Fixedly Five Cunning Cuts

All Platonic solids can be found in and around the cube. Take every second vertex of the cube and you get the vertices of a tetrahedron. The six midpoints of a cube’s faces are the vertices of an octahedron. Placing suitable identical tents on its faces gives the dodecahedron. And, even the icosahedron can be circumscribed around the cube in a symmetric fashion. And the cube is not special in this respect: take any of the other solids and the other four can be inscribed or circumscribed just as nicely. A regular polyhedron is one having identical regular polygons as faces, the same number of such faces around each vertex, and with no concavities. Why should there be exactly five of them? For several copies of a regular polygon to combine into a corner, they have to leave a gap when arranged around a point in the plane. This is only possible with three or four or five equilateral triangles, three squares, and three pentagons. This gives a total of five possible corners, every one of which corresponds to one of the Platonic solids