Approximating Planar Rotations

We study the problem of approximating a rotation of the plane, α :R ² R ² , α (x,y)=(x cos θ + y sin θ , y cos θ - x sin θ ), by a bijection β :Z ² Z ² . We show by an explicit construction that β may be chosen so that , where r= tan ( θ /2). The scheme is based on those invented and patented by the second author in 1994. Received November 21, 1996, and in revised form February 20, 1997.