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Discrete Differential Geometry pp 191–213Cite as

Polyhedral Surfaces of High Genus

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Part of the Oberwolfach Seminars book series (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).

Keywords

  • Polyhedral surfaces
  • high genus
  • neighborly surfaces
  • geometric construction
  • projected deformed cubes

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Authors and Affiliations

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

    Günter M. Ziegler

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  1. Günter M. Ziegler
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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

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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

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