Finding the Relationship
Now that the groups of the Rubik’s cube, the octahedron puzzle and the cuboctahedron puzzle have been found, the next goal is to relate them. The best way to show that groups are related is by mapping the generators of one group to elements in the other group. Our original hypothesis was that the Rubik’s cube group and the octahedron group are both subgroups of the cuboctahedron group. This would mean that the generators of both the Rubik’s cube group (F, R, L, U, D, B) and the octahedron group (UF, UR, UL, UB, DF, DR, DL, DB) can be mapped to elements in the cuboctahedron group. This is a logical hypothesis since the cuboctahedron has both square and triangular sides which come from the cube and octahedron, as shown in the truncation sequence below.