Symmetries of the plane: what group theory tells us about tilings in Escher prints? |
Summer Institute for Mathematics at the University of
Washington 2019
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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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We shall look at both periodic and non-periodic wallpaper
patterns and make a transition from a geometric notion of symmetry to an
algebraic notion of a group. All regular (or periodic) tilings
of the plane have been classified in the last century: essentially, there are
only 17
different wallpaper
patterns (or 17 crystallographic groups). All of them can be
supposedly found among Escher’s
extensive collection of symmetry prints which we shall attempt to do.
We
shall also observe that only 2,3,4, and 6 fold
symmetries exist in the regular plane tilings whereas
the 5 fold symmetry is missing! It turns out that this symmetry leads to an
entirely different way of tiling a plane – an irregular but not less
fascinating (and beautiful!) tiling. For many years it was believed that a set
of tiles which tile the plane totally non-periodically could not exist. In 1973, Roger Penrose of the University of Oxford
came up with a set of 6 tiles that would force non-periodic tiling. He has
later reduced the number of tiles to only two, now known as "kite"
and "dart". The Penrose tilings have
five-fold symmetries but unlike the regular tilings there
are infinitely many different patterns. We shall make an attempt to create our
own patterns and investigate how the famous golden section ratio is related in
various ways to the irregular tilings with 5-fold
symmetries.