## Regular Subdivision of a Sphere

A subdivision of a surface is called regular when every face has a boundary consisting of the same number of edges and the same number of edges meets at every vertex.

Euler's formula states (1) where is the number of vertices is the number of edges is the number of faces

If each face has edges, the total number of lines is If edges enter each vertex, we can also write Substitute these into (1) to obtain This equation has simple solutions in the table below.

 v f E V F v=E 2 E 2 F=E 2 f=E E V=E 2

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These solutions are shown below. Other solutions exist. If we obtain the trivial solutions or If the only solution is Suppose E&gt;2 then The only possible values of and are 3, 4, 5. We can draw up the following table.

 v f E V F Name 3 3 6 4 4 Tetrahedron 3 4 12 6 8 Cube 3 5 30 12 20 Dedecahedron 4 3 12 8 6 Octahedron 5 3 30 20 12 Icosahedron

In fact, if f=3 each face is a triangle. Triangles can be subdivided into triangles to form more regular subdivisions. Similarly, ff f=4 each face is a rectangle. Rectangles can be subdivided into rectangles to form more regular subdivisions. 