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arXiv · 2303.11446

The Torus of Triangles

Eric Brussel·Madeleine E. Goertz
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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.

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Eric Brussel, Madeleine E. Goertz. 2023-03-20. The Torus of Triangles. https://arxiv.org/abs/2303.11446

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