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
