Random Triangle Theory with Geometry and Applications
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 si...
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| Format: | Article |
| Language: | English |
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Springer US
2016
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| Online Access: | http://hdl.handle.net/1721.1/103361 https://orcid.org/0000-0001-7473-9287 https://orcid.org/0000-0001-7676-3133 |
| _version_ | 1811073976935383040 |
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| author | Edelman, Alan Strang, Gilbert |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Edelman, Alan Strang, Gilbert |
| author_sort | Edelman, Alan |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T09:41:07Z |
| format | Article |
| id | mit-1721.1/103361 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T09:41:07Z |
| publishDate | 2016 |
| publisher | Springer US |
| record_format | dspace |
| spelling | mit-1721.1/1033612022-09-30T16:10:46Z Random Triangle Theory with Geometry and Applications Edelman, Alan Strang, Gilbert Massachusetts Institute of Technology. Department of Mathematics Edelman, Alan Strang, Gilbert 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. National Science Foundation (U.S.) (NSF DMS 1035400) National Science Foundation (U.S.) (NSF DMS 1016125) National Science Foundation (U.S.) (NSF EFRI 1023152) 2016-06-27T19:54:12Z 2016-06-27T19:54:12Z 2015-03 2014-09 2016-05-23T12:14:25Z Article http://purl.org/eprint/type/JournalArticle 1615-3375 1615-3383 http://hdl.handle.net/1721.1/103361 Edelman, Alan, and Gilbert Strang. "Random Triangle Theory with Geometry and Applications." Foundations of Computational Mathematics 15:3 (2015), pp.681-713. https://orcid.org/0000-0001-7473-9287 https://orcid.org/0000-0001-7676-3133 en http://dx.doi.org/10.1007/s10208-015-9250-3 Foundations of Computational Mathematics Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ SFoCM application/pdf Springer US Springer US |
| spellingShingle | Edelman, Alan Strang, Gilbert Random Triangle Theory with Geometry and Applications |
| title | Random Triangle Theory with Geometry and Applications |
| title_full | Random Triangle Theory with Geometry and Applications |
| title_fullStr | Random Triangle Theory with Geometry and Applications |
| title_full_unstemmed | Random Triangle Theory with Geometry and Applications |
| title_short | Random Triangle Theory with Geometry and Applications |
| title_sort | random triangle theory with geometry and applications |
| url | http://hdl.handle.net/1721.1/103361 https://orcid.org/0000-0001-7473-9287 https://orcid.org/0000-0001-7676-3133 |
| work_keys_str_mv | AT edelmanalan randomtriangletheorywithgeometryandapplications AT stranggilbert randomtriangletheorywithgeometryandapplications |