Presentation on theme: "Activity 1-5: Kites and Darts"— Presentation transcript:
1Activity 1-5: Kites and Darts Activity 1-5: Kites and Darts
2A tessellation is called periodic if you can lift it up and shift it so that it sits exactly on top of itself again.
3Exactly the same tiles, but non-periodic this time. Sometimes a periodic tiling can be tweaked so that it becomes non-periodic.Exactly the same tiles, but non-periodic this time.
4The 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 withtwo tiles that fit this criteria in 1974.He called them ‘the Kite and the Dart’.
5Of course, kites and darts can be used on their own and together to generate periodic tilings.But… the matching rules for the tilesstop these tilings from counting.
6The matching rule is this: the tiles can only be placed with the Hs together and the Ts together.
7The only ways that tiles can legally meet at a point are as follows: StarSunAceKingQueenDeuceJackThe H-T rule can be enforced using red and green linesas above, or by using bumps and dents on the tiles.Some of these configurations ‘force’ other tiles around them.
8A sheet of tiles to cut up can be found here. 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.
9With these matching rules, it turns out that every infinite tiling that these tiles make is non-periodic.Task: find eachof the seven waysthat tilescan meetat a pointin this tiling.Note: every pointin the diagramis in an ace.
10Note too that every point in the tiling is in a cartwheel shape.
11The red shape at the centre here is called ‘Batman’. Sometimes Kite and Dart tilings demonstrate striking 5-fold and 10-fold symmetry.The red shapeat the centre hereis called ‘Batman’.
12There are many remarkable facts about Kite and Dart tilings. There are an infinitenumber of them,and they arealways 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 tilingit has been possibleto colour the tileswith only three coloursso that no two tilesof the same colourshare an edge.Is this possiblein any Kite and Dart tiling?
13Notice how the Darts ‘hold hands’ in this tiling (and every tiling) to form rings.
14You 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. WikipediaTo deflate,add these lineson the leftto everyKite andDart inyour tiling.Your tileswill get smaller,but they willall remain Kitesand Darts!
16How do we inflate a tiling? Cut every dart in half,and then glue together all the short edges of the original pieces.
17Inflate or deflate twice, and you get back to the tiling you started with (scaled differently).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 thatevery Kite and Dart tiling is non-periodic.Suppose we have a periodic such infinite tiling,with translation vector s.All inflations and deflations of the tilingmust also be periodic period s.Now simply inflate the diagram until s lies within a single tile.Now clearly periodicity is impossible.
18If 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.
19There are other pairs of shapes that always give non-periodic tilings too.This picture showsRoger Penroseon a tiled floorat Texas A&M university,showing annon-periodic tessellationemploying two rhombusesthat he discoveredafter the Kite and Dart.One last question; is there a single tile that only tiles non-periodically?
20Carom is written by Jonny Griffiths, email@example.com With thanks to: Roger Penrose. Wikipedia, for a brilliant article on Penrose tilings. John Conway for his talk on Kites and Darts back in 1979.Carom is written by Jonny Griffiths,