Alice in Triangleland: Lewis Carroll's Pillow Problem and Variants Solved on Shape Space of Triangles

Abstract

We provide a natural answer to Lewis Carroll's pillow problem of what is the probability that a triangle is obtuse, Prob(Obtuse). This arises by straightforward combination of a) Kendall's Theorem - that the space of all triangles is a sphere - and b) the natural map sending triangles in space to points in this shape sphere. The answer is 3/4. Our method moreover readily generalizes to a wider class of problems, since a) and b) both have many applications and admit large generalizations: Shape Theory. An elementary and thus widely accessible prototype for Shape Theory is thereby desirable, and extending Kendall's already-notable prototype a) by demonstrating that b) readily solves Lewis Carroll's well-known pillow problem indeed provides a memorable and considerably stronger prototype. This is a prototype of, namely, mapping flat geometry problems directly realized in a space to shape space, where differential-geometric tools are readily available to solve the problem and then finally re-interpret it in the original `shape-in-space' terms. We illustrate this program's versatility by posing and solving a number of variants of the pillow problem. We first find Prob(Isosceles is Obtuse). We subsequently define tall and flat triangles, as bounded by regular triangles whose base-to-median ratio is that of the equilateral triangle. These definitions have Jacobian and Hopfian motivation as well as entering Kendall's own considerations of `splinters': almost-collinear triangles. We find that Prob(Tall) = 1/2 = Prob(Flat) is immediately apparent from regularity's symmetric realization in shape space. However, Prob(Obtuse and Flat), Prob(Obtuse is Flat) and all other nontrivial expressions concerning having any two of the properties mentioned above, or having one of these conditioned on another, constitute nontrivial variants of the pillow problem, and we solve them all.