The Stack of Similarity Classes of Triangles

TL;DR

This paper constructs a smooth moduli space of similarity classes of labeled, oriented triangles, revealing its topological structure and its relation to known shape spaces, and studies the action of the dihedral group on this space.

Contribution

It introduces a new, smooth, compact moduli space for labeled, oriented triangles and relates it to classical shape spaces like the sphere and torus.

Findings

The moduli space is a connected sum of three projective planes.

It projects onto the sphere and the torus, classical shape spaces.

The dihedral group acts naturally, forming a quotient stack of classes.

Abstract

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 $[\mathfrak D/D_6]$ of absolute (unlabeled, unoriented) classes.