The Stack of Similarity Classes of Triangles
Abstract
We construct the smooth, compact moduli space of similarity classes of labeled, oriented triangles. The space, denoted , is a connected sum of three projective planes, and projects via blowdown to two shape spaces that have appeared in the literature: the well-known (Riemann) sphere (\cite{Kend84}, \cite{Beh}, \cite{Montgomery}, \cite{ES15}), and the less-well-known 2-torus (\cite{BG23}). A natural action by the dihedral group defines the quotient stack of absolute (unlabeled, unoriented) classes.
Keywords
Cite
@article{arxiv.2408.07792,
title = {The Stack of Similarity Classes of Triangles},
author = {Eric Brussel and Madeleine Goertz and Elijah Guptill and Kelly Lyle},
journal= {arXiv preprint arXiv:2408.07792},
year = {2025}
}
Comments
26 pages. Changes primarily include increased detail, more pictures, and better exposition. Semple and Fulton-MacPherson are more fully integrated. The order of sections is changed, some new ones added, some old ones deleted. No main results are added, and none removed