Euler's Formula

# Polygons and Polyhedra

Before we continue on with Euler's Formula, we're going to first look at some important definitions in geometry that will be used quite often on this page:

 Definition: A Polygon is a closed plane figure consisting of three edges at minimum, with each of these edges being straight. Some examples of polygons are various types of triangles, rectangles, squares, quadrilaterals, octagons, etc…
 Definition: A Polyhedron (or polyhedra for plurality) is a solid geometric figure constructed of polygons. For example, cubes, rectangular prisms, pyramids, etc…

# Euler's Formula

So suppose that we look at polyhedra in terms of their physical qualities, specifically the number of vertices, the number of edges, and the number of faces they contain. Note that a face of a polyhedra will be defined as being enclosed between edges, or in terms of graph depictions of these shapes, we will also count what is called an infinite face.

For example, let's look at the graph of a cube:

We can clearly see that this graph has $8$ vertices, $12$ edges, and $6$ faces. We will now construct a table of some other polyhedra:

Name of Polyhedra Number of Vertices ($v$) Number of Edges ($e$) Number of Faces ($f$)
Tetrahedron $4$ $6$ $4$
Triangular Prism $6$ $9$ $5$
Rectangular Prism $8$ $12$ $6$
Cube $8$ $12$ $6$
Square-based Pyramid $5$ $8$ $5$
Hexagonal Prism $12$ $18$ $8$

We should now be able to see a relationship between polyhedra which Euler first discovered:

(1)
$$v + f = e + 2$$

Note that this formula only words for polyhedra with polygonal faces NOT containing holes. Hence, the resulting polyhedra cannot have any ring or dented faces. Nevertheless, we can relate this formula to graphs, that is, for any graph $G$ representing a polyhedra meeting the criteria above:

(2)
\begin{align} \quad \mid V(G) \mid + f = \mid E(G) \mid + 2 \end{align}