Besides being interesting exercises in visualization and symmetry, these are fun examples for switching your mind back and forth between seeing the whole and seeing the parts. Each one also presents an individual challenge for you to imagine the shape of its interior before going in to look.
The compounds of cubes on this page and the mathematical relations between them are detailed in the book Symmetry Orbits by Hugo F. Verhayen, listed in the references. It also contains photos of intricate paper models of many of these models, made by Magnus Wenninger or Verheyen. For more compounds (including their duals --- compounds of octahedra which that book does not discuss) see my big list of cube and octahedra compounds. Many of these are derived by Harman's method. Four of the cube compounds on this page and several of the octahedral compounds are uniform; that notion is defined on a separate page about the uniform compounds of uniform polyhedra.
The next-to-last compound in the above list is analogous to the last, but the five axes are chosen to go through two opposite edge midpoints (rather than the face centers) of the five cubes in the "gear," before rotating the cubes on the axes [*].
In these compounds with prism symmetry, there is nothing special about the number five, and we could make similar compounds of any number of cubes. For example here is a compound of two cubes with a common 3-fold axis, resulting in 6-fold symmetry.
Another technique used in creating compounds is called splitting, meaning that pairs of cubes are counter-rotated along the desired symmetry axes: First, imagine six cubes, initially overlapping exactly on top of each other. Imagine also two copies of each of their three 4-fold symmetry axes---a total of six axes. Now attach each one of these six cubes to one of these six axes. Finally, rotate each axis through the same angle, but in each pair of axes, turn one clockwise and one counterclockwise. The result will be a compound of six cubes with a degree of rotational freedom, called Skilling's compound [*], being one of the uniform compounds first described in the 1976 paper by J. Skilling. Here is the same compound but with a different rotation angle. Note that with a rotation angle of 45 degrees, pairs of cubes would overlap and the compound would reduce to the compound of three cubes discussed above.
Splitting four cubes along the four 3-fold axes gives this compound of eight cubes [*], which is of interest because its dual is the uniform compound of eight octahedra [*]. And splitting six cubes along the six 2-fold axes gives this compound of twelve cubes [*]. Because a 4-fold axis is also a 2-fold axis (and then some), we can place six cubes each with a 4-fold axis along the directions of the compound's 2-fold axes to make the above compound of 6 cubes, and then split them to make this compound of 12 cubes [*].
Here is a kind of double splitting: We place the 2-fold axis of two cubes along the 4-fold axis of the desired compound; these two cubes are turned relative to each other by 90 degrees to give a sub-compound with 4-fold symmetry along that axis. Splitting these pairs to four cubes along each of the three 4-fold axes gives a compound of twelve cubes. Varying the free rotation angle causes this to unfold like a flower. Here are five progressively larger angles: 1, 2, 3, 4, 5.
There are two remarkable rigid compounds of ten cubes. To understand them, recall first from your knowledge of platonic relationships how one inscribes a cube in a dodecahedron so that eight of the vertices of the dodecahedron are vertices of the cube. That is the orientation of each cube in the above compound of five cubes. If instead we rotate the cube along one of the 3-fold axes common to it and the dodecahedron, there are two other angles of geometric importance: three edges of the cube (which are still in contact with a dodecahedron vertex) can point to intersect either three nearby 3-fold axes of the dodecahedron or three nearby 5-fold axes. Choosing either of these angles and putting one such cube along each of the ten 3-fold axes of the dodecahedron gives us two different compounds of ten cubes. First described in Harman's 1974 paper, they are called version A and version B, respectively. Dual to these two compounds of ten cubes are two uniform compounds of ten octahedra: version A, version B.
By splitting the ten cubes, starting from either version, we get a compound of 20 cubes with rotational freedom. Here are three different angles: 1, 2, 3. The respective duals to these are uniform compounds of twenty octahedra: 1, 2, 3. The first of these three angles is special in that the cubes' face planes align into 60 coplanar pairs, so the octahedra vertices coincide in pairs, resulting in a rigid compound of 20 octahedra.
By aligning a 2-fold axis of a cube with each of the fifteen 2-fold axes of the icosahedron, we obtain a rigid compound of fifteen cubes. (The cube edge can be made parallel or perpendicular to the icosahedron edge; either way the result is the same.) Splitting this gives a compound of 30 cubes with rotational freedom; here are five different angles: 1, 2, 3, 4, 5.
The first is a nonrigid compound of six cubes [*], constructed by aligning the 2-fold axis of a cube with each of the three 2-fold axes of the tetrahedron and using the splitting technique described above.
The second is rather interesting in that it is constructed by aligning four overlapping cubes so they each have a 3-fold axis in the direction of one of the tetrahedron's four 3-fold axes. Although these are the same directions as the original cube's four 3-fold axes, you must think of them as the tetrahedron's axes for the following step to make sense: Choose an angle and turn each cube on its axis by that angle clockwise as seen from a tetrahedron vertex. This will be counterclockwise as seen from a tetrahedron face at the other end of the axis. The result is a nonrigid compound of four cubes [*]. Find its 3-fold axes and see how every other one has the opposite handedness. At an angle of 60 degrees, it becomes the rigid Bakos compound listed above, with octahedral symmetry. At another angle, this becomes a compound of 4 of the 5 cubes from the standard icosahedral compound of 5 cubes. (The missing one from the five is the original unrotated cube. If we were to also include the original cube unrotated, to make a "compound compound" as described in the next section, the result would be the standard icosahedral compound of 5 cubes.)
Dual to this is the uniform nonrigid compound of 4 octahedra [*]. At the 60 degree angle it reduces to the uniform compound of four octahedra, and at the other angle, a compound of 4 of the 5 octahedra from the standard icosahedral compound of 5 octahedra.
Combining Bakos' compound of four cubes with one cube from which it is derived gives a compound of five cubes which Verheyen calls the Theosophical compound, after a 1905 "Theosophical" paper in which it is described. In its dual, the extra octahedron seems out of place compared to the uniform compound of four octahedra.
Combining the uniform compound of three cubes with one cube from which it is derived gives a compound of four cubes. Its dual is described in the paper "Some Interesting Octahedral Compounds" by Wenninger, listed in the references.
As a final example, one may compound all the rigid compounds with octahedral symmetry---the compound of three cubes, the compound of four cubes, and the compound of six cubes, along with the original cube---to make a compound of fourteen cubes with octahedral symmetry.