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I created this page because I feel it is necessary to have an article that gives a good overview over the aperiodic sets of tiles. This article still needs a lot of work, since I would like to iclude also a brief description of each set, rather than having only a plain list of links. In this way it can become a useful list that also gives a first, brief description of each set. Help on this project is very welcome. Toshio Yamaguchi (talk) 12:48, 28 July 2010 (UTC)[reply]

Here's a good start on more planar tiles:

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(This is more or less an email on the tiling list serve from 2007, just on planar tiles; a couple of new examples have been added and no effort has been made to cross check against the article. Also, no particular claim of completeness!!)

1) Hierarchical tilings; in each of these, one proves each individual tile (perhaps of a certain kind that is guaranteed to appear in every tiling) lies in some sort of non-periodic infinite hierarchy of structure:


Berger ca 1966, [20426] tiles, but reduced to [104]; Knuth further brings this down to [92], described in "Tilings and Patterns"   Läuchli [40], Robinson [52]/7 then [32]/6, Socolar X[X],  Goodman-Strauss [8]/2 give constructions based on a hierarchy of squares or L-tiles.  These are not all m.l.d. but there is a set of tiles admitting only tilings that can be locally decomposed into tilings by any of these.

Penrose P1: 6 or 5, P2:2, P3: 2; is P3 due to Ammann-Conway?, Ammann 3 (unmarked convex tiles), etc etc.

Mackay? 1975?

Ammann A1 6 [18?]

Ammann A2 2 [8]

Ammann A3 3 [12]

Ammann A4 2 [16]

Ammann A5 2 [24]

Socolar 3 [144] (twelve-fold symmetries)

Danzer 3 [168] (seven-fold symmetries)

Mozes 1988 gave a very general method for constructing hierarchical tilings based on the product of subst sequences. 

Radin 1995  large #, [infinity] (Matching rules for the Conway Pinwheel; first example that showed there really is a distinction between tiles that are translated only and those for which we allow rotations)

Goodman-Strauss 1998 generalized more or less completely; essentially all subst tiling species give an aperiodic set of tiles. However the sets of tiles tend to be fairly large. Fernique and Ollinger (2010) have a similar, much simpler construction that is more directly related to Mozes' 1988 method.

Penrose (hexagonal, 1995) 3 [72?], Socolar (1995) 12?, slightly modified by GS, 3. Socolar-Taylor have now a very similar tile with next-nearest-neighbor rules, that by itself is aperiodic.


2) Cut and Project structure: show the tiles must recreate the projection of  a plane slicing through a higher dimensional lattice ``irrationally". 

Begun by De Bruijn and definitively settled by Le TTQ in 1995. Were any small sets of tiles found that were of this kind, that were not hierarchical?


3)  Others

Mozes 1992 ?? (In Symbolic Dynamics and its Applications)

Kari [14], Kari-Culik [13, possibly 12?]

Hanf non-constructively proves the existence of tiles admitting only non-recursive tilings of the plane. 

130.184.198.164 (talk) 13:46, 2 August 2010 (UTC)[reply]

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are you aware of the substituiton tiling (2d) list of Dirk Frettlöh and Edmund Harris? http://tilings.math.uni-bielefeld.de/ —Preceding unsigned comment added by 141.44.225.35 (talk) 21:56, 2 August 2010 (UTC)[reply]

Yes, I am familiar with the Tilings Encyclopedia. One issue with it is that it does not explicitly give the aperiodic tile sets from which the tilings are generated. As Chaim Goodman-Strauss has already said in his papers and also on Wikipedia, the interest lies in the sets of tiles generating these tilings. Furthermore, there is no single nonperiodic tiling. Every aperiodic set of tiles admits, in fact, infinitely many distinct nonperiodic tilings. The tilings Encyclopedia does not mention the aperiodic set generating most of these tilings, they only mention the substitution rule. However, as Goodman-Strauss has noticed, for all substitution tilings there exists an aperiodic set of tiles that can produce the tilings obtained from the substitution rule. Thus it would be interesting to find the aperiodic set of tiles and the matching rules for every substitution on the Tilings Encyclopedia. Unfortunately I do not know how to transform each substitution tiling mentioned there into an aperiodic set of tiles. It would be a nice task to incorporate all the aperiodic tile sets for each substitution tiling there into this article. Toshio Yamaguchi (talk) 18:21, 3 August 2010 (UTC)[reply]

To answer your question about the Penrose P3 set: "Is P3 due to Ammann-Conway?" The answer is no: P2 and P3 both appear in U.S. patent #4133152 for which application was filed in 1976 by Penrose alone. - Ed Jeffery (talk) 19:42, 15 April 2013 (UTC)[reply]

Hard stuff

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I found this [1] which sounds interesting but which I do not really understand yet. Toshio Yamaguchi (talk) 16:14, 9 September 2010 (UTC)[reply]

Possibly worthy of addition

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Saving here so that myself or someone else can take a closer look later: http://arxiv.org/abs/1410.0592 ∴ ZX95 [discuss] 14:29, 7 October 2014 (UTC)[reply]

Wrong caption in first image

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The first image caption claims that the highlighted hexagon is a primitive cell. But I don't think that's true. A primitive cell is a smallest unit of translation symmetry, which in this case could be a parallelogram whose vertices are at the centers of four of the blue dodecagons. The hexagon is too big by a factor of three. —David Eppstein (talk) 08:17, 21 November 2014 (UTC)[reply]

I agree that the caption is wrong. Should the caption be corrected or should the figure be corrected? Eigenbra (talk) 19:48, 21 November 2014 (UTC)[reply]
The figure, I think, if it's easy enough to do. —David Eppstein (talk) 21:44, 21 November 2014 (UTC)[reply]
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Girih tiles

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Could someone please clarify the Mathematics of girih tilings? HLHJ (talk) 16:00, 18 August 2016 (UTC)[reply]

Interesting! As for the claim cited in that article is that "they precede Penrose by five centuries": to me, this misses the point that Penrose found matching rules and proved that his set of tiles could only tile the plane aperiodically. Penrose himself acknowledged that the tiles were from Kepler (four centuries earlier). You might claim that they "precede Kepler", though. --2607:FEA8:86DC:B0C0:7020:1DDB:EE47:C059 (talk) 02:36, 14 December 2021 (UTC)[reply]
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Figure changed

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Just to explain why the figure for the "trilobite/cross" example had to be changed: The black/white coloring of the cross tile must be different from the trilobite tile, as can be seen in the cited paper by Goodman-Strauss.--Nessaalk (talk) 10:45, 24 March 2021 (UTC) and 14:57, 18 March 2021 (UTC)[reply]

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