The Geometry of Triangles

TL;DR

This paper explores the mathematical structure of triangles through the lens of moduli spaces, establishing a formal framework for classifying oriented and non-oriented triangles and their geometric properties.

Contribution

It introduces a precise definition of continuous families of triangles and proves the existence of fine and coarse moduli spaces for oriented and non-oriented triangles, respectively.

Findings

Existence of a fine moduli space for oriented triangles

Existence of a coarse moduli space for non-oriented triangles

Potential applications to understanding stacks in geometry

Abstract

In this article we make the concept of a continuous family of triangles precise and prove the moduli functor classifying oriented triangles admits a fine moduli space but the functor classifying non-oriented triangles only admits a coarse moduli space. We hope moduli spaces of triangles can help understand stacks.