Skip to main content
SpringerLink
Log in

Polyhedral Surfaces of High Genus

  • Chapter
Discrete Differential Geometry

Part of the book series: Oberwolfach Seminars ((OWS,volume 38))

Abstract

The construction of the combinatorial data for a surface of maximal genus with n vertices is a classical problem: The maximal genus g = ⌊1/12(n − 3)(n − 4)⌋ was achieved in the famous “Map Color Theorem” by Ringel et al. (1968). We present the nicest one of Ringel’s constructions, for the case n ≡ 7 mod 12, but also an alternative construction, essentially due to Heffter (1898), which easily and explicitly yields surfaces of genus g ∼ 1/16 n 2.

For geometric (polyhedral) surfaces in ℝ3 with n vertices the maximal genus is not known. The current record is g ∼ 1/8n log2 n, due to McMullen, Schulz & Wills (1983). We present these surfaces with a new construction: We find them in Schlegel diagrams of “neighborly cubical 4-polytopes,” as constructed by Joswig & Ziegler (2000).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Dan Archdeacon and C. Paul Bonnington and Joanna A. Ellis-Monaghan, How to exhibit toroidal maps in space, Discrete Comput. Geometry, 38 (2007), 573–594.

    Article  MATH  MathSciNet  Google Scholar 

  2. Amos Altshuler, Polyhedral realizations in ℝ3 of triangulations of the torus and 2-manifolds in cyclic 4-polytopes, Discrete Math. 1 (1971), 211–238.

    Article  MATH  MathSciNet  Google Scholar 

  3. Amos Altshuler, Jürgen Bokowski, and Peter Schuchert, Neighborly 2-manifolds with 12 vertices, J. Combin. Theory Ser. A 75 (1996), no. 1, 148–162.

    Article  MATH  MathSciNet  Google Scholar 

  4. Nina Amenta and Günter M. Ziegler, Deformed products and maximal shadows, Advances in Discrete and Computational Geometry (South Hadley, MA, 1996) (Providence RI) (B. Chazelle, J.E. Goodman, and R. Pollack, eds.), Contemporary Mathematics, vol. 223, Amer. Math. Soc., 1998, pp. 57–90.

    Google Scholar 

  5. Eric K. Babson, Louis J. Billera, and Clara S. Chan, Neighborly cubical spheres and a cubical lower bound conjecture, Israel J. Math. 102 (1997), 297–315.

    Article  MATH  MathSciNet  Google Scholar 

  6. David W. Barnette, Peter Gritzmann, and Rainer Höhne, On valences of polyhedra, J. Combinatorial Theory, Ser. A 58 (1991), 279–300.

    Article  MATH  Google Scholar 

  7. Jürgen Bokowski and Ulrich Brehm, A new polyhedron of genus 3 with 10 vertices, Intuitive geometry (Siófok, 1985), Colloq. Math. Soc. János Bolyai, vol. 48, North-Holland, Amsterdam, 1987, pp. 105–116.

    Google Scholar 

  8. _____, A polyhedron of genus 4 with minimal number of vertices and maximal symmetry, Geom. Dedicata 29 (1989), no. 1, 53–64.

    Article  MATH  MathSciNet  Google Scholar 

  9. Jürgen Bokowski and Antonio Guedes de Oliveira, On the generation of oriented matroids, Discrete Comput. Geometry 24 (2000), 197–208.

    Article  MATH  Google Scholar 

  10. Ulrich Brehm and Jörg M. Wills, Polyhedral manifolds, Handbook of Convex Geometry (P. Gruber and J. Wills, eds.), North-Holland, Amsterdam, 1993, pp. 535–554.

    Google Scholar 

  11. Harold Scott MacDonald Coxeter, Regular skew polyhedra in 3 and 4 dimensions and their topological analogues, Proc. London Math. Soc. (2) 43 (1937), 33–62; Reprinted in “Twelve Geometric Essays,” Southern Illinois U. Press, Carbondale, 1968, 75–105.

    MATH  Google Scholar 

  12. Ákos Császár, A polyhedron without diagonals, Acta Sci. Math. (Szeged) 13 (1949/50), 140–142.

    Google Scholar 

  13. Richard A. Duke, Geometric embeddings of complexes, Amer. Math. Monthly 77 (1970), 597–603.

    Article  MATH  MathSciNet  Google Scholar 

  14. David Eppstein, Greg Kuperberg, and Günter M. Ziegler, Fat 4-polytopes and fatter 3-spheres, Discrete Geometry: In honor of W. Kuperberg’s 60th birthday (A. Bezdek, ed.), Pure and Applied Mathematics, vol. 253, Marcel Dekker Inc., New York, 2003, pp. 239–265; arXiv.org/math/0204007.

    Google Scholar 

  15. Branko Grünbaum, Convex Polytopes, Graduate Texts in Math., vol. 221, Springer-Verlag, New York, 2003, Second edition prepared by V. Kaibel, V. Klee and G.M. Ziegler (original edition: Interscience, London 1967).

    Google Scholar 

  16. Lothar Heffter, Ueber das Problem der Nachbargebiete, Math. Annalen 38 (1891), 477–508; www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN235181684_0038.

    Article  MathSciNet  Google Scholar 

  17. _____, Ueber metacyklische Gruppen und Nachbarconfigurationen, Math. Annalen 50 (1898), 261–268; www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN235181684_0050.

    Article  MathSciNet  MATH  Google Scholar 

  18. David Hilbert and Stephan Cohn-Vossen, Anschauliche Geometrie, Springer-Verlag, Berlin Heidelberg, 1932, Second edition 1996. English translation: Geometry and the Imagination, Chelsea Publ., 1952.

    MATH  Google Scholar 

  19. Stefan Hougardy, Frank H. Lutz, and Mariano Zelke, Polyhedra of genus 2 with 10 vertices and minimal coordinates, preprint, 3 pages, 2005; arXiv.org/math/0507592.

    Google Scholar 

  20. Michael Joswig and Thilo Rörig, Neighborly cubical polytopes and spheres, Israel J. Math 159 (2007), 221–242; arxiv.org/math/0503213.

    Article  MATH  MathSciNet  Google Scholar 

  21. Michael Joswig and Günter M. Ziegler, Neighborly cubical polytopes, Discrete & Computational Geometry (Grünbaum Festschrift: G. Kalai, V. Klee, eds.) 24:2–3 (2000), 325–344; arXiv.org/math/9812033.

    Google Scholar 

  22. Alfred B. Kempe, On the geographical problem of the four colours, American J. Math. 2 (1879), 193–200.

    Google Scholar 

  23. Frank 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; arXiv.org/math.CO/0506316.

    Google Scholar 

  24. Peter McMullen, Egon Schulte, and Jörg M. Wills, Infinite series of combinatorially regular polyhedra in three-space, Geom. Dedicata 26 (1988), 299–307.

    Article  MATH  MathSciNet  Google Scholar 

  25. Peter McMullen, Christoph Schulz, and Jörg M. Wills, Polyhedral 2-manifolds in E 3 with unusually large genus, Israel J. Math. 46 (1983), 127–144.

    Article  MATH  MathSciNet  Google Scholar 

  26. August F. Möbius, Mittheilungen aus Möbius’ Nachlass: I. Zur Theorie der Polyëder und der Elementarverwandtschaft, GesammelteWerke II (F. Klein, ed.), Verlag von S. Hirzel, Leipzig, 1886, pp. 515–559.

    Google Scholar 

  27. Isabella Novik, Upper bound theorems for simplicial manifolds, Israel J. Math. 108 (1998), 45–82.

    MATH  MathSciNet  Google Scholar 

  28. Julian Pfeifle and Günter M. Ziegler, Many triangulated 3-spheres, Math. Annalen 330 (2004), 829–837; arXiv.org/math/0304492.

    Article  MATH  MathSciNet  Google Scholar 

  29. Nicholas Pippenger and Kristin Schleich, Topological characteristics of random triangulated surfaces, Random Struct. Algorithms 28 (2006), 247–288; arXiv.org/gr-qc/0306049.

    Article  MATH  MathSciNet  Google Scholar 

  30. Gerhard Ringel, Über drei kombinatorische Probleme am n-dimensionalen Würfel und Würfelgitter, Abh. Math. Sem. Univ. Hamburg 20 (1955), 10–19.

    Article  MATH  MathSciNet  Google Scholar 

  31. _____, Map color theorem, Grundlehren Series, vol. 234, Springer-Verlag, New York, 1974.

    Google Scholar 

  32. Raman Sanyal, Thilo Rörrig, and Günter M. Ziegler, Polytopes and polyhedral surfaces via projection, in preparation, 2006.

    Google Scholar 

  33. Lars Schewe, Satisfiability Problems in Discrete Geometry, Dissertation, Technische Universität Darmstadt, 2007, 101 pages.

    Google Scholar 

  34. Ernst Steinitz, Über die Eulerschen Polyederrelationen, Archiv für Mathematik und Physik 11 (1906), 86–88.

    Google Scholar 

  35. _____, Polyeder und Raumeinteilungen, Encyklopädie der mathematischen Wissenschaften, Dritter Band: Geometrie, III.1.2., Heft 9, Kapitel IIIAB 12 (W.Fr. Meyer and H. Mohrmann, eds.), B.G. Teubner, Leipzig, 1922, pp. 1–139.

    Google Scholar 

  36. Ernst Steinitz and Hans Rademacher, Vorlesungen über die Theorie der Polyeder, Springer-Verlag, Berlin 1934; Reprint, Springer-Verlag 1976.

    Google Scholar 

  37. Dagmar Timmreck, Necessary conditions for geometric realizability of simplicial complexes, Discrete Differential Geometry (A.I. Bobenko, P. Schröder, J.M. Sullivan, G.M. Ziegler, eds.), Oberwolfach Seminars, vol. 38, Birkhäuser, 2008, this volume, pp. 215–233.

    Google Scholar 

  38. Günter M. Ziegler, Lectures on Polytopes, Graduate Texts in Mathematics, vol. 152, Springer-Verlag, New York, 1995, Revised edition, 1998; “Updates, corrections, and more” at www.math.tu-berlin.de/~ziegler.

    Google Scholar 

  39. _____, Projected products of polygons, Electronic Research Announcements AMS 10 (2004), 122–134; www.ams.org/era/2004-10-14/S1079-6762-04-00137-4/

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik, MA 6-2, Technische Universität Berlin, Str. des 17. Juni 136, 10623, Berlin, Germany

    Günter M. Ziegler

Authors
  1. Günter M. Ziegler
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Mathematik, MA 8-3, Technische Universität Berlin, Strasse des 17. Juni 136, 10623, Berlin, Germany

    Alexander I. Bobenko

  2. Institut für Mathematik, MA 3-2, Technische Universität Berlin, Strasse des 17. Juni 136, 10623, Berlin, Germany

    John M. Sullivan

  3. Department of Computer Science, Caltech, MS 256-80, 1200 E. California Blvd., Pasadena, CA, 91125, USA

    Peter Schröder

  4. Institut für Mathematik, MA 6-2, Technische Universität Berlin, Strasse des 17. Juni 136, 10623, Berlin, Germany

    Günter M. Ziegler

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Birkhäuser Verlag Basel/Switzerland

About this chapter

Cite this chapter

Ziegler, G.M. (2008). Polyhedral Surfaces of High Genus. 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_10

Download citation

Publish with us

Policies and ethics

Search

Navigation