A Universal Triangulation for Flat Tori

TL;DR

This paper adapts a classical proof to produce explicit PL isometric embeddings of flat tori and introduces a universal triangulation with 2434 triangles capable of representing any flat torus.

Contribution

It provides a constructive method for embedding flat tori and introduces a universal triangulation applicable to all flat tori.

Findings

Explicit PL embeddings for flat tori are achievable.

A universal triangulation with 2434 triangles can realize any flat torus.

The method produces distinct embeddings for different tori.

Abstract

A result due to Burago and Zalgaller (1960, 1995) states that every orientable polyhedral surface, one that is obtained by gluing Euclidean polygons, has an isometric piecewise linear (PL) embedding into Euclidean space $\mathbb{E}^3$. A flat torus, resulting from the identification of the opposite sides of a Euclidean parallelogram, is a simple example of polyhedral surface. In a first part, we adapt the proof of Burago and Zalgaller, which is partially non-constructive, to produce PL isometric embeddings of flat tori. Our implementation produces embeddings with a huge number of vertices, moreover distinct for every flat torus. In a second part, based on another construction of Zalgaller (2000) and on recent works by Arnoux et al. (2021), we exhibit a universal triangulation with 2434 triangles which can be embedded linearly on each triangle in order to realize the metric of any flat torus.