Randomness and Geometric Probability

• Buffon's needle. What is the probability that a dropped needle lands on a crack on a hardwood floor? From Kunkel's mathematics lessons.

• Geometric probability question. What is the probability that the shortest paths between three random points on a projective plane form a contractible loop?

• Geometric probability constants. From MathSoft's favorite constants pages.

• Green-haired geometric pre-hominids.

• Points on a sphere. Paul Bourke describes a simple random-start hill-climbing heuristic for spreading points evenly on a sphere, with pretty pictures and C source.

• Random domino tiling of an Aztec diamond and other undergrad research on random tiling.

• Random spherical arc crossings. Bill Taylor and Tal Kubo prove that if one takes two random geodesics on the sphere, the probability that they cross is 1/8. This seems closely related a famous problem on the probability of choosing a convex quadrilateral from a planar distribution. The minimum (over all possible distributions) of this probability also turns out to solve a seemingly unrelated combinatorial geometry problem, on the minimum number of crossings possible in a drawing of the complete graph with straight-line edges: see also "The rectilinear crossing number of a complete graph and Sylvester's four point problem of geometric probability", E. Scheinerman and H. Wilf, Amer. Math. Monthly 101 (1994) 939-943, rectilinear crossing constant, S. Finch, MathSoft, and Calluna's pit, Douglas Reay.

• Random polygons. Tim Lambert summarizes responses to a request for a good random distribution on the n-vertex simple polygons.

• Self-trapping random walks, Hugo Pfoertner.

• Zonohedron generated by 30 vectors in a circle, and another generated by 100 random vectors, Paul Heckbert, CMU. As a recent article in The Mathematica Journal explains, the first kind of shape converges to a solid of revolution of a sine curve. The second clearly converges to a sphere but Heckbert's example looks more like a space potato.

From the Geometry Junkyard, computational and recreational geometry pointers.
Send email if you know of an appropriate page not listed here.
David Eppstein, Theory Group, ICS, UC Irvine.
Semi-automatically filtered from a common source file.