Polyhedral 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... 

From these inequalities together with the polyhedral formula, we get... 

Observe the similarity of the polyhedral formula with the phase rule. We emphasize that the analogy to the phase rule to the polyhedral formula is emerging just for the three-dimensional case. 

Comparatively few papers are dealing with the similarity of polyhedral formula and phase rule [12-18]. We mentioned already that Eq. (7.6) is analogous... 

The idea can be easily generalized. For a thermodynamic system with K components and two additional variables the dimension of the hypersurface is S -1. Coexistence of P phases gives a hyperline of dimension K-l-(P-l) = K + 2-P, which is interpreted as the freedoms. This geometrical interpretation of the phase rule is completely different from the polyhedral formula. 

All the above mentioned authors did not suspect that the obtained solids of the general formula (XSiOj 5) might contain volatile polyhedral oligosilsesquioxanes. 

The nomenclature of oligosilsesquioxanes (as well as of their homo derivatives) has not been elaborated. These compounds are polyhedral oligocyclic systems of general formula (XiSiOi.5) with n = 2m (m 2), i.e., n = 4, 6, 8,10,12, etc. and X is the substituent at the silicon atom. In the simplest case, all the substituents Xj may be the same. On the basis of the existing lUPAC rules for naming polycyclic compounds, oligosilsesquioxanes may be termed in two ways ... 

The efforts to rationalize the formulas and structures of Zintl ions and related species predated extensive definitive structural information on anionic post-transition metal clusters obtained by Corbett and his group in the 1970s [8, 9]. After enough such structural information on the bare post-transition metal clusters became available, the resemblance of their polyhedra to the known polyhedral boranes became apparent. For this reason, the simple Zintl-Klemm concept has been largely superseded by newer, more advanced descriptions of chemical bonding in such clusters, initially those applied to the polyhedral boranes. 

Gosed polyhedral structure with triangular faces only known only as with molecular formula (B HJ2- ( + I skeletal electron-pairs for polyhedron. 

Nest-like, nonclosed polyhedral structure molecular formula B H + 4 ( + 2) skeletal electron-pairs n vertices of the parent (n + l)-atom c/oro-polyhedron occupied. 

Net-like, nonclosed polyhedral structure molecular formula B H,I + 8 ... 

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