The torus of triangles

Brussel, Eric; Goertz, Madeleine

doi:10.2140/involve.2026.19.575

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The torus of triangles

Eric Brussel and Madeleine Goertz

Vol. 19 (2026), No. 4, 575–591

DOI: 10.2140/involve.2026.19.575

Abstract

We prove the 2-torus $𝕋$ , an abelian linear algebraic group, is a fine moduli space of labeled, oriented, possibly degenerate inscribable similarity classes of triangles, where a triangle is inscribable if it can be inscribed in a circle. A natural action by the dihedral group $D_{6}$ defines a quotient stack $[𝕋 ∕ D_{6}]$ , which is the stack of absolute (unlabeled, unoriented) possibly degenerate inscribable classes. We show the main triangle types form distinguished algebraic substructures: subgroups, cosets, and elements of small order, and we apply the natural metric on $𝕋$ to compare them.

Keywords

triangles, moduli space, moduli space of triangles, shape space, shape space of triangles, torus, torus of triangles

Mathematical Subject Classification

Primary: 14C05, 51M05, 60D05

Milestones

Received: 7 December 2023

Revised: 18 September 2024

Accepted: 15 December 2024

Published: 13 October 2026

Communicated by Frank Morgan

Authors

Eric Brussel
Department of Mathematics California Polytechnic State University San Luis Obispo, CA United States

Madeleine Goertz
Department of Mathematics California Polytechnic State University San Luis Obispo, CA United States

Vol. 19, No. 4, 2026