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Tiling of the week 2: The Penrose tiling

These two tiles have a remarkable property. Just by fixing how they fit together locally they can force global behaviour. You can fit them together forever to cover an area as large as you want, yet whatever you make it will never be able to fit onto itself and just repeat. When we take this to infinity we form the Penrose tiling. This is very different to most tilings we think of, for example the squares on your bathroom wall.

These tiles were only discovered in the 1970's by Roger Penrose, there had been previous echos in the work of Kepler and in Islamic tilings. However neither of these captured exactly the same properties. Of course in the case of the Islamic craftsmen they had very different goals.

At the exhibit we will have these wooden tiles so you can have a go at trying to find patterns yourself!

Penrose tiles

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Comments

Thu, 06/18/2009 - 11:25 — Paul Halb (not verified)

I'm sure I've seen Penrose

I'm sure I've seen Penrose tiles before, but they were a different shape - do they always have those notches in them?

Sat, 06/20/2009 - 09:43 — exhibit0916

There are actually many

There are actually many different ways of making the Penrose tiles. You start with two rhombs but then have to restrict which edges can fit together. That is what the notches do. To complicate matters further there are two families of Penrose tiles, the Kite Darts:

http://tilings.math.uni-bielefeld.de/substitution_rules/penrose_kite_dart

and the rhombs:

http://tilings.math.uni-bielefeld.de/substitution_rules/penrose_rhomb

both have very similar properties and mathematically can be considered equivalent. However they produce subtly different images.

Edmund (www.mathematicians.org.uk/eoh)