Compounds of Cubes
This is a list of the many ways to assemble identical concentric cubes
or their dual octahedra into a compound with overall polyhedral symmetry.
The ordering and notation used here is that given in the 1996 book Symmetry
Orbits by Hugo F. Verhayen, listed in the references,
where many of these cube compounds are first described or illustrated.
That work should be consulted for further details. However, many of these
compounds have never been made in paper, and their pictures appear in no
other reference than here.
The notation can be understood approximately as follows: "n |
P x I / Q x I" indicates a compound of n cubes. The
overall symmetry is k-gonal (prism), tetrahedral, octahedral, or
icosahedral, according to whether the P term is Dk, A4, S4,
or A5, respectively. The Q term of D2 or C2, D3 or C3, and D4 or
C4 indicates alignment along a 2-fold, 3-fold, and 4-fold axis, respectively.
The initial n term may be followed by A or B if there are two variants.
Ten rigid compounds:
Fifteen with rotational freedom:
Note: The following entries each show one or more fixed angles within
a one-parameter degree of rotational freedom. You can see these compounds
in a 4D format, with rotation angle mapped into the time dimension, on
my list of spinning compounds.
6 | S4 x I / C4 x I (Skilling's uniform
compound) (smaller separation angle)
(dual: 6 octahedra)
2n | D4n x I / C4 x I (n=3: D12 x I / C4)
2n | D3n x I / C3 x I (n=1: D3 x I / C3
2n | D2n x I / D1 x I (n=1: D2 x I / D1
nA | Dn x I / C2 x I (n=5: D5 x I / C2
nB | Dn x I / C2 x I (n=5: D5 x I / C2
4 | A4 x I / C3 x I (dual: uniform
nonrigid compound of 4 octahedra)
6 | A4 x I / C2 x I
8 | S4 x I / C3 x I (dual: uniform
compound of 8 octahedra)
12 | S4 x I / D1 x I (varying angles: 1,
12A | S4 x I / C2 x I
12B | S4 x I / C2 x I
20 | A5 x I / C3 x I (varying angles: 1,
(dual to 1: the rigid uniform
compound of 20 octahedra meeting 2 per vertex), (duals to 2
and 3 are uniform compounds
of 20 octahedra with rotational freedom)
30A | A5 x I / C2 x I (varying angles: 1,
30B | A5 x I / C2 x I (varying angles: 1,
Five with central freedom: