The recent question about Wang tiles has led me to think that creating Penrose tilings might be an interesting popularity contest.
Wang tiles are tiles that can tile the plane, but only aperiodically (see Fill the Screen with Wang Tiles for a full description). Penrose tiles give the affirmative answer to a famous problem in mathematics: is it possible to have this property with only two tiles?
Here are two different pairs of tiles, both known as Penrose tiles:
In the second pair of tiles, the thin rhombus has angles 36 and 144 degrees; the fat rhombus has angles of 72 and 108 degrees.
In either pair, the edges of the tiles must be matched up so that the colored circles continue across an edge between two tiles. The colored circles and nodules are there to enforce the edge-matching rules (do not draw the nodules). These matching rules ensure that the tiling created cannot be periodic.
The problem is to create a Penrose tiling and output it in some graphical form. The Wikipedia article about Penrose tiles gives some ideas for how to construct them.
- One of the two pairs of tiles above must be used, or else the third set of 3 Penrose tiles described in the Wikipedia page (The original Penrose tiling P1)
- You do not need to draw the nodules on the righthand set (they are usually omitted). I know this isn't much of a rule, but the pictures will be prettier if you don't draw the bumps.
- You must fill a rectangular region with at least 20 tiles
- As long as the output is recognizably a Penrose tiling, you can take any artistic license you want (e.g. only drawing the colored arcs)
I used the deflation method described on Wikipedia (in my opinion, the easiest method) to produce the following example:
Please include a picture with your submission. Rule 4 allows for a lot of license to be taken, so hopefully there are some interesting takes on the challenge!