Talk:Aperiodic tiling

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Mathematics grading:	B Class	Mid Importance	Field: Geometry and topology

How do we know that the assembly is necessarily non-local? Who showed this? --131.215.220.112 23:14, 3 August 2005 (UTC)

[edit] Help rewrite

I have attempted to rewrite the article, but nevertheless I believe that it still needs the attention of an expert and serious clean up. The old paragraphs have been incorporated in the text, even if I don't think an expert would agree.al 15:54, 17 January 2007 (UTC) Removed an addition which does not fit in 'Mathematical Considerations': a new section at the end suggests that it was a first case of practical use.al 17:55, 3 March 2007 (UTC)

It is getting better, but rewriting is a tricky business and now the introduction is rather awkward. Why the restriction 'in geometry'? A more general approach includes Meyer, Delone, and model sets but their mention has disappeared. (There are entries for Voronoi cell and Dual graphs). Substitution is a key concept but it is algebric.

The definition of aperiodic as non-periodic looks like a tautology.

An important point would be to distinguish between two approaches:

• aperiodic tilings as generated or produced by aperiodicity-enforcing tiles
• aperiodic tilings as illustrations of a more general idea

which are not exactly synonymous with 'local'and 'global'.

The definition of aperiodicity by a set of aperiodic-enforcing tiles is standard one, but I do not know how it works in one dimension e.g. for the tiling of the line by segments according to the Fibonacci word. In a Penrose tiling, generated by its decorated tiles, it is possible to introduce local 'defects' which do not affect its aperiodicity. al 21:49, 27 March 2007 (UTC)

The "in geometry" is simply to set the scene for the context in which the term "aperiodic tiling" is defined, as in WP:LEAD. The lead section should also try to be accessible. Making the distinction between nonperiodic and aperiodic clear in a way accessible to non-specialists is a tricky problem (but an important one to solve here so that readers do understand what the article is about), and Senechal points out (p169) that the terms are frequently misused even by those familiar with the distinction.

Following the standard terminology, you have the aperiodic tilings (by aperiodic protosets), and then the various constructions that may be used to generate or describe them, and related concepts; these constructions may also generate objects of interest but outside the scope of aperiodic tilings, and those and more general discussion of related concepts belong in other articles such as substitution tiling. This article needs to discuss the various ideas relating to aperiodic tilings; but further information about these ideas in more general contexts belongs in those other articles. More detail regarding Penrose tilings with defects belongs in Penrose tiling.

Some of the links to related concepts are present here, others may need to be added. I deliberately left the tag for expert attention, since it's still needed for those sections relating to other concepts. Joseph Myers 00:38, 28 March 2007 (UTC)

[edit] Terminology

The articles periodic function and substitution tiling both use aperiodic as synonymous with nonperiodic. A distinction, disregarded even by those who understand it, is probably inappropriate, pace Senechal.

As I read it:

• (1)any tiling by that set (..) is nonperiodic

('that set'= set of tiles said to be aperiodic)

• (2)A tiling by an aperiodic protoset is said to be an aperiodic tiling.

As it stands, we have to deal with: not periodic, nonperiodic, aperiodic and quasiperiodic. And we note that 'periodic' is originally temporal and hence one-dimensional.al 18:50, 28 March 2007 (UTC)

Well, we can't invent new terminology here, so all we can do is establish (here or at Wikipedia:WikiProject Mathematics/Conventions) which existing terminology to use in Wikipedia for which concepts, and fix pages not following that terminology, while making sure pages note that there is confusion and ambiguity in the terminology in practical use. I might quote Goodman-Strauss on the tilings mailing list today [1]:

This confusion is widespread, and I think it is incumbent on all that are interested in tilings to make a real effort to correct any conflation of the terms "aperiodic" and "non-periodic", at least when discussing tilings.

to illustrate the established understanding that these terms have particular meanings in the field of tiling but are nevertheless sometimes confused. Joseph Myers 19:55, 28 March 2007 (UTC)