Selected Open and Solved Problems in Computational Synthetic Geometry

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


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).


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



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.


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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

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