A Universal Triangulation for Flat Tori

TL;DR

The paper addresses the problem of isometrically embedding every flat torus into 3-space via piecewise-linear maps. It advances two complementary strategies: (i) adapting the Burago–Zalgaller construction to obtain PL isometric embeddings of flat tori, yielding embeddings with very large triangulations, and (ii) constructing a universal triangulation of 2434 triangles that works linearly on each triangle to realize any flat torus, by combining a long-torus universal scheme (270 triangles) with short-torus diplotori from Arnoux–Tsuboi. The result is a finite, universal combinatorial skeleton that can realize all flat tori up to isometry in $\,\mathbb{E}^3$, together with explicit conformal embeddings for rectangular and general tori and acute-triangulation techniques. This has practical implications for robust PL representations of toroidal geometries and provides a framework for extending universal triangulations to broader moduli spaces. The work thus bridges classical isometric embedding theory with modern explicit constructions, delivering a concrete, finite triangulation that uniformly captures the entire flat-torus moduli space.

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.

Figures (29)

• Figure 1: A point (in red) in a fundamental domain of the moduli space of tori (in light grey) with the corresponding parallelogram. Note that the boundary of the domain should be identified adequately to represent this moduli space faithfully, leading to a non-compact orbifold.
• Figure 2: The prism above $t$.
• Figure 3: Uniform subdivision of a triangle. The vertices of the subdivision have barycentric coordinates $(i/n,j/n,k/n)$ for $i,j,k\in \mathbb{N}$ and $i+j+k=n$ for some fixed $n$.
• Figure 4: The subtriangle $\Omega A_1A_2$ is folded above $t$.
• Figure 5: a, the reflection plane $\Pi_1$. b, after reflection in $\Pi_1$, and the plane $\Pi_2$. c, reflection in $\Pi_2$. d, after an even number of reflections the point $\Omega$ is sent to $\omega'$.

Theorems & Definitions (16)

• Theorem 1
• Remark 1
• Theorem 2: Burago and Zalgaller bz-iplit-95
• Lemma 3: bz-iplit-95
• Proposition 5
• Lemma 6
• proof
• Lemma 7
• proof
• Lemma 10