Counting Escher's m£m Ribbon Patterns

Using a construction scheme originally devised by M.C. Escher, one can generate doubly-periodic patterns of the xy-plane with the operations of ro- tation, re∞ection and translation acting on an asymmetric square motif. Rotating and/or re∞ecting the original motif yields eight distinct aspects. By selecting m 2 (not necessarily distinct) motif aspects and arranging them in an m £ m Escher tile, one can then tile the xy-plane by translating the Escher tile by integer mul- tiples of m in the x and/or y direction to create wallpaper patterns. Two wallpaper patterns are considered equivalent if there is some isometry between the two. Previously, the general formula was given by the second au- thor (Gethner) in (6) for the number of inequivalent patterns generated by m£ m Escher tiles composed of the four rotated aspects of a single asymmetric motif by applying Burnside's Lemma. Here we extend that formula to include the four additional re∞ected aspects when composing m£m Escher tiles with which to tile the plane.