Advertisement

Selected Open and Solved Problems in Computational Synthetic Geometry

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 148)

Abstract

Computational Synthetic Geometry was the title of the Springer Lecture Notes of the first author with Bernd Sturmfels in 1989. During the last 25 years combinatorial structures such as abstract point-line configurations in the sense of Branko Grünbaum’s book from 2009 [17], \((d-1)\)-spheres of questionable convex d-polytopes, or regular maps have been studied in view of their possible geometric realization. We present selected open and solved problems from these areas. Oriented matroids have played an essential role in most of these problems. The topological representation of oriented matroids as sphere systems leads to pseudoline arrangements in the rank 3 case. We show in particular the application of a new topological representation of all combinatorial point-line configurations (quasiline arrangements).

Keywords

Computational synthetic geometry Point-line configurations Oriented matroids Pseudoline arrangements Quasiline arrangements 

Notes

Acknowledgments

The data structure for the Hurwitz map of genus 7 was produced by Marston Conder. We would like to thank him for fruitful discussions. We would also like to thank Jarke J. van Wijk for the pictures used in Figs. 3 and 4.

References

  1. 1.
    A. Björner, M. Las Vergnas, B. Sturmfels, N. White, G. Ziegler, Oriented Matroids (Cambridge University Press, Cambridge, 1996)Google Scholar
  2. 2.
    J. Bokowski, Abstract complexes with symmetries, in Symmetry of Discrete Mathematic Peteral Structures and Their Symmetry Groups, A Collection of Essays, ed by K.H. Hoffmann, R. Wille, Reseach and exposition in mathematics, vol. 15 (Heldermann, Berlin, 1991)Google Scholar
  3. 3.
    J. Bokowski, A geometric realization without self-intersections does exist for Dyck’s regular map. Discrete Comput. Geom. 4(6), 583–589 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    J. Bokowski, Computational Oriented Matroids (Cambridge University Press, Cambridge, 2006)Google Scholar
  5. 5.
    J. Bokowski, P. Cara, S. Mock, On a self-dual 3-sphere of Peter McMullen. Period. Math. Hungar. 39(1–3), 17–32 (1999)MathSciNetzbMATHGoogle Scholar
  6. 6.
    J. Bokowski, B. Grünbaum, L. Schewe, Topological configurations \(n_4\) exist for all \(n \ge 17\). European J. Combin. 30, 1778–1785 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    J. Bokowski, S. King, S. Mock, I. Streinu, The topological representation of oriented matroids. Discrete Comput. Geom. 33(4), 645–668 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    J. Bokowski, J. Kovič, T. Pisanski, A. Žitnik, Combinatorial configurations, quasiline arrangments, and systems of curves on surfaces, submitted. arxiv.org/pdf/1410.2350.pdf
  9. 9.
    J. Bokowski, V. Pilaud, On topological and geometric \((19_4)\)-configurations. To appear in Eur. J. Combin. arXiv:1309.3201
  10. 10.
    J. Bokowski, T. Pisanski, Oriented matroids and complete-graph embeddings on surfaces. J. Combin. Theory Ser. A 114(1), 1–19 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    J. Bokowski, P. Schuchert, Equifacetted 3-spheres as topes of non-polytopal matroid polytopes. Discrete Comput. Geom. 13(3–4), 347–361 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    U. Brehm, Maximally symmetric polyhedral realizations of Dyck’s regular map. Mathematika 34(2), 229–236 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    M. Conder, Hurwitz groups: a brief survey. Bull. Amer. Math. Soc. (N.S.) 23(2), 359–370 (1990)Google Scholar
  14. 14.
    M. Conder, P. Dobcsányi, Determination of all regular maps of small genus. J. Combin. Theory Ser. B 81, 224–242 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    J. Folkman, J. Lawrence, Oriented matroids. J. Combin. Theory Ser. B 25(2), 199–236 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    G. Gévay, T. Pisanski, Kronecker covers, \(V\)-construction, unit-distance graphs and isometric point-circle configurations. Ars Math. Contemp. 7, 317–336 (2014)MathSciNetzbMATHGoogle Scholar
  17. 17.
    B. Grünbaum, Configurations of Points and Lines, Graduate Studies in Mathematics, vol 103 (American Mathematical Society, Providence, RI, 2009)Google Scholar
  18. 18.
    A.M. Macbeath, On a curve of genus 7. Proc. London Math. Soc. 15(3), 527–542 (1965)Google Scholar
  19. 19.
    P. McMullen, On the combinatorial structure of convex polytopes. Ph.D. thesis, University of Birmingham, 1968Google Scholar
  20. 20.
    T. Pisanski, A. Žitnik, Representing graphs and maps. in Topics in Topological Graph Theory, Encyclopedia of mathematics and its applications, vol. 128, (Cambridge University Press, Cambridge, 2009) pp. 151–180Google Scholar
  21. 21.
    E. Schulte, J.M. Wills, A polyhedral realization of Felix Klein s map \(\{3,7\}_8\) on a Riemannian manifold of genus 3, J. London Math. Soc. 32(2), 539–547 (1985)Google Scholar
  22. 22.
    J.J. van Wijk, Visualization of regular maps: The chase continues. IEEE Trans. Visual Comput. Graphics no. 1. doi: 10.1109/TVCG.2014.2352952 Google Scholar
  23. 23.
    J.J. van Wijk, Symmetric tiling of closed surfaces: Visualization of regular maps. in ACM Transactions on Graphics Proceedings ACM SIGGRAPH’09 28(3), Article 49, 12, August 2009Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of MathematicsTechnische Universität DarmstadtDarmstadtGermany
  2. 2.Institute of Mathematics, Physics and MechanicsLjubljanaSlovenia
  3. 3.Andrej Marušič InstituteUniversity of PrimorskaKoperSlovenia
  4. 4.University of Primorska FAMNITKoperSlovenia
  5. 5.Faculty for Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia

Personalised recommendations