Visual Topology and Variational Problems on Two-Dimensional Surfaces

  • Anatoly T. Fomenko
  • Alexandr O. Ivanov
  • Alexey A. Tuzhilin


Minimal Surface Soap Film Soap Bubble Steiner Minimal Tree Minimal Network 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    D. Cieslik, Steiner Minimal Trees, Kluwer Academic Publishers, Dordrecht-Boston-London, 1998.Google Scholar
  2. [2]
    Dao Chong Thi, A. T. Fomenko, Minimal Surfaces and Plateau Problem, Nauka, Moscow, 1987 (in Russian). (Engl. translation Amer. Math. soc., Providence, RI, 1991.)Google Scholar
  3. [3]
    B. A. Dubrovin, A. T. Fomenko, S. P. Novikov, Modern Geometry, Nauka, Moscow, 1986. English translation: Part 1, GTM 93, 1984; Part 2, GTM 104, 1985, Springer-Verlag, New York.Google Scholar
  4. [4]
    M. Emmer, Bolle di sapone: un viaggio tra matematica, arte e fantasia, La Nuova Italia, Firenze, 1991.Google Scholar
  5. [5]
    M. Emmer, Soap Bubbles in art and science, M. Emmer, ed., The Visual Mind, MIT press, 1993.Google Scholar
  6. [6]
    M. Emmer, Soap Bubbles, video, english version, 27 minutes, Rome (1984).Google Scholar
  7. [7]
    A. T. Fomenko, and A. A. Tuzhilin, Elements of geometry and topology of minimal surfaces in three-dimensional space, Translations of Math. Monographs, AMS, v. 93, 1992.Google Scholar
  8. [8]
    R. L. Francis, A note on the optimum location of new machines in existing plant layouts, J. Indust. Engrg., v. 14, pp. 57–59, 1963.Google Scholar
  9. [9]
    M. R. Garey, and D. S. Johnson, The Rectilinear Steiner Problem is NP-Complete, SIAM J. Appl. Math., v. 32, pp. 826–834, 1977.MathSciNetGoogle Scholar
  10. [10]
    M. Hanan, On Steiner’s Problem with Rectilinear Distance, SIAM J. Appl. Math., v. 14, pp. 255–265, 1966.MATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    S. Hildebrandt, and A. Tromba, The Parsimonious Universe, Springer-Verlag, New York, 1996.Google Scholar
  12. [12]
    F. K. Hwang, On Steiner minimal trees with rectilinear distance, SIAM J. of Appl. Math., v. 30, pp. 104–114, 1976.MATHGoogle Scholar
  13. [13]
    F. K. Hwang, D. Richards, and P. Winter, The Steiners Tree Problem, Elsevier Science Publishers, 1992.Google Scholar
  14. [14]
    A. O. Ivanov, and A. A. Tuzhilin, Minimal Networks. The Steiner Problem and Its Generalizations, CRC Press, N.W., Boca Raton, Florida, 1994.Google Scholar
  15. [15]
    A. O. Ivanov, and A. A. Tuzhilin, Branching Solutions to One-Dimensional Variational Problems, World Scientific Publ., 2001.Google Scholar
  16. [16]
    V. Jarnik, and M. Kössler, O minimalnich grafeth obeahujicich n danijch bodu, Cas. Pest. Mat. a Fys., v. 63, pp. 223–235, 1934.Google Scholar
  17. [17]
    L. S. Polak, Variational principals in mechanics, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1959 (in Russian).Google Scholar
  18. [18]
    J. E. Taylor, The structure of singularities in soap-bubble-like and soapfilm-like minimal surfaces, Ann. Math., v. 103, pp. 489–539, 1976.MATHGoogle Scholar
  19. [19]
    J. A. Thorpe, Elementary topics in differential geometry, Springer-Verlag, New York, 1979.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Anatoly T. Fomenko
    • 1
  • Alexandr O. Ivanov
    • 1
  • Alexey A. Tuzhilin
    • 1
  1. 1.Department of Mechanics and MathematicsMoscow State UniversityMoscowRussia

Personalised recommendations