The Torus of Triangles

2023/03/20 by Eric Brussel, Brussel, Eric, Madeleine Goertz +1
Mathematics · #14C05 #51M05 #60D05 #Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Mathematical Dynamics and Fractals #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.2303.11446

openalex publication_date 2023/03/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove the 2-torus \mathbb 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 \it inscribable if it can be inscribed in a circle. A natural action by the dihedral group D₆ defines a quotient stack [\mathbb T/D₆], 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 \mathbb T to compare them.

Related

• The Stack of Similarity Classes of Triangles
• The Geometry of Triangles
• Torsion classes in the equivariant Chow groups of algebraic tori
• Degenerations of representations and thin triangles
• Topological characterization of torus groups