## Experimental Mathematics

### Surface Realization with the Intersection Segment Functional

#### Abstract

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

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

Dates
First available in Project Euclid: 12 March 2010

https://projecteuclid.org/euclid.em/1268404804

Mathematical Reviews number (MathSciNet)
MR2649986

Zentralblatt MATH identifier
1191.52010

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

#### Citation

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