A universal triangulation for flat tori (2022)
The paper "A Universal Triangulation for Flat Tori" presents a method for embedding flat tori into Euclidean space, introducing a universal triangulation of 2434 triangles for accurate metric representation.
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The paper titled "A Universal Triangulation for Flat Tori" by Francis Lazarus and Florent Tallerie discusses advancements in the field of computational geometry, specifically regarding the isometric embedding of flat tori into Euclidean space. Building on the foundational work of Burago and Zalgaller, the authors adapt a previously non-constructive proof to create piecewise linear (PL) embeddings of flat tori, which are surfaces formed by identifying opposite sides of a Euclidean parallelogram. Their implementation results in embeddings characterized by a large number of vertices, unique to each flat torus. Furthermore, the authors present a universal triangulation consisting of 2434 triangles, which can be linearly embedded to accurately represent the metric of any flat torus. This work not only revises previous constructions to reduce the number of triangles but also contributes to the understanding of geometric topology and metric geometry.
- The paper presents a method for embedding flat tori into Euclidean space.
- It builds on earlier work by Burago and Zalgaller, adapting their proof for practical applications.
- A universal triangulation with 2434 triangles is introduced, allowing for accurate metric representation.
- The authors' approach results in unique embeddings for each flat torus.
- The research contributes to the fields of computational geometry and geometric topology.
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