Random Triangle Theory with Geometry and Applications
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10208_2015_9250_ReferencePDF.pdf
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1.35 MB
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Adobe PDF
Checksum (MD5)
e76a2240e210af8100ed04012380f8ac
Author(s)
Edelman, Alan
Strang, Gilbert
Date Issued
March 2015
Journal
Foundations of Computational Mathematics
Publisher
Springer US
Citation
Edelman, Alan, and Gilbert Strang. "Random Triangle Theory with Geometry and Applications." Foundations of Computational Mathematics 15:3 (2015), pp.681-713.
Version
Author's final manuscript
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 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 2 × 2 random matrix problem. In this note, we develop shape theory in its historical context for a wide audience. We hope to encourage others 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 × 2 matrices. A new theorem describes three similar triangles hidden in the hemisphere. Many triangle properties are reformulated as matrix theorems, providing insight into 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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1007/s10208-015-9250-3
