Because Pi Day fell on the Friday before break, many of you
may have missed out on the festivities. We’ll develop a bit of
integral geometry (a.k.a. geometric probability) that lets us compute
Pi by a random process – the Buffon Needle Problem – as well as the
lengths of curves in the plane and the sphere, eventually leading to
the now-classic theorem of Fenchel-Fary-Milnor about the total
curvature of space curves.
Start with a graph whose edges are labeled by positive integers. Label each vertex with an integer so that if two vertices are joined by an edge the vertex labels are congruent modulo the edge label. A set of vertex labels satisfying this condition is called a spline. Much of our research pertains to splines on n-cycles. We use the Chinese Remainder Theorem to find particularly nice splines, and we identify two bases for the set of splines on n-cycles.
In 1917 a Japanese mathematician named Kakeya posed “the needle problem,” asking: what is the area of the smallest figure in the plane in which a unit line segment (a “needle”) can be rotated 180 degrees? It was conjectured that the smallest such “Kakeya set” was a deltoid with area $\pi/8$. However, in 1928 a Russian mathematician named Besicovitch published a very surprising result: Kakeya sets can be arbitrarily small. More than just a historic and geometric curiosity, Kakeya sets have come to play an important role in analysis today. In this talk, we will construct a Kakeya set with arbitrarily small area and then discuss some problems that are unexpectedly related to these sets.