Question

Explain the relationship between the cube and octahedron (and dodecahedron and icosahedron) which implies they have the same symmetry group?

Answer 1

The cube, octahedron, dodecahedron, and icosahedron have the same symmetry group because they have the same number of faces, edges, and vertices.

Step-by-step explanation:

The cube and the octahedron (as well as the dodecahedron and icosahedron) have the same symmetry group because they share the same number of faces, edges, and vertices.

Both the cube and the octahedron have 8 faces, 12 edges, and 6 vertices. This shared characteristic makes them members of the same symmetry group, known as the cubic symmetry or the octahedral symmetry.

Another way to think about the relationship between these shapes is that the octahedron is the dual of the cube, meaning that the vertices of the octahedron correspond to the faces of the cube, and vice versa.