Polyhedral tori with minimal coordinates

TL;DR

This paper provides explicit minimal-coordinate realizations of all triangulated tori with up to 12 vertices, including specific constructions in small integer cubes and cuboids, advancing understanding of polyhedral tori geometry.

Contribution

It offers explicit minimal coordinate realizations for all triangulated tori with up to 12 vertices, including realizability in small integer cubes and cuboids, and analyzes the realizability of associated oriented matroids.

Findings

Explicit minimal coordinate realizations for all triangulated tori up to 12 vertices.

Realizability of all 7-vertex triangulations in the 6x6x6 cube.

Construction of polyhedral tori with 8, 9, and 10 vertices in small cuboids.

Abstract

We give explicit realizations with small integer coordinates for all triangulated tori with up to 12 vertices. In particular, we provide coordinate-minimal realizations in general position for all triangulations of the torus with 7, 8, 9, and 10 vertices. For the unique 7-vertex triangulation of the torus we show that all corresponding 72 oriented matroids are realizable in the 6x6x6-cube. Moreover, we present polyhedral tori with 8 vertices in the 2x2x2-cube, general position realizations of triangulated tori with 8 vertices in the 2x2x3-cuboid as well as polyhedral tori with 9 and 10 vertices in the 1x2x2-cuboid.