Symmetries of the plane: regular and irregular tilings
Symmetries of the plane: regular and irregular tilings |
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Summer Institute for Mathematics at the University of Washington 2008
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| Algebra is nothing but written geometry; | L'algèbre n'est qu'une géométrie écrite; |
| Geometry is nothing but pictured algebra. | la géométrie n'est qu'une algèbre figurée. |
| Sophie Germain |
We shall look at both periodic and non-periodic wallpaper patterns and make a transition from a very geometric notion of symmetry of two dimensional objects to a very 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).
It is claimed that all regular wallpaper patterns can be found among Escher's extensive collection of symmetry prints, whereas the question of whether all patters can be found on the wall of the Alhambra palace is more controversial. Only 2-, 3-, 4- and 6-fold symmetries can be found in the regular plane tilings. The 5-fold symmetry is mysteriously missing. It turns out that this symmetry can lead to an entirely different though no less fascinating way of tiling a plane - an irregular tiling. For many years, it was believed that a set of tiles that tiled only 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.back to SIMUW main page