Describe all of the rotational symmetries of a cube.
a. Find four things that are permuted when the cube undergoes a rotational symmetry. (the things can’t be the vertices, edges, or faces. But, they could involve combinations of these elements.)
b. Label the four things a, b, c, and d. For each rotational symmetry of the cube, express the permutation of the four things produced by the rotation.
The correct answer and explanation is:
Correct Answer:
a. One possible set of four things that are permuted under cube rotations are the four space diagonals of the cube. A space diagonal connects two opposite corners (vertices) of the cube. A cube has exactly four space diagonals, and these are distinct from edges, faces, or vertices.
b. Label the four space diagonals as follows:
- a: Diagonal from vertex 1 to 7
- b: Diagonal from vertex 2 to 8
- c: Diagonal from vertex 3 to 5
- d: Diagonal from vertex 4 to 6
Each rotational symmetry of the cube results in a permutation of the four diagonals. The group of all rotational symmetries of the cube is isomorphic to the symmetric group S₄, meaning that each rotation corresponds to a permutation of the four diagonals {a, b, c, d}.
There are 24 distinct rotational symmetries of a cube. These include:
- Identity rotation (no movement): (a)(b)(c)(d)
- Rotations of 90°, 180°, 270° around face axes
- Rotations of 120°, 240° around body diagonals
- Rotations of 180° around edge axes
Each rotation permutes the diagonals in a different way. Some sample permutations:
- A 120° rotation around the space diagonal a sends b → c, c → d, d → b, a → a. So the permutation is (bcd)
- A 180° rotation about a line through midpoints of opposite edges may swap two diagonals and fix the others, for example: (a b)(c d)
- A 90° rotation around the vertical axis could result in (a b c d)
These permutations can be listed as elements of the group S₄, showing how the cube’s rotational symmetries act on the set {a, b, c, d}. Since the cube’s symmetry group is isomorphic to S₄, all 24 permutations arising from its rotations correspond to the 24 elements in S₄. Thus, each rotation results in a unique rearrangement of the four labeled space diagonals.