A Spatial Problem Solved with Stereographic Projection

The IMO Jury looked for a relatively simple geometric question for the IMO paper, and decided to use only the planar version of the problem. The answer in the planar variant is not surprising: any completely symmetric set consists of the vertices of a regular polygon. The straightforward generalization to space would read: a completely symmetric set consists either of the vertices of a regular polygon, or the vertices of a regular polyhedron. Surprisingly however, this is not the correct answer for the 3D{version: a completely symmetric nonplanar set consists of the vertices of a regular tetrahedron or a regular octahedron. The other regular polyhedrons, namely the cube, the regular dodecahedron and the regular icosahedron, are ruled out. This counterintuitive result is not easy to see, particularly when looking at the problem strictly as one of space geometry. We show here how stereographic projection helps to reduce the problem to a planar one, and thus, makes it easier to understand.