Combinatorial Properties
A convex polyhedron is a Platonic solid if and only if
- all its faces are congruent convex regular polygons,
- none of its faces intersect except at their edges, and
- the same number of faces meet at each of its vertices.
Each Platonic solid can therefore be denoted by a symbol {p, q} where
- p = the number of edges of each face (or the number of vertices of each face) and
- q = the number of faces meeting at each vertex (or the number of edges meeting at each vertex).
The symbol {p, q}, called the Schläfli symbol, gives a combinatorial description of the polyhedron. The Schläfli symbols of the five Platonic solids are given in the table below.
Polyhedron | Vertices | Edges | Faces | Schläfli symbol | Vertex configuration |
|
---|---|---|---|---|---|---|
tetrahedron | 4 | 6 | 4 | {3, 3} | 3.3.3 | |
cube / hexahedron | 8 | 12 | 6 | {4, 3} | 4.4.4 | |
octahedron | 6 | 12 | 8 | {3, 4} | 3.3.3.3 | |
dodecahedron | 20 | 30 | 12 | {5, 3} | 5.5.5 | |
icosahedron | 12 | 30 | 20 | {3, 5} | 3.3.3.3.3 |
All other combinatorial information about these solids, such as total number of vertices (V), edges (E), and faces (F), can be determined from p and q. Since any edge joins two vertices and has two adjacent faces we must have:
The other relationship between these values is given by Euler's formula:
This nontrivial fact can be proved in a great variety of ways (in algebraic topology it follows from the fact that the Euler characteristic of the sphere is 2). Together these three relationships completely determine V, E, and F:
Note that swapping p and q interchanges F and V while leaving E unchanged (for a geometric interpretation of this fact, see the section on dual polyhedra below).
Read more about this topic: Platonic Solid
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