"Imagine that you want to know the most efficient way to make a torusa doughnut-shaped mathematical objectfrom origami paper. But this torus, which is a surface, looks drastically different than the outside of a glazed bakery doughnut. Instead of seeming almost perfectly smooth, the torus that you envision is jagged with many faces, each of which is a polygon. In other words, you want to construct a polyhedral torus with faces that are shapes such as triangles or rectangles."
"The complexity of the problem only grows if you decide that you want to envision constructing something similar but in four or more dimensions. Mathematician Richard Evan Schwartz of Brown University tackled the problem in a recent study by working backward from an existing polyhedral torus to answer questions about what would be needed to construct it from scratch. He posted his findings to a preprint server in August 2025."
"Schwartz was able to find a solution to a long-standing question: What's the minimum number of vertices (corners) needed to make polyhedral tori with a property called intrinsic flatness? The answer, Schwartz found, is eight vertices. He first demonstrated that seven vertices aren't enough. He then discovered an example of an intrinsically flat polyhedral torus with eight vertices. It's very striking that Rich Schwartz was able to entirely solve this well-known problem, says Jean-Marc Schlenker, a mathematician at the University of Luxembourg."
A solution demonstrates that the minimum number of vertices needed for an intrinsically flat polyhedral torus is eight, and seven vertices are insufficient. The method determines construction requirements by working backward from an existing polyhedral torus and produces an explicit eight-vertex example of an intrinsically flat polyhedral torus. The complexity of constructing polyhedral tori increases when generalizing to four or more dimensions. The result resolves a long-standing, elementary-seeming question by establishing a sharp lower bound on vertices for intrinsically flat polyhedral tori.
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