Experimental Mathematics

Surface Realization with the Intersection Segment Functional

Stefan Hougardy, Frank H. Lutz, and Mariano Zelke

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Deciding realizability of a given polyhedral map on a (compact, connected) surface belongs to the hard problems in discrete geometry from the theoretical, algorithmic, and practical points of view.

In this paper, we present a heuristic algorithm for the realization of simplicial maps, based on the intersection segment functional. This heuristic was used to find geometric realizations in ${\mathbb R}^3$ for all vertex-minimal triangulations of the orientable surfaces of genera $g=3$ and $g=4$. Moreover, for the first time, examples of simplicial polyhedra in ${\mathbb R}^3$ of genus 5 with 12 vertices have been obtained.

Article information

Experiment. Math., Volume 19, Issue 1 (2010), 79-92.

First available in Project Euclid: 12 March 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 52B70: Polyhedral manifolds 57Q15: Triangulating manifolds

Triangulated surface polyhedral realization intersection segment functional


Hougardy, Stefan; Lutz, Frank H.; Zelke, Mariano. Surface Realization with the Intersection Segment Functional. Experiment. Math. 19 (2010), no. 1, 79--92. https://projecteuclid.org/euclid.em/1268404804

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