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Activity 1-5: Kites and Darts
A tessellation is called periodic if you can lift it up and shift it so that it sits exactly on top of itself again.
Sometimes a periodic tiling can be tweaked so that it becomes non-periodic. Exactly the same tiles, but non-periodic this time.
The question arises – is there a tile or a set of tiles so that EVERY infinite tiling of the plane they make is non-periodic? Roger Penrose came up with two tiles that fit this criteria in He called them the Kite and the Dart.
Of course, kites and darts can be used on their own and together to generate periodic tilings. But… the matching rules for the tiles stop these tilings from counting.
The matching rule is this: the tiles can only be placed with the Hs together and the Ts together.
The only ways that tiles can legally meet at a point are as follows: The H-T rule can be enforced using red and green lines as above, or by using bumps and dents on the tiles. Some of these configurations force other tiles around them. Star Ace Sun Jack King QueenDeuce
Task: have a play with some Kites and Darts, and get a feel for how they tile together. A sheet of tiles to cut up can be found here.here
With these matching rules, it turns out that every infinite tiling that these tiles make is non-periodic. Task: find each of the seven ways that tiles can meet at a point in this tiling. Note: every point in the diagram is in an ace.
Note too that every point in the tiling is in a cartwheel shape.
Sometimes Kite and Dart tilings demonstrate striking 5-fold and 10-fold symmetry. The red shape at the centre here is called Batman.
There are many remarkable facts about Kite and Dart tilings. There are an infinite number of them, and they are always non-periodic. In any infinite Kite and Dart tiling, the ratio of Kites to Darts is to 1, where is the Golden Ratio. You notice in this tiling it has been possible to colour the tiles with only three colours so that no two tiles of the same colour share an edge. Is this possible in any Kite and Dart tiling?
Notice how the Darts hold hands in this tiling (and every tiling) to form rings.
You can inflate or deflate any Kite and Dart tiling to give another Kite and Dart tiling with bigger or smaller tiles. This shows that the Penrose tiling has a scaling self-similarity, and so can be thought of as a fractal. Wikipedia To deflate, add these lines on the left to every Kite and Dart in your tiling. Your tiles will get smaller, but they will all remain Kites and Darts!
How do we inflate a tiling? Cut every dart in half, and then glue together all the short edges of the original pieces. Inflate
One consequence of the inflation/deflation property is that any finite Kite and Dart tiling must appear in any infinite Kite and Dart tiling. We can prove now that every Kite and Dart tiling is non-periodic. Suppose we have a periodic such infinite tiling, with translation vector s. Inflate or deflate twice, and you get back to the tiling you started with (scaled differently). Now simply inflate the diagram until s lies within a single tile. Now clearly periodicity is impossible. All inflations and deflations of the tiling must also be periodic period s.
If we start with either the Star (left) or the Sun (right) and insist on perfect five-fold symmetry, then every tile is forced as above... If we inflate or deflate one of these tilings, we get the other.
There are other pairs of shapes that always give non-periodic tilings too. This picture shows Roger Penrose on a tiled floor at Texas A&M university, showing an non-periodic tessellation employing two rhombuses that he discovered after the Kite and Dart. One last question; is there a single tile that only tiles non-periodically?
With thanks to: Roger Penrose. Wikipedia, for a brilliant article on Penrose tilings. John Conway for his talk on Kites and Darts back in Carom is written by Jonny Griffiths,