Geometric Probability on a Lattice with the 15th Type of Convex Pentagon as a Fundamental Region

Abstract

In 2015, a group of mathematicians at the University of Washington, Bothell, discovered the 15th pentagon that can cover a plane, with no gaps and overlaps. However, research on its containment measure theory or geometric probability is limited. In this study, the Laplace extension of Buffon’s problem is generalized to the case of the 15th pentagon. In the solving process, the explicit expressions for the generalized support function and containment function of this irregular pentagon are derived. In addition, the chord length distribution function and density function of random distance of this pentagon are obtained in terms of the containment function.