I have a question:

If one has a sufficient supply of Penrose tile and starts from a nice five-symmetric start and then continues putting down tiles outwards in a spiral fashion, is it relatively easy/likely to continue without problems or would it be very likely to run into contradictions?

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    $\begingroup$ It depends on the particular tiles you're using (there are three or four 'standard' sets of Penrose tiles). If the tiles have their local matching rules then it's not possible to come to a contradiction. If not, then sure: there are plenty of pattern-inadmissible configurations that can nevertheless be physically placed, such as with the thin and fat rhombs tiles or the Robinson triangles or the kite and darts $\endgroup$ – Dan Rust Dec 3 '17 at 2:36
  • $\begingroup$ If someone writes a complementary second answer (e.g. relating the current answer to this comment) I promise to give both answers a full bounty. $\endgroup$ – Phira Jun 21 '18 at 6:48
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    $\begingroup$ I don't know if we are understanding the same for 'local matching rules'. If we do then I disagree with the previous comment. There are many local configurations of Penrose tiles that obbey the local matching rules but whose forced tiles are contradictory. I'll edit my answer with one of those. $\endgroup$ – Melquíades Ochoa Jun 22 '18 at 18:13

You are very likely to run into contradictions.

This happens because the Penrose empires (the tiles forced by any configuration of Penrose tiles) are disconnected and not local. This means that every initial configuration would force you to place some tiles not neighbouring any of the already placed tiles.

One way of seeing this is through the so-called Ammann bars. These are families of parallel lines in the directions of a pentagon whose types of intersections enforce the placement and orientation of a tile in a Penrose tiling.

enter image description here

You can find out more about Penrose empires and Ammann bars in Grumbaum & Shephard's book, and there's also a thesis and a software devoted to them (from which I took the screenshot).


There are many configurations of Penrose tiles that obey their local matching rules but their forced tiles are contradictory. For example the following configuration

enter image description here

would enforce enter image description here

which is contradictory.


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