Six combinatorial clases of maximal convex tropical polyhedra

TL;DR

This paper classifies six combinatorial types of convex maximal tropical tetrahedra, linking tropical linear algebra with convex geometry, and explores their properties, symmetries, and examples.

Contribution

It introduces a classification of convex maximal tropical tetrahedra into six classes based on combinatorial properties, a novel connection between tropical linear algebra and convex bodies.

Findings

Six classes of convex maximal tropical tetrahedra identified

Only one class contains symmetric solids, one contains chiral ones

Convex maximal tropical tetrahedra are generally not vertex-transitive

Abstract

In this paper we bring together tropical linear algebra and convex 3-dimensional bodies. We show how certain convex 3-dimensional bodies having 20 vertices and 12 facets can be encoded in a $4\times 4$ integer zero-diagonal matrix $A$. A tropical tetrahedron is the set of points in $\R^3$ tropically spanned by four given tropically non-coplanar points. It is a near-miss Johnson solid. The coordinates of the points are arranged as the columns of a $4\times 4$ real matrix $A$ and the tetrahedron is denoted $\spann(A)$. We study tropical tetrahedra which are convex and maximal, computing the extremals of $\spann(A)$ and the length (tropical or Euclidean) of its edges. Then, we classify convex maximal tropical tetrahedra, combinatorially. There are six classes, up to symmetry and chirality. Only one class contains symmetric solids and only one contains chiral ones. In the way, we show that the combinatorial type of the regular dodecahedron does not occur here. We also prove that convex maximal tropical tetrahedra are not vertex-transitive, in general. We give families of examples, circulant matrices providing examples for two classes. A crucial role is played by the $2\times 2$ minors of $A$.