CECM Colloquium

**Wednesday January 31, 2001 From
3:30 - 4:30 in K9509, SFU
**

**
**

** Ellen Gethner, University of British Columbia
Talks on
**

**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.

** References:**

[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.