Fullerene-Like Spheres with Faces of Negative Curvature

  • Michel Deza
  • Mathieu Dutour Sikirić
  • Mikhail Shtogrin
Part of the Carbon Materials: Chemistry and Physics book series (CMCP, volume 6)


Given \(R \subset\mathbb{N}\), an (R, k)-sphere is a k-regular map on the sphere whose faces have gonalities iR. The most interesting/useful are (geometric) fullerenes, that is, ({5, 6}, 3)-spheres. Call \({\kappa }_{i} = 1 + \frac{i} {k} - \frac{i} {2}\) the curvature of i-gonal faces. (R, k)-spheres admitting κ i < 0 are much harder to study. We consider the symmetries and construction for three new instances of such spheres: ({a, b}, k)-spheres with p b ≤ 3 (they are listed), icosahedrites (i.e., (3, 4, 5)-spheres), and, for any \(c \in\mathbb{N}\), fullerene c-disks, that is, ({5, 6, c}, 3)-spheres with p c = 1.


  1. Brinkmann G, McKay BD (2007) Fast generation of planar graphs. MATCH Commun Math Comput Chem 58:333–367Google Scholar
  2. Brinkmann G, Delgado-Friedrichs O, Dress A, Harmuth T (1997) CaGe – a virtual environment for studying some special classes of large molecules. MATCH Commun Math Comput Chem 36:233–237Google Scholar
  3. Brinkmann G, Harmuth T, Heidemeier O (2003) The construction of cubic and quartic planar maps with prescribed face degrees. Discrete Appl Math 128:541–554CrossRefGoogle Scholar
  4. Deza M, Dutour Sikirić M (2005) Zigzag structure of simple two-faced polyhedra. Comb Probab Comput 14:31–57CrossRefGoogle Scholar
  5. Deza M, Dutour Sikirić M (2008) Geometry of chemical graphs: polycycles and two-faced maps. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  6. Deza M, Dutour Sikirić M (2012) Zigzag and central circuit structure of ({1, 2, 3}, 6)-spheres. Taiwan J Math 16:913–940Google Scholar
  7. Deza M, Shtogrin M (2003) Octahedrites. Symmetry Cult Sci 11:27–64Google Scholar
  8. Deza M, Huang T, Lih KW (2002) Central circuit coverings of octahedrites and medial polyhedra. J Math Res Expos 22:49–66Google Scholar
  9. Deza M, Dutour Sikirić M, Shtogrin M (2003) 4-valent plane graphs with 2-, 3- and 4-gonal faces. In: Li KW (ed) Advances in algebra and related topics. World Scientific, Singapore, pp 73–97Google Scholar
  10. Deza M, Dutour Sikirić M, Fowler P (2009) The symmetries of cubic polyhedral graphs with face size no larger than 6. MATCH Commun Math Comput Chem 61:589–602Google Scholar
  11. Dutour Sikirić M, Deza M (2004) Goldberg-Coxeter construction for 3- or 4-valent plane graphs. Electron J Comb 11:R20Google Scholar
  12. Dutour Sikirić M, Deza M (2011) 4-regular and self-dual analogs of fullerenes. In: Graovac A (ed) Mathematics and topology of fullerenes. Springer, Berlin, pp 103–116CrossRefGoogle Scholar
  13. Higuchi Y (2001) Combinatorial curvature for planar graphs. J Graph Theory 38:220–229CrossRefGoogle Scholar
  14. Thurston WP (1998) Shapes of polyhedra and triangulations of the sphere. Geom Topol Monogr 1:511–549CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Michel Deza
    • 1
  • Mathieu Dutour Sikirić
    • 2
  • Mikhail Shtogrin
    • 3
  1. 1.Ecole Normale SuperieureParisFrance
  2. 2.Department of Marine and Environmental ResearchInstitut Rudjer BoskovicZagrebCroatia
  3. 3.Steklov Mathematical InstituteDemidov Yaroslavl State UniversityYaroslavl, MoscowRussia

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