Counting Escher's m x m Ribbon Patterns

J. J. Fowler ; E. Gethner

Journal for geometry and graphics, Tome 10 (2006) no. 1, pp. 1-13

Résumé

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.

Détail

Classification : 51F15, 52C20
Mots-clés : Motif, wallpaper pattern, Escher tile, symmetry, group action, geometric structure