Project III 2023-24

Random Triangles and Polygons

Martin Kerin

A random 100,000-gon in the plane. Source: Clayton Shonkwiler, Stiefel manifolds and polygons, Proceedings of Bridges 2019: Mathematics, Art, Music, Architecture, Education, Culture, Linz, Austria, 187-194.

Description In 1895, the mathematician Charles L. Dodgson, better known as Lewis Carroll, posed the following problem: Three points are taken at random on an infinite plane. Find the chance of their being the vertices of an obtuse-angled triangle. The problem admits several interpretations and different solutions have been given by many authors, including Carroll himself. Moreover, one may also ask a similar question for more general figures, such as polygons. The solutions to these questions bring together differential geometry and probability theory and lead to applications in shape classification and computer vision. Prerequisites and Corequisites Initially, the project will focus on classical interpretations of Carroll's problem in Euclidean space and requires nothing more than an understanding of Euclidean (planar) geometry and some basic understanding of probability. As the project progresses, it would be useful to be familiar with some parts of (or taking) the modules Probability II, Analysis III, Geometry III, Topology III and Differential Geometry III, but most things can be covered as needed. In particular, the project will touch on basic notions in topology (open sets, closed, sets, compactness, connectedness) covered in Topology III; basic notions in geometry (spherical geometry) covered in Geometry III; basic analytic and algebraic notions (definition of a group, smooth functions, multivariate analysis); basic notions of measure theory (probability measures) covered in Probability II/Analysis III. Some Resources Jason Cantarella, Tom Needham, Clayton Shonkwiler, and Gavin Stewart, Random triangles and polygons in the plane, Amer. Math. Monthly 126 (2019), no. 2, 113-134. (Here's a short video in which Clayton Shonkwiler describes their basic approach to the problem.) Laney Bowden, Andrea Haynes, Clayton Shonkwiler, and Aaron Shukert, Spherical geometry and the least symmetric triangle , Geom. Dedicata 198 (2019), 19-34. Clayton Shonkwiller, The geometry of constrained random walks and an application to frame theory, 2018 IEEE Statistical Signal Processing Workshop (SSP), Freiburg, Germany, 343-347. Clayton Shonkwiller, Stiefel Manifolds and Polygons , Proceedings of Bridges 2019: Mathematics, Art, Music, Architecture, Education, Culture; Linz, Austria, 187-194.

email: M Kerin