Topology

Euler’s Formula for Polyhedra

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Variables:

• \(V is the number of vertices, E the number of edges, F the number of faces of a convex polyhedron (or planar connected graph).\)

Description/Usage:

A classic result relating the counts of fundamental elements of polyhedra (e.g., a cube has \(V=8, E=12, F=6\), and indeed \(8-12+6=2\)). This is often shown by deforming the polyhedron to a network on a sphere or plane. It generalizes to \(V-E+F=2-2g\) for surfaces of higher genus \(g\).

Euler Characteristic of Surfaces

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Variables:

Description/Usage: