Fractals with Positive Length and Zero Buffon Needle Probability

It turns out that H1 is a measure, now called one-dimensional Hausdorff measure because it was generalized by Hausdorff [10] to the whole family of measures Hα, where α is any positive number (integer or noninteger). The modern theory of “fractals” is largely based on the notion of the Hausdorff dimension dimH (F) of a set F , defined by dimH (F) = inf{α > 0 : Hα(F) = 0}. We recommend the book by Falconer [8] for an introduction to fractals. Here we consider only α = 1, since we are interested in the generalizations of length. If F is a rectifiable curve, then H1(F) is exactly its length; therefore in modern analysis, it is standard to refer to H1(F) for any compact set F as the “length of F .” Another way to measure length goes back even further, to the eighteenth century. In 1733 Georges-Louis Leclerc, the “Comte de Buffon,” posed the following problem, which became known as “Buffon’s needle problem”: Given a collection of parallel lines in the plane, with distance d between adjacent lines, determine the probability that a needle of length < d will cross one of these lines when dropped at random on the plane. The answer 2 /(πd) was given by Buffon himself in 1777 and can be found in many probability texts (for example, in [9]). It follows from Buffon’s formula that if a polygon of perimeter p and diameter less than d is dropped on the same plane, then the expected number of points at which it will cross one of the parallel lines is 2p/(πd). This idea was formalized and extended in 1868 by Crofton (see [6] and [19]), as follows. Count the number of intersections of a given set F with a straight line, and then integrate this number over the space of all lines. The result is denoted by I1(F) and called the integral-geometric measure of F . More precisely, let θ be