Discrete Differential Geometry

Volume 38 of the series Oberwolfach Seminars pp 191-213

Polyhedral Surfaces of High Genus

  • Günter M. ZieglerAffiliated withInstitut für Mathematik, MA 6-2, Technische Universität Berlin

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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).


Polyhedral surfaces high genus neighborly surfaces geometric construction projected deformed cubes