Euler - Poincare Formula

Euler s formula was generalized by Schldfli [8]. The Euler - Poincare formula is an extension of Euler s polyhedral formula. It takes into account more complicated polyhedra with penetrating holes g that are addressed as the genus of the body, internal voids (chambers) with shells s, and loops Z that are closed paths surrounding the edges. For this case 

A conventional polyhedron has as many loops as surfaces, no holes but one inner void which is the volume itself that give rise to only one outer shell. Equation (7.8) reduces then to Eq. (7.5). 

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Another method, applied to evaluate the connectivity of two-phase materials, is based on the Euler-Poincare formula that establishes for a single solid, independently on its complexity, the following rule ... 

According to the Euler-Poincar formula 3 ), we have for K that (number of vertices) — (number of 1-simplexes)=rBTo(fiQ — riTi(iif).Thus, in our case. 

Carbon boron duality Lipscomb and co-workers proposed boron nanotubes, one year after our prediction. They claimed that boron nanotubes can be derived from the dual geometries of boron and carbon cylinders (nanotubes) and based upon Euler-Poincare formula for a cylinder P F=E P for vertices, Ffor faces, and E for edges). The boron duals may be imagined as arising from a correspondence between a boron atom and the center of each carbon face in the carbon nanotube. If the formula for a given carbon nanotube is taken to be C/>, then the formula for its boron dual nanotube is Bp Hp. This means that the numbers P and F are interchanged in going from carbon nanotube to boron nanotube partner. 

The Euler-Poincare formula invokes the use of Betti numbers [10] which may be calculated as the count of the number of critical points, of various types, associated with the geometrical structure of nanotori. The theory of Morse flmctions [11] relates critical points to topological structure. We shall show, an alternating sum of Betti numbers defines the Euler characteristic of a torus to be zero. This connects the topology of a nanotorus, nanotube, and plan sheet, which have the same Euler characteristic. We show that for every possible carbon nanotorus there is a geometrical dual boron nanotorus. 

Carbon nanotubes are cylindrical structures related to carbon fullerene structures. Indeed, carbon nanotube cylinders are often capped at their ends with hemispherical carbon fiillerenes, illustrating the close relation of the two types of structure. Nanotube structures are of great interest because of their mechanical and one-dimensional electrical properties [2]. In our discussion of boron nanotubes [8], the duals of carbon nanotubes, we employ a generalization of Descartes-Euler formula, viz., the Euler-Poincare formula for a cylinder,... 

In Table V we display a count of P, C, and F for each of the molecules displayed in Figure 7. It may be seen how the Euler-Poincare formula is satisfied... 

Table V. Carbon and Boron Dual Nanotubes. Listed are the factors of the Euler-Poincare formula for a cylinder, viz., P - C + F = 0.

The Descartes-Euler formula, equation 1, has been used to define the class of molecules called boron fullerenes as the topological duals of carbon fiillerenes. In order to extend the concept of duality to nanotubes, however, the Descartes-Euler formula must be generalized to the Euler-Poincare formula as in Equation 2. One may understand why the right side of Equation 2 is zero by use of Betti numbers [10]. Betti numbers may be calculated as a count of the number of critical points of each type (i.e., minima, saddle points, maxima) associated with the geometrical structure of a molecule. The Euler-Poincare formula [11] may be written in a very general way in terms of Bptti numbers as... 

The Euler-Poincare formula implies a duality which we have explored in the context of molecular nanotori. We show that for every possible carbon nanotorus there is a topological dual boron nanotorus. Moreover the alternating sum of Betti numbers is also zero for an open nanotube, and a planar network, which is consistent with the possibility of converting a nanotorus into an open nanotube, and thence into a planar network. 

When additionally the complex A is pure, it follows from Theorem 12.3(2) that the reduced Betti number is nonzero only in the top dimension. Therefore, by the Euler-Poincare formula, in this case the cohomology groups can be computed simply by computing the Euler characteristic. In the even more special case that A is an order complex of a poset A = A P), by Hall s theorem, it suffices to compute the value of the Mobius function pp 0,1). 

A special place in the theory of vertex-colorings of a graph is occupied by the so-called four-color problem the question whether there is a four-coloring of a planar map such that every pair of coimtries that share a (nonpoint) boundary segment receive different colors. Let us show the weaker five-color theorem. Before we can prove it, we need a standard fact, which is a special case of the Euler-Poincare formula. 

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