Random Triangle Theory with Geometry and Applications

TL;DR

This paper investigates the probability of a random triangle being acute using modern mathematical tools like shape theory, random matrix theory, and the Hopf fibration, providing new proofs, insights, and applications.

Contribution

It introduces a modern shape-theoretic approach to triangle probability problems, offering new proofs, connecting to random matrix theory, and developing software for applications.

Findings

Triangles are uniformly distributed on a hemisphere under Gaussian vertex distribution.

A new proof links the Hopf map to the SVD of 2x2 matrices.

The smallest singular value serves as an effective test for uniformity.

Abstract

What is the probability that a random triangle is acute? We explore this old question from a modern viewpoint, taking into account linear algebra, shape theory, numerical analysis, random matrix theory, the Hopf fibration, and much much more. One of the best distributions of random triangles takes all six vertex coordinates as independent standard Gaussians. Six can be reduced to four by translation of the center to $(0,0)$ or reformulation as a 2x2 matrix problem. In this note, we develop shape theory in its historical context for a wide audience. We hope to encourage other to look again (and differently) at triangles. We provide a new constructive proof, using the geometry of parallelians, of a central result of shape theory: Triangle shapes naturally fall on a hemisphere. We give several proofs of the key random result: that triangles are uniformly distributed when the normal distribution is transferred to the hemisphere. A new proof connects to the distribution of random condition numbers. Generalizing to higher dimensions, we obtain the "square root ellipticity statistic" of random matrix theory. Another proof connects the Hopf map to the SVD of 2 by 2 matrices. A new theorem describes three similar triangles hidden in the hemisphere. Many triangle properties are reformulated as matrix theorems, providing insight to both. This paper argues for a shift of viewpoint to the modern approaches of random matrix theory. As one example, we propose that the smallest singular value is an effective test for uniformity. New software is developed and applications are proposed.