The Elliptic Integral Machine: A Collision-based Model of Computation

In this work we will show how any elliptic integral can be computed by analyzing the asymptotic behavior of idealized mechanical models. Specifically, our results reveal how a set of circular billiard systems computes the canonical set of three elliptic integrals defined by Legendre. We will treat these Newtonian systems as a particular application of the billiard-ball model, a ballistic computer idealized by Eduard Fredkin and Tommaso Toffoli. Initially, we showed how to define the initial conditions in order to encode the computation of a set of integral functions. We then combined our first conclusions with results established in the 18th and 19th centuries mostly by Euler, Lagrange, Legendre and Gauss in developing the theory of integral functions. In this way, we derived collision-based methods to compute elementary functions, integrals functions and mathematical constants. In particular, from the Legendre identity for elliptic integrals, we were able to define a new collision-based method to compute the number $\pi$, while an identity demonstrated by Gauss revealed a new method to compute the arithmetic-geometric mean. In order to explore the computational potential of the model, we admitted a hypothetical device that measures the total number of collisions between the balls and the boundary. There is even the possibility that the methods we are about to describe could one day be experimentally applied using optical phenomena, as recent studies indicate that it is possible to implement collision-based computation with solitons.