Penrose tilings constitute a class of non-periodic tilings of the plane. A tiling of the plane, as the name suggests, is a covering of the entire plane by shapes (tiles), no two of which overlap. A tiling can have almost any imaginable form, but the most interesting and most carefully studied tilings are those in which all of the tiles are identical copies of just a few different tiles. A tiling of this type is said to be periodic if there is a pattern that repeats itself--more precisely, if there is some small block of the tiling that, when it is shifted about by translations, will cover the entire tiling (a translation is an operation on the plane that shifts the position of every point by some fixed amount in some fixed direction). Many of the designs of the celebrated artist M. C. Escher are periodic...