		DF6
X/~		BBC2
#|/~$		R56
O		 CCC20
T/$		DEE25
T|~		G1
O|/~$
I5>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
C4<∥∥∥∥∥∥∥>∥>∥>∥∥∥∥∥>∥∥>∥>∥∥∥>>∥∥>
B6<∥∥∥∥∥>>>>∥>∥>>∥∥∥>>>>>>>>∥>>>>>
CF6<∥∥∥∥∥∥∥∥∥>>∥∥∥∥∥>∥∥∥∥>>>∥>>∥∥>>
F5<∥∥∥∥∥∥∥∥∥>∥>>∥∥>∥∥>>∥∥>∥∥>>∥>∥>
E4<∥∥∥∥∥∥∥∥∥∥>>∥>∥∥∥∥>∥>>>>∥∥>∥∥>>
D7<∥∥∥∥∥∥∥∥∥∥∥∥∥∥>>>∥∥∥∥∥∥∥>>∥>>>>
A12<∥<∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥>>∥∥>∥∥∥∥>>∥∥>
J10<∥<∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥>>>∥∥∥∥>∥>∥>
BC10<<<∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥>∥∥>∥>∥∥∥>>∥∥>
BB10<∥<∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥>∥>∥∥∥>∥∥>>∥∥>
CK6<<∥<<∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥>∥∥∥>∥∥∥>
BE4<∥<<∥<∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥>>>∥∥>∥∥>>
CE3<<∥∥<<∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥>∥∥∥>∥∥∥>
BF6<∥<∥<∥∥∥∥∥∥∥∥∥∥∥∥∥∥>>∥∥>∥∥∥>∥>∥>
EE4<∥<∥∥<∥∥∥∥∥∥∥∥∥∥∥∥∥>∥>>>>∥∥>∥∥>>
CD10<∥∥∥∥∥<∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥>>∥>>>>
FF4<∥∥∥<∥<∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥>∥∥>∥>
H12<∥∥<∥∥<∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥>∥∥∥>>
AB16<<<∥∥∥∥<∥<<∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥>>∥∥>
EF6<∥<∥<<∥<∥∥∥∥∥∥<<∥∥∥∥∥∥∥∥∥∥∥>∥∥∥>
BFF8<∥<∥<∥∥∥<∥<∥∥∥<∥∥∥∥∥∥∥∥∥∥∥∥>∥∥∥>
CJ6<<<∥∥<∥∥<<∥∥∥∥∥<∥∥∥∥∥∥∥∥∥∥∥>∥∥∥>
AE8<∥<<∥<∥<<∥∥∥<∥∥<∥∥∥∥∥∥∥∥∥∥∥>∥∥∥>
EFF7<<<<<<∥∥∥<∥<<<<<∥∥∥∥∥∥∥∥∥∥∥>∥∥∥>
EEE6<∥<<∥<∥∥∥∥<∥<∥∥<∥∥∥∥∥∥∥∥∥∥∥>∥∥∥>
BD34<∥<∥∥∥<∥∥∥∥∥∥∥∥∥<∥∥∥∥∥∥∥∥∥∥∥>>>>
DF6<∥∥<<∥<∥∥∥∥∥∥∥∥∥<<<∥∥∥∥∥∥∥∥∥∥∥∥>
BBC2<<<<<<∥<<<<<<<<<∥∥∥<<<<<<<∥∥∥∥∥>
R56<<<∥∥∥<<∥<<∥∥∥∥∥<∥∥<∥∥∥∥∥∥<∥∥∥∥>
CCC20<∥<∥<∥<∥<∥∥∥∥∥<∥<<∥∥∥∥∥∥∥∥<∥∥∥∥>
DEE25<∥<<∥<<∥∥∥∥∥<∥∥<<∥<∥∥∥∥∥∥∥<∥∥∥∥>
G1<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
I5
E		C4
V		B6
I		CF6
E~		F5
E/		E4
E|		D7
Y		A12
L		J10
I$		BC10
K		BB10
*		CK6
V/~		BE4
I|~		CE3
V|/		BF6
I/		EE4
I|		CD10
X		FF4
Y/		H12
Y~		AB16
#		EF6
L|/		BFF8
*/$		CJ6
K|$		AE8
L|~$		EFF7
K|/~		EEE6
*|~		BD34
T		DF6
X/~		BBC2
#|/~$		R56
O		 CCC20
T/$		DEE25
T|~		G1
O|/~$

Puzzle

What is the largest number n such that no polycube with exactly n cells has full symmetry?

Last revised 2024-05-12.

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