Full Euclidean Algorithm by Means of a Steady Walk

Let x and y be two positive real numbers with < y. Consider a traveler, on the interval [0, y/2], departing from 0 taking steps of length equal to x. Every time step reaches an endpoint interval, traveler rebounds off in order complete length. We show that footprints are output full Euclidean algorithm for y, whenever y/x is rational number. In case irrational, is, theoretically, not finite; however, it new tool study its irrationality.