Exploring Polyhedra and Discovering Euler's Formula

Summary

Summary

The activities and exercises collected here provide an introduction to Euler's formula, an introduction to interesting related topics, and sources for further exploration. While Euler's formula applies to any planar graph, a natural and accessible context for the study of Euler's formula is the study of polyhedra.

The article includes an introduction to Euler's formula, four student activities, and two appendices containing useful information for the instructor, such as an inductive proof of Euler's theorem and several other interesting results that may be proved using Euler's theorem. Each activity includes a discussion of connections to discrete mathematics and notes to the instructor. Three of the activities have worksheets at the end of this article, followed by solutions and a template for a toroidal polyhedron.

A brief introduction to Euler's formula

Theorem (Euler's formula, polyhedral version). Let P be any polyhedron topologically equivalent to a sphere. Let V be the number of vertices, E the number of edges and F the number of faces of P. Then V - E + F = 2.

A graph is said to be a simple graph if it is an undirected graph containing neither loops nor multiple edges. A graph is a plane graph if it is embedded in the plane without crossing edges. A graph is said to be planar if it admits such an embedding.