The Secret of Gambling with Irregular Dice: Estimating the Face Statistics of Polyhedra

Abstract

This paper is devoted to the description of a new geometric algorithm to estimate the “face statistics” of an irregularly-shaped polyhedron, i.e. the probability that it rests on a given face after being dropped onto a horizontal surface. A recently  published paper of the author discussed existing and new methods to estimate this probability, together with a comparison of their reliability, and the challenges of benchmarking them against data obtained via computer simulation. The present work focuses on a new estimation algorithm, with computational complexity linearly proportional to the number of vertices. The main industrial application area of this result is automated assembly: the throughput of a part feeder is heavily influenced by the face statistics of the parts processed by the feeder.