# Euler's formula?

#### Joppy

##### Well-known member
MHB Math Helper
Hi, I have been working through a book on algebraic topology (an introduction) and am (embarrassingly) stuck on the following exercise.

Given a polyhedron (doesn't have to be regular), let $F_n$ be the number of $n$-gon faces, and let $V_n$ be the number of vertexes at which exactly $n$ edges meet. Verify the following (there are several similar relations, but I'll just give one):

$(2V_3 + 2V_4 + 2V_5 + ...) - (F_3 + 2F_4 + 3F_5 + ...) = 4$.

I don't understand what we mean when we are given a polyhedron and then we sum an infinite number of vertices and faces? Would someone mind providing insight into what these sums are representing?

#### Euge

##### MHB Global Moderator
Staff member
Hi Joppy ,

In those sums, all but finitely many terms are zero. So the sums are really finite sums.

#### Joppy

##### Well-known member
MHB Math Helper
Hi Joppy ,

In those sums, all but finitely many terms are zero. So the sums are really finite sums.
Thanks Euge.

Do you have any ideas on how to verify the expression?

Because $F_n$ and $V_n$ represent an arbitrary number of $n$-gon faces and vertices (at which $n$ edges meet) I'm quite confused as to how we even work with these expressions.

#### Euge

##### MHB Global Moderator
Staff member
The way the question is phrased, the author must be using a more general definition of polyhedron. What is the definition of polyhedron in your text?

Euler's polyhedral formula states that for a polyhedron with $V$ vertices, $F$ faces, and $E$ edges, $V - E + F = 2$.

#### Joppy

##### Well-known member
MHB Math Helper
Thanks!

The way the question is phrased, the author must be using a more general definition of polyhedron. What is the definition of polyhedron in your text?
A polyhedron is a complex that is topologically equivalent to a sphere. A complex is built from cells (topologically equivalent to a disk) by 'gluing' the edges of cells together.

Euler's polyhedral formula states that for a polyhedron with $V$ vertices, $F$ faces, and $E$ edges, $V - E + F = 2$.
I was expecting to apply this.

#### Opalg

##### MHB Oldtimer
Staff member
The total number of vertices in the polyhedron is $V_3 + V_4 + V_5 + \ldots$. The total number of faces is $F_3 + F_4 + F_5 + \ldots$. To apply Euler's formula, you then need to know how many edges there are.

Notice that an $n$-gon face has $n$ edges, and that each edge in the polyhedron is shared between two faces.

#### Euge

##### MHB Global Moderator
Staff member
A polyhedron is a complex that is topologically equivalent to a sphere. A complex is built from cells (topologically equivalent to a disk) by 'gluing' the edges of cells together.

OK. Then show that $2E = 3F_3 + 4F_4 + 5F_5 + \cdots$. Using $2V - 2E + 2F = 4$, obtain the result.

#### HallsofIvy

##### Well-known member
MHB Math Helper
The simplest example is a tetrahedron. It has 4 faces, 4 vertices, and 6 edges. 4- 6- 4= 2.

There is also "Euler's formula" on a surface such as a plane. With F, E, and V the faces, edges, and vertices of a polygon, F- E+ V= 1. For example, a triangle has 1 face, 3 edges, and 3 vertices: 1- 3+ 3= 1. Given Euler's formula for a surface, you can then prove Euler's formula for three dimensions, as Joppy suggested. In the surface of a sphere we would need to treat the "outside" as a face so that "F" is increased by 1 and instead of F- E+ V= 1, we have F- E+ V= 2.