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The recent question about Wang tiles has led me to think that creating Penrose tilings might be an interesting popularity contest.

Background

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?

The tiles

Here are two different pairs of tiles, both known as Penrose tiles:

Kites and darts enter image description here

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 goal

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.

Rules:

  1. 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)
  2. 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.
  3. You must fill a rectangular region with at least 20 tiles
  4. 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:

Tiling

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!

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  • \$\begingroup\$ The third tiling you mention is The original pentagonal Penrose tiling (P1), correct? \$\endgroup\$ – Calvin's Hobbies Aug 5 '14 at 23:55
  • \$\begingroup\$ Yes. I omitted it in the writeup because it's not as concise, but it's fine to use for purposes of the challenge. I've edited the text to be more explicit. \$\endgroup\$ – Eric Tressler Aug 6 '14 at 0:09
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    \$\begingroup\$ Even if there were programming-related criteria to guide voting, I doubt that many people would take them into account; but as is this is certain to work out in practice as an art contest rather than a programming contest, and so doesn't belong here. \$\endgroup\$ – Peter Taylor Aug 6 '14 at 8:24
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    \$\begingroup\$ Let's see, I think it is already a challenge to make a program that outputs that tiling. \$\endgroup\$ – flawr Aug 6 '14 at 11:29