A Simple 3D Isometric Embedding of the Flat Square Torus

TL;DR

This paper presents a straightforward method for embedding a flat square torus in three-dimensional space using an isometric polyhedral model derived from a modified envelope polyhedron, illustrating its geometric properties.

Contribution

It introduces a simple, explicit 3D isometric embedding of the flat square torus based on a modified envelope polyhedron construction.

Findings

The embedding has zero Gaussian curvature on faces and edges.

All meridian and latitudinal circumferences are equal at 4 units.

The polyhedron has 8 faces, 8 vertices, and 16 edges.

Abstract

Start with Gott (2019)'s envelope polyhedron (Squares-4 around a point): a unit cube missing its top and bottom faces. Stretch by a factor of 2 in the vertical direction so its sides become (2x1 unit) rectangles. This has 8 faces (4 exterior, 4 interior), 8 vertices, and 16 edges. F-E+V = 0, implying a (toroidal) genus = 1. It is isometric to a flat square torus. Like any polyhedron it has zero intrinsic Gaussian curvature on its faces and edges. Since 4 right angled rectangles meet at each vertex, there is no angle deficit and zero Gaussian curvature there as well. All meridian and latitudinal circumferences are equal (4 units long).