A cube has three different sorts of rotational symmetry.  There are the three fourfold axes which pass through the faces.  There are the four threeforld axes which pass through the corners.  And there are the six twofold axes that pass through the edges.

In my "Three Cubes" animation, three identical cubes exist at a single location, penetrating each other.  At first they coincide perfectly in space.  Then they begin to rotate, each about a different fourfold axis, corresponding to the x, y, and z cartesian axes.  After a 90-degree turn, the cubes are again in perfect coincidence.  There is exactly one unique combination of directions for the rotations.

In my "Four Cubes" animations, four cubes coexist in one location and each rotates about a different threefold axis.  After a 120-degree turn, the cubes match up again.  There are three versions of this animation, corresponding to the three distinct ways we can choose directions for the rotations.  The "Four Cubes Vectors" images shows the vectors that correspond (via righthand rule) to the rotations in the animations.  These images are numbered in correspondence to the animations.

In my "Six Cubes" animations, six cubes begin in one common position before each cube rotates about a different twofold axis.  After a 180-degree turn, the cubes are back to their original position.  As with "Four Cubes", there are again three possible distict choices for the combination of directions of rotations.  This is visually represented by the "Six Cubes Vectors" images, which are numbered corresponding to the animations.

Each of the cube animations is accompanied by an "Intersection" animation.  Each frame shows the volume common to all the cubes in the corresponding frame of the cube animation.

The "Stills" images are the midpoint frame of each animation.  Thus "Three Cubes Still" shows the cubes after a 45-degree rotation; "Four Cubes Still" shows the cubes after 60 degrees; "Six Cubes" after 90 degrees.

It is exciting to note where the "Six Cubes" animations resemble the "Four Cubes Still" for an instant.  This additional unexpected symmetry certainly warrants further study.