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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2303.11446 |
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| _version_ | 1866929662009016320 |
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| author | Brussel, Eric Goertz, Madeleine E. |
| author_facet | Brussel, Eric Goertz, Madeleine E. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_11446 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Torus of Triangles Brussel, Eric Goertz, Madeleine E. Metric Geometry Algebraic Geometry 14C05, 51M05, 60D05 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. |
| title | The Torus of Triangles |
| topic | Metric Geometry Algebraic Geometry 14C05, 51M05, 60D05 |
| url | https://arxiv.org/abs/2303.11446 |