On the digits of \(\pi\) (Q393485)

The magic number \(\pi\) has been the subject of numerous studies since antiquity. In 2001, Galperin suggested that successive digits of \(\pi\) may be obtained from a dynamical system. The model is described in terms of billiard balls which are point masses \(m\) and \(M\) that undergo perfectly elastic collisions with each other or with a wall in frictionless motion that is strictly one-dimensional. In the paper under review, this idealized dynamical system is further analyzed. Several interesting findings were obtained by the author by using direct proof, by observation of asymptotic trends or through limited numerical computations with Mathematica. In particular, it has been proved that a discrete invariant exists for ball-ball and ball-wall collisions, reversal of motion for mass \(M\) must occur at a ball-ball impact, and a terminal ball-ball impact yields an odd digit of \(\pi,\) whereas a terminal ball-wall impact gives an even digit. The analysis of the asymptotic trends provides information on the maximal velocity \(\left| u_{\max }\right| \) of particle \(m,\) minimum position \(y_{\min}\) of particle \(M\) and reversal time \(t_{r}\) to reach \(y_{\min}.\) Finally, numerical experiments show that two qualitatively distinct types of reversal are encountered.