Wednesday January 31, 2001 From 3:30 - 4:30 in K9509, SFU
Ellen Gethner, University of British Columbia
M.C. Escher Inspires a Coloring Problem of a Different Colour: Art, Computer Science, and Mathematics Collide
(Joint work with David G. Kirkpatrick and Nicholas Pippenger, University of British Columbia)
Abstract: The works of Dutch graphic artist M.C. Escher are familiar friends to, among others, Artists, Computer Scientists, and Mathematicians. Recently, Doris Schattschneider [Sc1,Sc2,D,G1], has called attention to a combinatorial technique used by Escher to create periodic patterns in the plane. The technique was inspired by a game that Escher played with his children [E]. Ultimately, a square tile is decorated by a finite set of overlapping polygonal regions, which intersect the boundary of the tile symmetrically and aesthetically. The plane is then tiled by standard horizontal and vertical translations, yielding a doubly periodic wallpaper pattern. That the original tile is made up of overlapping regions lends an air of mystery to the final outcome. Escher deepened the mystery by adding color to the mix, and in doing so aroused the curiosity of Rick Mabry, Stan Wagon, and Doris Schattschneider [MWS]. Their question: is there a rectangular prototile (concatenated copies of the original square tile), which can be colored so that colors on the sides and top and bottom match appropriately? Can this be done in such a way that overlapping components in the resulting plane pattern always receive different colors? The answer is "yes:" such a prototile always exists. The proof is constructive: it is an algorithm that draws from geometry, graph theory, and number theory. The output of the algorithm: the dimensions of the prototile together with coloring instructions. Moreover, a classification of all prototiles for a given square tile is in progress, and uses tools from computational and classical number theory, graph theory, and graph algorithms [GKP].
I will tell the story of this ongoing problem and solution, and of the many pleasant expected and unexpected generalizations that are now beginning to surface.
[D] Dan Davis, "On a tiling scheme from M.C. Escher", The Electronic Journal of Combinatorics, v. 4, no. 2(1997) #R23.
[E] George A. Escher, "Potato Printing, A Game for Winter Evenings," M.C. Escher: Art and Science, H.S.M. Coxeter, M. Emmer, R. Penrose, M. Teuber, eds. Amsterdam: North Holland, 1986, 9-11.
[GKP] Ellen Gethner, David Kirkpatrick, and Nicholas Pippenger, "The classification of Escher-tiles," in preparation, 2001.
[G1] Ellen Gethner, " On the exact number of inequivalent mxm tiles in up to four and in up to eight aspects," forthcoming.
[MWS] Rick Mabry, Stan Wagon, and Doris Schattschneider, " Automating Escher's combinatorial patterns," Mathematica in Education and Research, v. 5, no. 4(1996), 38-52 .
[Sc1] Doris Schattschneider, "Escher's combinatorial patterns", The Electronic Journal of Combinatorics, v. 4, no. 2(1997) #R17.
[Sc2] Doris Schattschneider, Visions of Symmetry: Notebooks, Periodic Drawings, and Related Work of M.C. Escher, W.H. Freeman & Co, 1990, 44-52.