Show that there can be no more than 5 regular polyhedra. (b) Find volume and sur

Question

Show that there can be no more than 5 regular polyhedra. (b) Find volume and surface of a regular octahedron of edge e. (c) For each of the 5 regular polyhedra, enumerate the number of vertices v, edges e, and faces f, and then evaluate the quantity v - e +f f. One of the most interesting theorems relating to any convex (or more generally any simply connected) polyhedron, is that v - e + f = 2. This may have been known to Archimedes (ca. 225 B.C.), and was very nearly stated by Descartes about 1635. Since Euler later independently announced it in J 752, the result is often referred to as the Euler-Descartes formula.

Explanation / Answer

A) It is impossible to have more than 5 regular polyhedra (platonic solids) because any other possibility will violate rules (Euler's Formula) about the number of edges, faces and corners that we can have.

For example: for any convex polyhedron (also for platonic solids), no. of faces (f) + no. of vertices or corner points (v) - no. of edges (e) will always equal to 2.

Now if we have more than 5 regular polyhedra this rule violates.

C) For a cube f= 6, v = 8, and e = 12

v + f - e = 2.

For a tetrahedron f= 4, v = 4, and e = 6

v + f - e = 2.

For a octahedron f= 8, v = 6, and e = 12

v + f - e = 2.

For a icosahedron f= 20, v = 12, and e = 30

v + f - e = 2.

For a dodecahedron f= 12, v = 20, and e = 30

v + f - e = 2.

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