Euler Characteristic of Polyhedra

 

Leonhard Euler was a Swiss mathematician of the 18^th century. One of his many contributions to mathematics was his polyhedron formula:

V+F-E=2

This formula states there is a fixed relationship between the vertices (V), faces (F), and edges (E) of any solid, convex polyhedron.

Here are examples of the formula for some common, simple shapes:

Tetrahedron
4 vertices
4 faces
6 edges

4+4-6=2

Four-sided pyramid
5 vertices
5 faces
8 edges

5+5-8=2

Cube
8 vertices
6 faces
12 edges

8+6-12=2

The “2” at the end of the equation is also known as the Euler characteristic (Euler characteristic - Wikipedia). In the study of topology, the Euler characteristic of all solid, convex polyhedra is 2. The Euler characteristic of a toroidal polyhedron (a doughnut-like solid with a hole) is 0.