The Torus of Triangles

TL;DR

The paper addresses the moduli problem for labeled, oriented similarity classes of inscribable triangles, including degenerate cases, by constructing a fine moduli space ${\mathbb T}$ isomorphic to the space of inscribable classes. It parameterizes classes via angle triples modulo $\pi$, shows ${\mathbb T}$ is a 2-torus with a natural abelian group law, and relates degenerates to boundary points; it also proves a natural universal family $\sigma$ and identifies a quotient stack $[{\mathbb T}/D_6]$ to capture absolute (unlabeled, unoriented) classes. Key contributions include explicit descriptions of subgroups and cosets for triangle types, metric-based ratios among types, and a rigorous moduli-theoretic framework that uses both a fine moduli space and a quotient stack to account for symmetries. Overall, the work provides a robust geometric and algebraic structure for the space of inscribable triangle shapes, with connections to classical Porisma-type families and compactifications of configuration spaces.

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 $D_6$ defines a quotient stack $[\mathbb T/D_6]$, 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 $\mathbb T$ to compare them.

Theorems & Definitions (16)

• Definition 2.1
• Lemma 3.1
• proof
• Corollary 3.2
• proof
• Theorem 4.1
• proof
• Lemma 4.3
• proof
• Theorem 4.4