Journal of Mathematical Chemistry

, Volume 15, Issue 1, pp 143–156 | Cite as

Curved graphite and its mathematical transformations

  • Humberto Terrones


Mathematical transformations for graphite with positive, negative and zero Gaussian curvatures are presented. When the Gaussian curvatureK is zero, we analyse a bending transformation from a planar sheet into a cone. The Bonnet, the Goursat and a mixed transformation are studied for graphitic structures with the same topologies as triply periodic minimal surfaces (K < 0). We have found that using the Kenmotsu equations for surfaces of constant mean curvature it is possible to invert spherical and cylindrical graphite. A bending transformation for surfaces of revolution is also studied; during this transformation the helical arrangement of cylinders changes. All these transformations can give an insight into kinematic processes of curved graphite and into new shapes.


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  1. [1]
    S. Andersson, S.T. Hyde, K. Larsson and S. Lidin, Chem. Rev. 88 (1988) 221.Google Scholar
  2. [2]
    O. Bonnet, Note sur la théorie générale des surfaces, C.R. Acad. Sei. Paris 37 (1853) 529–532.Google Scholar
  3. [3]
    A. Borchardt, A. Fuchicello, K.V. Kilway, K.K. Baldridge and J.S. Siegel, J. Am. Chem. Soc. 114 (1992)1921.Google Scholar
  4. [4]
    M. Do Carmo,Differential Geometry of Curves and Surfaces (Prentice-Hall, New Jersey, 1976).Google Scholar
  5. [5]
    J. Emsley, New Scientists 134 (9 May 1992) 14.Google Scholar
  6. [6]
    F. Gackstatter, Colloq. Physique C7, Suppl. 23, 51 (1990) 163.Google Scholar
  7. [7]
    E. Goursat, Sur un mode de transformation des surfaces minima, Acta Mathematica 11 (1l Feb. 1888) 135–186.Google Scholar
  8. [8]
    E. Goursat, Sur un mode de transformation des surfaces minima (Second Mémoire), Acta Mathematica 11(29 March 1888) 257–264.Google Scholar
  9. [9]
    N. Hamada, S. Sawada and A. Oshiyama, Phys. Rev. Lett. 68 (1992) 1579.Google Scholar
  10. [10]
    S.T. Hyde and S. Andersson, Z. Kristallogr. 170 (1985) 225.Google Scholar
  11. [11]
    S.T. Hyde and S. Andersson, Z. Kristallogr. 174 (1986) 225.Google Scholar
  12. [12]
    S. Iijima, Nature 354 (1991) 56.Google Scholar
  13. [13]
    S. Iijima, T. Ichihashi and Y. Ando, Nature 356 (1992) 776.Google Scholar
  14. [14]
    S.P. Kelty, Chen Chia-Chun and C.M. Leiber, Nature 352 (1991) 223.Google Scholar
  15. [15]
    K. Kenmotsu, Math. Ann. 245 (1979) 89.Google Scholar
  16. [16]
    J.J. Koenderink,Solid Shape (The MIT Press, Cambridge, Massachusetts, 1990).Google Scholar
  17. [17]
    W. Krätschmer, L.D. Lamb, K. Fostiropoulos and D.R. Huffman, Nature 347 (1990) 354.Google Scholar
  18. [18]
    H.W. Kroto, J.R. Heath, S.C. O'Brien, R.F. Curl and R.E. Smalley, Nature 318 (1985) 162.Google Scholar
  19. [19]
    H.W. Kroto, Nature 239 (1987) 529.Google Scholar
  20. [20]
    H. Kroto, Science 242 (1988) 1139.Google Scholar
  21. [21]
    T. Lenosky, X. Gonze, M. Teter and V. Elser, Nature 355 (1992) 333.Google Scholar
  22. [22]
    A.L. Mackay, Physica B 131 (1985) 300.Google Scholar
  23. [23]
    A. L. Mackay, Nature 314 (1985) 604.Google Scholar
  24. [24]
    A.L. Mackay and H. Terrones, Nature 352 (1991) 762.Google Scholar
  25. [25]
    J.C.C. Nitsche,Lectures on Minimal Surfaces, Vol. l (Cambridge University Press, 1989).Google Scholar
  26. [26]
    M. O'keeffe, G.B. Adams and O.F. Sankey, Phys. Rev. Lett. 68 (1992) 2325.Google Scholar
  27. [27]
    A.H. Schoen, Infinite periodic minimal surfaces without self-intersections, NASA Technical Note D-5541(1970).Google Scholar
  28. [28]
    H.A. Schwarz,Gesammelte Mathematische Abhandlungen, Vol. 1 (Springer, Berlin, 1890).Google Scholar
  29. [29]
    T.L. Scott, M.M. Hashemi and M.S. Bratcher, J. Am. Chem. Soc. 114 (1992) 1920.Google Scholar
  30. [30]
    M. Spivak,A Comprehensive Introduction to Differential Geometry, Vols. 1–5 (Publish or Perish, Berkeley, 1979).Google Scholar
  31. [31]
    D.J. Struik, Differential Geometry (Addison-Wesley, Reading, Massachusetts, 1961).Google Scholar
  32. [32]
    H. Terrones, Colloq. Physique C7, 51 (1990) 345.Google Scholar
  33. [33]
    H. Terrones, Mathematical surfaces and invariants in the study of atomic structures, Ph.D. Thesis, University of London (1992).Google Scholar
  34. [34]
    H. Terrones and A.L. Mackay, J. Math. Chem. 15 (1994) 183.Google Scholar
  35. [35]
    H. Terrones and A.L. Mackay, Micelles and foams: 2-manifolds arising from local interactions,NATO Advanced Research Workshop: Growth Patterns in Physics and Biology, eds. E. Louis, L. Sander and P. Meakin (Plenum Press, to be published).Google Scholar
  36. [36]
    D. Vanderbilt and J. Tersoff, Phys. Rev. Lett. 68 (1992) 511.Google Scholar
  37. [37]
    K. Weierstrass, Untersuchungen über die Flächen, deren mittlere Krümmung überall gleich null ist, Monatsber. d. Berliner Akad. (1866) pp. 612–625.Google Scholar

Copyright information

© J.C. Baltzer AG, Science Publishers 1994

Authors and Affiliations

  • Humberto Terrones
    • 1
  1. 1.Department of CrystallographyBirkbeck College, University of LondonLondonUK

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