Definition:Euler Characteristic of Surface

Definition

Let $S$ be a surface.

Let $T$ be a triangulation of $S$.

The Euler characteristic of $S$ is written $\map \chi S$ and is defined as:

$\map \chi S = v - e + f$

where:

$v = \size V$ is the number of vertices of $T$

$e = \size E$ is the number of edges of $T$

$f$ is the number of faces of $T$.

Also see

• Definition:Euler Characteristic of Finite Graph

• Euler's Theorem for Planar Graphs

• Euler Polyhedron Formula

• Euler Characteristic is not Dependent upon Triangulation

• Results about the Euler characteristic of a surface can be found here.

Source of Name

This entry was named for Leonhard Paul Euler.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Euler characteristic
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Euler's theorem: 1. (for polyhedra)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Euler characteristic