Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
D. Archdeacon, C.P. Bonnington, and J.A. Ellis-Monaghan, How to exhibit toroidal maps in space, Discrete Comput. Geometry, 38 (2007), 573–594.
A. Altshuler, J. Bokowski, and P. Schuchert, Spatial polyhedra without diagonals, Isr. J. Math. 86 (1994), 373–396.
_____, Neighborly 2-manifolds with 12 vertices, J. Comb. Theory, Ser. A 75 (1996), 148–162.
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.
J. Bokowski, A geometric realization without self-intersections does exist for Dyck’s regular map, Discrete Comput. Geom. 4 (1989), 583–589.
_____, 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.
_____, Computational Oriented Matroids, Cambridge University Press, Cambridge, 2006.
J. Bokowski and A. Guedes de Oliveira, On the generation of oriented matroids, Discrete Comput. Geom. 24 (2000), 197–208.
J. Bokowski and B. Sturmfels, Computational Synthetic Geometry, Lecture Notes in Mathematics, vol. 1355, Springer-Verlag, Berlin, 1989.
S. Fendrich, Methoden zur Erzeugung und Realisierung von triangulierten kombinatorischen 2-Mannigfaltigkeiten, Diplomarbeit, Technische Universität Darmstadt, 2003, 56 pages.
B. Grünbaum, Arrangements and Spreads, Conference Board of the Mathematical Sciences. Regional Conference Series in Mathematics, vol. 10, American Mathematical Society, Providence, RI, 1972.
S. Hougardy, F.H. Lutz, and M. Zelke, Surface realization with the intersection edge functional, arXiv:math.MG/0608538, 2006, 19 pages.
S.A. Lavrenchenko, Irreducible triangulations of the torus, J. Sov. Math. 51 (1990), 2537–2543, translation from Ukr. Geom. Sb. 30 (1987), 52–62.
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.
L. Schewe, Satisfiability Problems in Discrete Geometry, Dissertation, Technische Universität Darmstadt, 2007, 101 pages.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Birkhäuser Verlag Basel/Switzerland
About this chapter
Cite this chapter
Bokowski, J. (2008). On Heuristic Methods for Finding Realizations of Surfaces. In: Bobenko, A.I., Sullivan, J.M., Schröder, P., Ziegler, G.M. (eds) Discrete Differential Geometry. Oberwolfach Seminars, vol 38. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8621-4_13
Download citation
DOI: https://doi.org/10.1007/978-3-7643-8621-4_13
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-8620-7
Online ISBN: 978-3-7643-8621-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)