After about two thousand years that mathematicians had been contemplating polyhedra (pyramids, cubes, etc.) the Swiss mathematician Leonhard Euler (1707-1783) discovered an amazing regularity among them. If the numbers of vertices and faces are added together and the number of edges subtracted the result for all polyhedra is 2 if there are no holes in the polyhedra. Later another Swiss mathematician, Simon Lhuilier (L'Huillier) (1750-1840), found that for all polyhedra having one hole the Euler computation yields the value of 0. The value for the Euler computation for a polyhedron came to be known as the Euler characteristic of the polyhedron. It was symbolized with the Greek letter Chi, Χ. Lhuilier found a formula the Euler characteristic of a polyhedron with G holes; i.e., Χ=2(1-G).

The vertices of a polyhedron are its elements of zero dimension. Edges are the polyhedron's elements of one dimension and faces are the elements of two dimensions. If N_n denotes the number of elements of n dimensions then the Euler characteristic is defined as

Χ = N₀ - N₁ + N₂
 

With this scheme the Euler characteristic can be generalized to

Χ_m = N₀ - N₁ … ± N_m
or, equivalently,
Χ_m = Σ_i=0^m(-1)ⁱN_i
 

This generalization is often called the Euler-Poincaré characteristic.

The Euler computation was carried out for the two dimensional surface of the polyhedra, the hull of the three dimensional figures. If P_m stands for the m dimensional analogue of a polyhedron with no holes then Euler's result was that

Χ₂(P₃) = 2
 

If the three dimensional cell contained within the polyhedral hull is included in the computation then

Χ₃(P₃) = 1
 

The analogous computations can be carried out for geometric figures of other dimensions. A square has four vertices and four edges and one square surface. Therefore Χ₁(P₂) = 4 - 4 = 0 and Χ₂(P₂) = 4 - 4 + 1 = 1.

Likewise a line segment has two vertices and one edge. (Hereafter a line segment will be referred to simply as a line.) Thus Χ₀(P₁) = 2 and Χ₁(P₁) = 2 - 1 = 1. A point is a zero dimensional figure and has just one vertex. Thus Χ₀(P₀) = 1.

The only reasonable definition for the hull of a point is the elimination of the point and the creation of a null figure; i.e., no vertices, no edges, etc. The characteristic for the null figure is zero.

The four dimensional analogue of a cube is called a tessaract. It has 16 vertices, 32 edges, 24 faces and 8 cells. It also has its nameless four dimensional interior which could be called a tessaract solid. Thus

Χ₃(P₄) = 16 - 32 + 24 - 8 = 0
and
Χ₄(P₄) = 16 - 32 + 24 - 8 + 1 = 1
 

The above results are compiled in the following table.

The pattern is clear; i.e.,

Χ_m(P_m) = 1
and
Χ_m-1(P_m) = 1 - (-1)^m
 

Holes

It is very simple to see the effect of holes in a line or a square. A hole in a line creates two additional vertices and two lines where there was only one before. Let P_m(G) denote a geometric figure of m dimensions with G holes. Then

Χ₁(P₁(1)) = 4 - 2 = 2
and
Χ₀(P₁(1)) = 4
 

Stated differently

ΔΧ₁ = 1
and
ΔΧ₀ = 2
 

For a square the creation of a hole results in a figure of 8 vertices, 8 edges and 2 faces. Thus

Χ₂(P₂(1)) = 8 - 8 + 2 = 2
and
Χ₁(P₂(1)) = 8 - 8 = 0
 

For the hull of a square:

For a solid square:

Thus

ΔΧ₂ = 1
and
ΔΧ₁ = 0
 

It was found elsewhere that for a cube

ΔΧ₃ = -1
and
ΔΧ₂ = -2
 

Again the only reasonable definition for a hole in a point is the elimination of the point and the creation of a null figure, which of course has the characteristic zero.

These results are tabulated in the following two tables

Lhuilier's formula is that Χ₂(P₃(G)) = 2(1 - G). Its extension to the solid polyhedra is Χ₃(P₃(G)) = (1 - G). For lines Χ₁(P₁(G)) =(1 + G) and Χ₀(P₁(G)) = 2(1 + G). For squares the relations are the same; i.e., Χ₂(P₂G)) = (1 + G) and Χ₂(P₂(G)) = 2(1+ G). For a point the formula would be Χ_-1(P₀(G)) = 0. The reduction of these to one master formula awaits the analysis of the case of the tessaract.

The Euler Poincaré Characteristic of Tessaract with a Hole

In the case of the hulls for a prism or a polygon the creation of a pit does not alter the topologically invariant characteristic. The change in the characteristic comes only when pits become holes. When two pits in a prism come together to make a hole two faces are eliminated. Since the coefficient for the number of faces in the computation of the characteristic is +1 the elimination of two faces reduces the characteristic by two; i.e., ΔΧ = -2. In the case of a polygon hull the joining of two pits to make a hole eliminates two edges. Since the coefficient of the number of edges in the computation of the characteristic is -1 the elimination of two edges increases the characteristic for the polygon hull by two; i.e., ΔΧ = +2.

Therefore the creation of pits in a tessaract hull does change the characteristic but when the pits come together to form a hole two cells are eliminated. Since the coefficient of cells in the computation of the characteristic is -1 the elimination of two cells increases the characteristic by 2; i.e., ΔΧ = +2.

Thus

(To be continued.)

For more on the Euler-Poincaré Characteristic of geometric figures see Euler.