Buffon's problem with a star of needles and a lattice of parallelograms

TL;DR

This paper extends Buffon's needle problem to a star of needles intersecting a lattice of parallelograms, deriving intersection probabilities and the asymptotic distribution as the number of needles increases.

Contribution

It introduces a novel geometric approach to analyze a star of needles intersecting a lattice and derives the limit distribution function independent of lattice angle.

Findings

Calculated intersection probabilities for odd n

Derived the limit distribution function as n approaches infinity

Showed asymptotic independence of intersections with each lattice family

Abstract

A star of n (n greater than or equal to 2) line segments (needles) of equal length with common endpoint and constant angular spacing is randomly placed onto a lattice which is the union of two families of equidistant lines in the plane with angle alpha between the nonparallel lines. For odd n, we calculate the probabilities of exactly i intersections between the star and the lattice (for even n, see [3]). Using a geometrical method, we derive the limit distribution function of the relative number of intersections as n tends to infinity. This function is independent of alpha. We show that the relative numbers for each of the two families are asymptotically independent random variables.