On Heuristic Methods for Finding Realizations of Surfaces

  • Jürgen Bokowski
Part of the Oberwolfach Seminars book series (OWS, volume 38)


This article discusses heuristic methods for finding realizations of oriented matroids of rank 3 and 4. These methods can be applied for the spatial embeddability problem of 2-manifolds. They have proven successful in previous realization problems in which finally only the result was published.


Geometric realization heuristic methods oriented matroids pseudoline arrangents 


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  1. [1]
    D. Archdeacon, C.P. Bonnington, and J.A. Ellis-Monaghan, How to exhibit toroidal maps in space, Discrete Comput. Geometry, 38 (2007), 573–594.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    A. Altshuler, J. Bokowski, and P. Schuchert, Spatial polyhedra without diagonals, Isr. J. Math. 86 (1994), 373–396.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    _____, Neighborly 2-manifolds with 12 vertices, J. Comb. Theory, Ser. A 75 (1996), 148–162.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    A. Björner, M. Las Vergnas, B. Sturmfels, N. White, and G.M. Ziegler, Oriented Matroids, 2nd ed., Encyclopedia of Mathematics and Its Applications, vol. 46, Cambridge University Press, Cambridge, 1999.MATHGoogle Scholar
  5. [5]
    J. Bokowski, A geometric realization without self-intersections does exist for Dyck’s regular map, Discrete Comput. Geom. 4 (1989), 583–589.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    _____, Effective methods in computational synthetic geometry, Automated Deduction in Geometry, Proc. 3rd Internat. Workshop (ADG 2000), Zürich, 2000 (Berlin) (J. Richter-Gebert and D. Wang, eds.), Lecture Notes in Computer Science, vol. 2061, Springer-Verlag, 2001, pp. 175–192.Google Scholar
  7. [7]
    _____, Computational Oriented Matroids, Cambridge University Press, Cambridge, 2006.Google Scholar
  8. [8]
    J. Bokowski and A. Guedes de Oliveira, On the generation of oriented matroids, Discrete Comput. Geom. 24 (2000), 197–208.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    J. Bokowski and B. Sturmfels, Computational Synthetic Geometry, Lecture Notes in Mathematics, vol. 1355, Springer-Verlag, Berlin, 1989.Google Scholar
  10. [10]
    S. Fendrich, Methoden zur Erzeugung und Realisierung von triangulierten kombinatorischen 2-Mannigfaltigkeiten, Diplomarbeit, Technische Universität Darmstadt, 2003, 56 pages.Google Scholar
  11. [11]
    B. Grünbaum, Arrangements and Spreads, Conference Board of the Mathematical Sciences. Regional Conference Series in Mathematics, vol. 10, American Mathematical Society, Providence, RI, 1972.Google Scholar
  12. [12]
    S. Hougardy, F.H. Lutz, and M. Zelke, Surface realization with the intersection edge functional, arXiv:math.MG/0608538, 2006, 19 pages.Google Scholar
  13. [13]
    S.A. Lavrenchenko, Irreducible triangulations of the torus, J. Sov. Math. 51 (1990), 2537–2543, translation from Ukr. Geom. Sb. 30 (1987), 52–62.MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    F.H. Lutz, Enumeration and random realization of triangulated surfaces, Discrete Differential Geometry (A.I. Bobenko, P. Schröder, J.M. Sullivan, G.M. Ziegler, eds.), Oberwolfach Seminars, vol. 38, Birkhäuser, 2008, this volume, pp. 235–253.Google Scholar
  15. [15]
    L. Schewe, Satisfiability Problems in Discrete Geometry, Dissertation, Technische Universität Darmstadt, 2007, 101 pages.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • Jürgen Bokowski
    • 1
  1. 1.Fachbereich Mathematik AG 7: Diskrete OptimierungTechnische Universität DarmstadtDarmstadtGermany

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