Spherical Geometry and the Least Symmetric Triangle

TL;DR

This paper identifies the least symmetric triangle using geometric and algebraic methods, providing exact calculations and insights into polygon and shape spaces relevant to both pure mathematics and chemistry.

Contribution

It introduces a novel approach to determine the least symmetric triangle via Grassmannian and group action analysis, with exact computations for various triangle types.

Findings

Exact determination of the least symmetric triangle.

Identification of least symmetric obtuse and acute triangles.

Methodology applicable to polygon and shape space computations.

Abstract

We study the problem of determining the least symmetric triangle, which arises both from pure geometry and from the study of molecular chirality in chemistry. Using the correspondence between planar $n$-gons and points in the Grassmannian of 2-planes in real $n$-space introduced by Hausmann and Knutson, this corresponds to finding the point in the fundamental domain of the hyperoctahedral group action on the Grassmannian which is furthest from the boundary, which we compute exactly. We also determine the least symmetric obtuse and acute triangles. These calculations provide prototypes for computations on polygon and shape spaces.