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Abstract:
Suppose you have a convex polygon placed at some position in the XY-plane
(for instance, the origin). You pause to answer your cell phone when the
polygon wanders away from its initial position; it does so by reflections
in its sides. Your object is to now move it back to its initial position
by again using reflections in the sides of the polygon. This is
particularly easy when the polygon is an equilateral triangle, a square or
a regular hexagon. This is because these are the only regular polygons
that tile the plane without overlapping i.e., the group of reflections in
the sides acts discretely on the plane. Things get much more interesting
when our chosen polygon is the regular pentagon, which we call "Lucy" (or
more generally any regular n-gon where n is not 3, 4 or 6). In this
case, one cannot tile the plane using pentagons, but more dramatically,
one may move the pentagon by a series of reflections so that it is
centered essentially anywhere in the plane i.e., the set of possible
centers is dense in the plane. This is in stark contrast to the case of n
= 3,4,6. So, given that Lucy has moved away from the origin, the
probability of our finding a path back is 0.
We now perform a magic trick: we open a window into an alternate universe
that contains another pentagon, "Lily". The two pentagons are linked in a
precise way so that when one moves, so does the other (Lily's sides are a
permutation of Lucy's sides). Now the following strategy works: move Lucy
directly towards the origin (Lily will move about, but we are not
concerned). When Lucy gets close to the origin, we shift attention and
move Lily to her origin, and then back to Lucy etc. Within a few
switches, both pentagons land exactly where they are supposed to. In this
talk we will describe the game and explain why the magic trick works.
This game was created by Rich Schwartz of the University of Maryland and
is a small part of the beautiful theory of arithmetic groups; the game is
named for his two daughters. You can check out the game for yourself:
(Applets 12, 13 and 14 are about
Lucy and Lily); it's very, very cool mathematics!!!
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