ISSN:
1572-8951
Keywords:
Platonic solid
;
Archimedean solid
;
topology
;
Euler formula
;
barycentric subdivision
;
combinatorial explosion
Source:
Springer Online Journal Archives 1860-2000
Topics:
Chemistry and Pharmacology
Notes:
Abstract Structures of polyoxometalates frequently are discovered to be based upon regular convex polyhedra, including the Platonic and Archimedean solids. A topological approach involving barycentric subdivision of the faces of such polyhedra, leads to their description as combinations of triangular building blocks assembled according to systematic rules. An analysis of the Keggin structure of [Mo12O36(PO4)]3−, is presented. As it turns out, it is the only spherical polyhedral structure of T symmetry built up from six 8-gons and eight 6-gons. Similarly, there is only one spherical polyhedral structure of T symmetry built up from eight 6-gons and twenty-four 4-gons satisfying also some obvious chemical combinatorial constraints. Such a structure is observed for [H9V18O42(VO4)]6−. Analysis of possible structures of lower symmetry (D3, D4), e.g. as observed for [V15O36(Hal)]6− and [H4V18O42(Hal)]9−, reveals the onset of combinatorial explosion. For example, there are 67 D3-structures satisfying the chemical condition.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00999621
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