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Journal for Geometry and Graphics 10 (2006), No. 1, 001--013
Copyright Heldermann Verlag 2006

Counting Escher's m x m Ribbon Patterns

J. Joseph Fowler
Dept. of Computer Science, University of Arizona, 1040 E. 4th Street, Tucson, AZ 85721-0077, U.S.A.

Ellen Gethner
Dept. of Computer Science and Engineering, University of Colorado, Denver, CO 80217-173364, U.S.A.


Using a construction scheme originally devised by M.C. Escher, one can generate doubly-periodic patterns of the $xy$-plane with the operations of rotation, reflection and translation acting on an asymmetric square motif. Rotating and/or reflecting the original motif yields eight distinct aspects. By selecting $m^2$ (not necessarily distinct) motif aspects and arranging them in an $m \times m$ Escher tile, one can then tile the $xy$-plane by translating the Escher tile by integer multiples 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 author [Proc. 32nd Southeastern Conf. on Combinatorics, Graph Theory and Computing, Baton Rouge, vol. 153 (2001) 77--96] for the number of inequivalent patterns generated by $m \times m$ 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 reflected aspects when composing $m \times m$ Escher tiles with which to tile the plane.

Keywords: Motif, wallpaper pattern, Escher tile, symmetry, group action, geometric structure.

MSC: 51F15; 52C20

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