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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...
MACROCOSM, MICROCOSM, AND ANALOGY
"... The analogy between microcosm and macrocosm has in its time been many things, varying from an inspiring axiom to a tired cliché, but it deserves to be considered as something more than a pretty figure of speech. Analogy is so rarely taken seriously, except as a literary or ..."
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The analogy between microcosm and macrocosm has in its time been many things, varying from an inspiring axiom to a tired cliché, but it deserves to be considered as something more than a pretty figure of speech. Analogy is so rarely taken seriously, except as a literary or
Modedependent Differences in Chord Classification under an Original Computational Method of Tonal Structure Analysis
"... Abstract — Basing upon original computational analytic method (Majchrzak 2005, 2007), the present work aims, at: 1) Showing the differences for the major key and the minor (harmonic) key in the classification of chords, as an aspect of importance for interpreting a piece’s tonal structure diagram; 2 ..."
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Abstract — Basing upon original computational analytic method (Majchrzak 2005, 2007), the present work aims, at: 1) Showing the differences for the major key and the minor (harmonic) key in the classification of chords, as an aspect of importance for interpreting a piece’s tonal structure diagram; 2) Drawing attention to the subordination of the minor key as versus the major key in the chord classification, using the same algorithm. The relations between chords appearing in the major and minor (harmonic) key are shown by applying the comparisons of: 1) thirdbased chords; 2) degrees in the C major and A minor keys, on which the same diatonic chords appear.