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Fullerenes and Coordination Polyhedra versus HalfCubes Embeddings
, 1997
"... A fullerene F n is a 3regular (or cubic) polyhedral carbon molecule for which the n vertices  the carbons atoms  are arranged in 12 pentagons and ( n 2 \Gamma 10) hexagons. Only a finite number of fullerenes are expected to be, up to scale, isometrically embeddable into a hypercube. Looking fo ..."
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A fullerene F n is a 3regular (or cubic) polyhedral carbon molecule for which the n vertices  the carbons atoms  are arranged in 12 pentagons and ( n 2 \Gamma 10) hexagons. Only a finite number of fullerenes are expected to be, up to scale, isometrically embeddable into a hypercube. Looking for the list of such fullerenes, we first check the embeddability of all fullerenes F n for n ! 60 and of all preferable fullerenes C n for n ! 86 and their duals. Then, we consider some infinite families, including fullerenes with icosahedral symmetry, which describe virus capsids, onionlike metallic clusters and geodesic domes. Quasiembeddings and fullerene analogues are considered. We also present some results on chemically relevant polyhedra such as coordination polyhedra and cluster polyhedra. Finally we conjecture that the list of known embeddable fullerenes is complete and present its relevance to the Katsura model for vesicles cells. Contents 1 Introduction and Basic Properties 2 1...
Zigzag structure of complexes
, 2008
"... Inspired by Coxeter’s notion of Petrie polygon for dpolytopes (see [Cox73]), we consider a generalization of the notion of zigzag circuits on complexes and compute the zigzag structure for several interesting families of dpolytopes, including semiregular, regularfaced, Wythoff Archimedean ones, C ..."
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Inspired by Coxeter’s notion of Petrie polygon for dpolytopes (see [Cox73]), we consider a generalization of the notion of zigzag circuits on complexes and compute the zigzag structure for several interesting families of dpolytopes, including semiregular, regularfaced, Wythoff Archimedean ones, Conway’s 4polytopes, halfcubes, folded cubes. Also considered are regular maps and Lins triality relations on maps.