单词

Euler's Theorem

释义

Euler's Theorem

If a finite planar graph G is drawn in the plane, so that no two edges cross, the plane is divided into a number of regions (including the exterior region) called faces. Euler's Theorem (for planar graphs) is the following:

Theorem

Let G be a connected planar graph drawn in the plane. If there are V vertices, E edges and F faces, then V − E + F = 2.

An application of this gives Euler's Theorem (for polyhedra):

Theorem

If a convex polyhedron has V vertices, E edges and F faces, then V − E + F = 2.

For example, a cube has V = 8, E = 12, F = 6, and a tetrahedron has V = 4, E = 6, F = 4.

The two theorems are essentially the same as a convex polyhedron is topologically a sphere, and a sphere with a point removed is topologically the plane. See Euler characteristic.

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