Descartes-Euler formula

By the way, do you know what a dual polyhedron is If you take a boron octahedron and I put a carbon in the center of each of the faces and throw away the borons, then I get cubane. Now, here is a proposal I made in 1978, before buckminsterfullerene was discovered it has 32 borons and 60 faces. If I put a carbon in the center of each of its faces and throw away the borons, then I get buckminsterfullerene. This B32H32 is the dual polyhedron of buckminsterfullerene. When the discovery of buckminsterfullerene happened in 1985, I remembered that I d done something along those lines. Then later Lou Massa and I looked at these analogues in general so we could go on to propose other examples, all related by the Descartes-Euler formula, which can be found in Coxeter s book. 

Imitation of carbon fullerenes In an analogy to carbon fullerenes, Lipscomb and co-workers proposed boron fullerenes as large closo boron hydride based on Descartes-Euler formula P + F=E- -2 P for vertices, Ffor faces. 

For example, for a cube P-C + F = 8-12 + 6 = 2, as required. Similarly for an octahedron P-C + F = 6-12 + 8=2. The octahedron, moreover, is a geometrical dual of the cube, because the role of P and F are interchanged in the two structures. Thus the 8 ppints of the cube correspond to the 8 faces of the octahedron, and the 6 faces of the cube correspond to the 6 points of the octahedron. The duals share exactly the same value C (=12), and exactly, the same point group symmetry (Oh). For any case where the Descartes-Euler formula applies, duals are defined by the interchange of P and F, C held constant. Thus, Figure 1 indicates the dodecahedron and icosahedron are duals, and the tetrahedron is self-dual. 

The geometry of open SWNT s is qualitatively different than that of the single-cage and multi-cage fullerenes. In the latter cases, each of the fullerenes may be topologically deformed onto the surface of a sphere. But, an open cylinder nanotube may not be so deformed. The Descartes-Euler formula does not hold for an open-ended nanotube. What then becomes of the dual concept, based upon the Descartes-Euler formula, when we come to open nanotube geometry Evidently a generalization of the Descartes-Euler formula is required since it occurs that... 

In Table I we list a few examples of the geometrical correspondence which prevails between closo boron hydrides and the carbon fiillerenes [3]. The geometrical structure of Buckminsterflillerene maps into that of the 32-vertex closo boron hydride, B32H32. Both molecules of symmetry Ih display correspondence of the geometrical centers of the 32 carbon (polygon) faces to the 32 boron vertices, the 60 boron faces to the 60 carbon vertices, and the 90 carbon contacts to the 90 boron contacts. In accordance with the Descartes-Euler formula, for both molecules the sum of the vertices and faces exceeds by 2 the number of contacts. A cursory review of Table I indicates that similar correspondences prevail for all the examples listed. Clearly the list of examples may be readily extended. 

We mention that the Descartes-Euler formula may be used to predict which closo boron hydride is the analog of each experimentally known fullerene. Thus, for a given fullerene, if the number of carbons is multiplied by 3/2 (a formal carbon contact number) to give the total number of carbon contacts, one may then calculate the number of carbon faces whose equality with the number of Won vertices yields immediately the molecular formula desired. 

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,... 

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... 

We have shown how a simple idea, the Descartes-Euler formula, defines the boron fullerenes as the geometrical duals of the carbon fullerenes. As the carbon fullerenes are actually existing molecules, the Descartes-Euler formula is immediately suggestive of the possible existence of their boron duals. This is confirmed by quantum chemical molecular orbital calculations which indicate considerable stability for boron flillerene cage and multi-cage geometries. Therefore we encourage attempts at the synthesis of these proposed compounds. 

Carbon nanotori spontaneously self assemble, in the same type of experiments, which lead to carbon fUllerenes and nanotubes. We have shown a generalization of the Descartes-Euler formula, called the Euler-Poincare... 

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