2017/11/30 by Edward Anderson, Anderson, Edward · 5 citations
Engineering · Mathematics · #Advanced Numerical Analysis Techniques #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #History and Overview (math.HO) #History and Theory of Mathematics #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.1711.11492
openalex publication_date 2017/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We provide a natural answer to Lewis Carroll's pillow problem of what is the\nprobability that a triangle is obtuse, Prob(Obtuse). This arises by\nstraightforward combination of a) Kendall's Theorem - that the space of all\ntriangles is a sphere - and b) the natural map sending triangles in space to\npoints in this shape sphere. The answer is 3/4. Our method moreover readily\ngeneralizes to a wider class of problems, since a) and b) both have many\napplications and admit large generalizations: Shape Theory. An elementary and\nthus widely accessible prototype for Shape Theory is thereby desirable, and\nextending Kendall's already-notable prototype a) by demonstrating that b)\nreadily solves Lewis Carroll's well-known pillow problem indeed provides a\nmemorable and considerably stronger prototype. This is a prototype of, namely,\nmapping flat geometry problems directly realized in a space to shape space,\nwhere differential-geometric tools are readily available to solve the problem\nand then finally re-interpret it in the original `shape-in-space' terms. We\nillustrate this program's versatility by posing and solving a number of\nvariants of the pillow problem. We first find Prob(Isosceles is Obtuse). We\nsubsequently define tall and flat triangles, as bounded by regular triangles\nwhose base-to-median ratio is that of the equilateral triangle. These\ndefinitions have Jacobian and Hopfian motivation as well as entering Kendall's\nown considerations of `splinters': almost-collinear triangles. We find that\nProb(Tall) = 1/2 = Prob(Flat) is immediately apparent from regularity's\nsymmetric realization in shape space. However, Prob(Obtuse and Flat),\nProb(Obtuse is Flat) and all other nontrivial expressions concerning having any\ntwo of the properties mentioned above, or having one of these conditioned on\nanother, constitute nontrivial variants of the pillow problem, and we solve\nthem all.\n