Structural Chemistry

, Volume 28, Issue 1, pp 225–234 | Cite as

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

  • Victor M. Buchstaber
  • Nikolay Yu. Erokhovets
Original Research


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 C 20). 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.


Fullerenes Growth operations Finite sets Polytopes Edge-truncations Dodecahedron 



This work is supported by the Russian Science Foundation under grant 14-11-00414. The second author is a Young Russian Mathematics award winner.

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Victor M. Buchstaber
    • 1
  • Nikolay Yu. Erokhovets
    • 1
  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia

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