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:

    • 1 | S4 x I / S4 x I (this is just the trivial case of 1 cube)
    • 5 | A5 x I / A4 x I (the uniform compound of 5 cubes, 5-color version)(dual: uniform compound of 5 octahedra, 5-color version)
    • 3 | S4 x I / D4 x I (the uniform compound of three cubes)(dual: 3 octahedra)(compound of compounds of cubes and octahedra)
    • 4 | S4 x I / D3 x I (Bakos' compound) (dual: rigid uniform compound of 4 octahedra)
    • n | D4n x I / D4 x I (n=5: D20 x I / D4 x I) (dual: 5 octahedra)
    • n | D3n x I / D3 x I (n=2: D6 x I / D3 x I) (n=5: D15 x I / D3 x I) (dual: 5 octahedra)
    • n | D2n x I / D2 x I (n=5: D10 x I / D2 x I) (dual: 5 octahedra)
    • 6 | S4 x I / D2 x I (dual: 6 octahedra)
    • 10 | A5 x I / D3 x I (version A) (version B) (20 cubes of versions A+B combined) (dual to A: uniform compound of 10 octahedra) (dual to B: uniform compound of 10 octahedra) (dual to A+B: 20 octahedra)
    • 15 | A5 x I / D2 x I (dual: 15 octahedra)

    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 x I)
    • 2n | D2n x I / D1 x I (n=1: D2 x I / D1 x I)
    • nA | Dn x I / C2 x I (n=5: D5 x I / C2 x I)
    • nB | Dn x I / C2 x I (n=5: D5 x I / C2 x I)
    • 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, 2, 3, 4, 5)
    • 12A | S4 x I / C2 x I
    • 12B | S4 x I / C2 x I
    • 20 | A5 x I / C3 x I (varying angles: 1, 2, 3) (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, 2, 3, 4, 5)
    • 30B | A5 x I / C2 x I (varying angles: 1, 2, 3, 4, 5)

    Five with central freedom:

    • n | Cn x I / E x I (n=5: C5 x I / E x I)
    • 2n | Dn x I / E x I (n=5: D5 x I / E x I)
    • 12 | A4 x I / E x I
    • 24 | S4 x I / E x I
    • 60 | A5 x I / E x I
    Other:
    • Four cubes (S4 x I / D4 x I plus original)
    • "Theosophical" Compound (5 cubes: S4 x I / D3 x I plus original)
    • The cube of 14 cubes (all the rigid S4 cubes combined) (dual: 14 octahedra)
    Virtual Polyhedra, Copyright 1996, George W. Hart