Euler's Formula for Polyhedra

The regular polyhedra were known at least since the time of the ancient Greeks. The names of the more complex ones are purely Greek. But despite their being known for close to two millenia no one apparently noticed the fact that the sum of the number of faces F and the number of vertices V less the number of edges E is equal to two for all of them; i.e.,

V - E + F = 2
 

It was the Swiss mathematician Leonhard Euler who recognized and published this fact. The value of two is said to be the Euler characteristic of each of the polyhedra. This value is not changed by stretching or shrinking any side or face or even shrinking a side or face to zero. This means that the Euler characteric is a topological invariant because it is not altered by any continuous mapping.

The quantity (V-E+F) is called the Euler characteristic of the polyhedron and is denoted as by the Greek letter Chi, Χ. Thus Χ(cube)=2.

The Generalization of Euler's Formula

The French-Swiss mathematician, Simon Lhuilier (1750-1840), found a slight generalization of Euler's formula to take into account polyhedra having holes. Lhuilier's formula is

V - E + F = 2 − 2G = 2(1− G)
 

where G is the number of holes in the polyhedron. Thus the Euler characteristic is 2 for a regular polyhedron but 0 for a torus-like polyhedron.

For a simple treatment of the effect of holes and handles on the Euler characteristic see Euler Characteristic.

It was the French mathematician, Henri Poincaré, who fully generalized Euler's formula.

Consider an n-dimensional polytope K. It is made up of simplexes of equal or lesser dimensions. A two dimensional simplex (2-simplex) is a triangle; a one dimensional simplex (1-simplex) is a line and a zero dimensional simplex (0-simplex) is a point. Let α_m be the number of m-dimensional simplexes in K, where m runs from 0 to n. For example, the surface of a tetrahedron is two dimensional and it is composed of 4 triangles (2-simplexes), 6 line segment edges (1-simplexes) and 4 points (0-simplexes).

Chains

Consider a sequential labeling of the m-simplexes of K taking into account their orientation; i.e., choose an m-simplex with a particular orientation and call it number 1, another as number 2 and so on. Now construct vectors with integer components of the form

C = (c₁, c₂, ..., c_αm)
 

where the c_i's are integers (c_i∈Z),

One of these vectors is called a chain and the set of them form a mathematical group under the operation of component-wise addition with the identity being the vector of zeroes and the inverse of a vector being the vector of the negatives of the components. The group is abelian and its dimension is α_m.

A chain may be represented as a formal sum of the form

C = c₁σ₁^m + c₂σ₂^m + ... + c_αmσ_αm^m
 

where σ_i^m is the unit vector for the i-th simplex; i.e.,

σ_i^m = (0,0,..,1,..,0,0) with the "1" in the i-th place.
 

This is essentially the same as representing a vector in 3-space as as the formal sum

V = xi + yj + zk
 

where i, j and k are the unit vectors in the three directions and equivalent to (1,0,0), (0,1,0) and (0,0,1).

The Boundary Operator ∂

The boundary for σ_i^m is defined as

∂(σ_i^m) = Σ [σ_i^m, σ_j^m-1]σ_i^m
 

where [σ_i^m, σ_j^m-1] is the incidence number of the σ_i^m simplex with the σ_j^m-1 simplex; i.e.,

[σ_i^m, σ_j^m-1] = 0 if σ_j^m-1 is not a face of σ_i^m
[σ_i^m, σ_j^m-1] = ±1 if σ_j^m-1 is a face of σ_i^m
 

with the sign of the incidence number depending upon the orientation of σ_j^m-1 with respect to the orientation of σ_i^m.

The boundary of C = Σc_iσ_j^m is then

∂C = Σc_i∂(σ_j^m)
= Σ_ic_iΣ_j[σ_i^m, σ_j^m-1]σ_j^m-1
 

The Boundary of a Boundary

Because of a property of the incidence numbers that for all i and k

Σ [σ_i^m, σ_j^m-1] [σ_j^m-1, σ_k^m-2] = 0
where the sum is over all values of j.
 

it follows that

∂(∂C) = 0
the boundary of a boundary is empty.
 

In other words, boundaries have no boundary.

Cycles and m-Boundaries

The boundary operation defines a homomorphism of the group of chains on m-simplexes, C_m(K) onto the group of chains on the (m-1)-simplexes, C_m-1(K):

∂:C_m(K)→C_m-1(K).
 

Within the group of chains on m-simplexes there are special chains that have an empty boundary; i.e. ∂(C)=0. These boundary-less chains are called m-cycles and they form a subgroup of C_m(K) denoted as Z_m(K).

Within the group of m-cycles there are the special cycles that are the boundary of some chain in C_m+1(K). They form a subgroup of Z_m(K) and hence a subgroup of C_m(K). They are called m-boundaries and their group is denoted as B_m(K).

Homomorphism of Groups

The subgroup relationship of C_m(K), Z_m(K) and B_m(K) determines a factor group of Z_m(K) with respect to B_m(K). This is called the m-th homology group of the polytope K and is denoted as H_m(K)

The boundary operator ∂ maps C_m(K) onto B_m-1(K)(K) and it maps Z_m(K) onto the identity element of B_m-1(K), which is also the identity element of C_m-1(K) and Z_m-1(K). Z_m(K) is said to be the kernal of the homomorphism. This means that Z_m(K) is a factor group of C_m(K) with respect to B_m-1(K); i.e.,

Z_m(K) = C_m(K)/B_m-1(K)
or, in additive notation
Z_m(K) = C_m(K) - B_m-1(K)
 

From the theorems for abelian free groups it follows then for m>0

rank(Z_m(K)) = rank(C_m(K)) - rank(B_m-1(K)
 

Letting rank(Z_m(K))=ζ_m, rank(C_m(K))=α_m, and rank(B_m(K))=β_m the above condition is

ζ_m = α_m - β_m-1
 

For the special case of m=0, rank(B_m-1(K))=0 so

ζ₀ = α₀
 

Let rank(H_m(K))=p_m. The number p_m is known as the m-th Betti number of K, so named after the Italian mathematician Enrico Betti (1923-1892) who believed that the structure of a topological space could be completely characterized by a set of numerical invariants. Now the characterization of a topological space is in terms of the topologically invariant groups H_m(K). Since H_m(K)=Z_m(K)-B_m(K) then

p_m = ζ_m - β_m
in particular
p₀ = ζ₀ - β₀ c
which, since β₀=0, reduces to
p₀ = ζ₀
 

The relevant equations for m>0 placed together are:

α_m = ζ_m - β_m-1
p_m = ζ_m - β_m
 

Subtracting the second equation from the first gives

α_m - p_m = β_m-1 + β_m
for m>0
and
α₀ - p₀ = 0
 

The Euler-Poincaré Formula

Consider now the alternating sum

Σⁿ_m=0(-1)^m(α_m-p_m) = Σⁿ_m=0(-1)^m(β_m-1 + β_m)
 

The right hand side of the above reduces to

+β₀ - β₁ - β₀ + β₂ + β₁
which collapses to
(-1)ⁿβ_n
 

Since C_n+1(K)=0 it follows that B_n(K)=0 and thus β_n=0 so

Σⁿ_m=0(-1)^m(α_m-p_m) = 0
and hence
Σⁿ_m=0(-1)^mα_m = Σⁿ_m=0(-1)^mp_m
 

This latter equation is the Euler-Poincaré Formula. The left hand side is the Euler characteristic of the polytope K.