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.


Plane Graph Negative Curvature Plane Partition Regular Plane Plane Tiling 
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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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