The mathematics behind Escher's prints: a round trip
journey from symmetry to groups and back
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Summer Institute for Mathematics at the University of
Washington 2011
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Algebra
is nothing but written geometry; |
L'algèbre n'est
qu'une géométrie
écrite; |
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Geometry
is nothing but pictured algebra. |
la géométrie
n'est qu'une algèbre figurée. |
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Sophie Germain
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In this class we shall make a transition
from a very geometric notion of symmetry of two dimensional objects and
wallpaper patterns to a very algebraic notion of a group. Once we lay down the
foundations of the abstract group theory we shall go back to symmetries and
classify all possible wallpaper patterns. We shall give a (hopefully,
complete!) proof of the amazing fact that there are only 17 different
wallpaper patterns (or 17 crystallographic groups).
Our case study examples will be drawn from Escher's extensive collection of symmetry prints. We shall devote one class to Eshcher's life and work researched and presented entirely by the students.