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 hasedges, the total number of lines is

Ifedges 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 |

>

These solutions are shown below.

Other solutions exist. Ifwe obtain the trivial solutionsor

Ifthe only solution is

Suppose E>2 then

The only possible values ofandare 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.