MATHEMATICS COLLOQUIUM

 WHEN:
On Thursday, February 22, 2001 at 2.30PM

WHERE:
Durham Science Center, Room 255

WHAT:

Ellen Gethner
University of British Columbia

will give a talk on

Abstract:
The well-known Dutch artist M.C. Escher is a familiar friend to many communities of scholars; included among them are artists, mathematicians, and computer scientists. Doris Schattschneider has recently drawn attention to some Combinatorial problems posed and partially solved by Escher.

Place an interesting asymmetric MOTIF (an abstract design) in a square. Consider the 0, 90, 180, and 270 degree rotations of the design about the center of the square. These are called the four ASPECTS of the motif. Create a decorated TILE T by selecting any four aspects of the original stamp, and placing them in a 2x2 grid formation. Tile the plane with T to create a periodic WALLPAPER PATTERN. With this set-up, how many inequivalent wallpaper patterns can be produced from the collection of 256 possible tiles? (Exercise: Why 256?) Alternatively, how many inequivalent 2x2 tiles with motifs in up to four aspects are there? Escher proved that the answer to this question is 23. His technique was to use brute force and a sketchpad!

The talk will be about solutions to generalizations of Escher's problem; the harmony of techniques used are drawn from Combinatorics, Elementary Number Theory, Geometry, and Group Theory. If time permits, a Coloring Problem of a Different Colour will be mentioned.