Finite sets of operations sufficient to construct any fullerene from C ₂₀

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• 作者：Victor M. Buchstaber ; Nikolay Yu. Erokhovets
• 关键词：Fullerenes ; Growth operations ; Finite sets ; Polytopes ; Edge ; truncations ; Dodecahedron
• 刊名：Structural Chemistry
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：28
• 期：1
• 页码：225-234
• 全文大小：
• 刊物类别：Chemistry and Materials Science
• 刊物主题：Computer Applications in Chemistry; Physical Chemistry; Theoretical and Computational Chemistry;
• 出版者：Springer US
• ISSN：1572-9001
• 卷排序：28

文摘

We study the well-known problem of combinatorial classification of fullerenes. By a (mathematical) fullerene we mean a convex simple three-dimensional polytope with all facets pentagons and hexagons. We analyse approaches of construction of arbitrary fullerene from the dodecahedron (a fullerene C₂₀). A growth operation is a combinatorial operation that substitutes the patch with more facets and the same boundary for the patch on the surface of a simple polytope to produce a new simple polytope. It is known that an infinite set of different growth operations transforming fullerenes into fullerenes is needed to construct any fullerene from the dodecahedron. We prove that if we allow a polytope to contain one exceptional facet, which is a quadrangle or a heptagon, then a finite set of growth operations is sufficient. We analyze pairs of objects: a finite set of operations, and a family of acceptable polytopes containing fullerenes such that any polytope of the family can be obtained from the dodecahedron by a sequence of operations from the corresponding set. We describe explicitly three such pairs. First two pairs contain seven operations, and the last – eleven operations. Each of these operations corresponds to a finite set of growth operations and is a composition of edge- and two edges-truncations.