The Stack of Similarity Classes of Triangles

We construct the smooth, compact moduli space of similarity classes of labeled, oriented triangles. The space, denoted $\mathfrak{D}$, 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 $D_6$ defines the quotient stack $[\mathfrak{D}\mathrm{/}{D}_6]$ of absolute (unlabeled, unoriented) classes.