Talk:Substitution tiling

Request for improved diagrams
The diagrams of substitution tilings should highlight the presence of the initial prototile in the interior of the expanded tiling. Without this, the substitution tiling might not tile the entire plane. For instance, if the second substitution tiling mapped the prototile to its placement in the lower left of the expanded tiling, then the system would only tile the first quadrant of the plane.–Dan Hoeytalk 13:39, 16 June 2007 (UTC)

Notation, terms and correct definition / Discussion Dan Hoey - Dirk Frettl&ouml;h
I want to remove the tag 'This article does not cite any references or sources.', and add references. The once I find most appropriate are Kellendonk, Johannes; Putnam, Ian F. Tilings, $C\sp *$-algebras, and $K$-theory. Directions in mathematical quasicrystals, 177--206, CRM Monogr. Ser., 13, Amer. Math. Soc., Providence, RI, 2000 Frettloeh, D. Duality of model sets generated by substitutions. Rev. Roumaine Math. Pures Appl. 50 (2005),  no. 5-6, 619--639 (Do you have another suggestion?)

No, unless Gruenbaum and Shepard have something. I really haven't read that much. I generally tried to straighten out the article so it was explicit enough to express the covering condition (below), though perhaps I went too close to OR in doing so.

If I use the two sources above, I have to rechange the definition of a substitution tiling. It seems to be agreed now that this (my old one) is the appropriate one: 'Given a tile-substitution sigma, a tiling T consisting of copies of the prototiles T_1,T_2,...,T_m is called substitution tiling (for the tile substitution sigma), if each finite pattern in T is a copy of some pattern in sigma^k(T_i) (for some k,i).'

I agree that "copies" may be a better term than "placements". I think I got the "expanding map" from the copy I started with, and I agree that calling it a "tile substitution" may be a better term. I'm not sure the condition that Q be linear with positive eigenvalues is right--perhaps it must be a similarity map. Still, I think the language needs to be defined, so that when we speak of a "copy of a prototile" we can expect to extract the isometry that takes the prototile to its copy.

Why? 1. The theory of nonperiodic tilings is now enriched by methods of topology and dynamical systems. The definition above is compatible with the definition of a tiling space, or a tiling dynamical system, whenever the substitution is primitive.

This is very good. I take it that we now wish to consider substitution tilings that may not cover R^d (or not even live in a Euclidean space at all).

Something should be done to cover conditions that ensure that the tiling covers R^d, though. For instance, the limit tilings of the example diagrams might not cover R^2--they might cover a half-plane, a quadrant, or an octant. It is the condition that a finite pattern is a copy of a pattern in the interior of sigma^k(T_i) that I wanted to be able to express. It was certainly a mistake to restrict substitution tilings with that condition, but the condition is still important to present. I don't know how it is usually treated in substitution theory, but I expect you do, and I encourage you to present it.

2. Your definition may define some tiling T as a substitution tiling, but a translate T+x may not be one. For instance, the canonical square tiling T of the plane is, but T shifted by (sqrt(2),0) is not.

I do not believe that is true. If phi_0 were a translation by (sqrt(2),0), I think that would work. But it is more likely that I am missing your point.

Another point is, that now the definition is even longer and more technical than mine. I would prefer, in agreement with the references above, a simpler formulation.

I agree that I overcomplicated things. Frettloetalk 11:39, 8 Aug 2007 (GMT), Dan Hoeytalk 19:49, 9 Aug 2007 )

Definition of substitution tiling
I am completely happy with 'placements' and will use it from now on in my papers instead of 'copies'. Seems indeed clearer.

I'm not sure the condition that Q be linear with positive eigenvalues is right--perhaps it must be a similarity map.

First: ... all eigenvalues larger than one. Second: No, that's the point. Here is a tile substitution using an expanding map Q which stretches by a factor of approx 1.5 in the vertical direction, and by a factor of approx 1.9 in the horizontal direction: http://www.math.uni-bielefeld.de/baake/frettloe/sl-mar.pdf, page 13 This is not a similarity map, but it should be called a tile substitution. (in the literature, this is usually called self-affine substitution... in contrast to self-similar substitution...).

Something should be done to cover conditions that ensure that the tiling covers R^d, though. For instance, the limit tilings of the example diagrams might not cover R^2--they might cover a half-plane, a quadrant, or an octant. It is the condition that a finite pattern is a copy of a pattern in the interior of sigma^k(T_i) that I wanted to be able to express. It was certainly a mistake to restrict substitution tilings with that condition, but the condition is still important to present. I don't know how it is usually treated in substitution theory, but I expect you do, and I encourage you to present it.

I guess I can clarify this point. What you have in mind is what used to be somehow the definition of a substitution tiling, or better, of a self-similar tiling. The current definition:

Every tiling of {\mathbb R}^d, where any finite part of it is congruent to a subset of some \sigma(Ti) is called a substitution tiling (for the tile substitution (Q,\sigma)).

rules out slight difficulties, and is just shorter (maybe more elegant). First, it says 'A tiling of R^d', so one has not to bother whether the whole space is filled, or only part of it.

Frettloetalk 10 Aug 2007 15:19 (GMT)

ambiguous expression
I am trying to translate article to Russian.

I found, that expression which do not require the tiles to be geometrically rigid is ambiguous.

What exactly it means? Tiles inequality (but with the same shape)? Or something else? Jumpow (talk) 19:02, 14 January 2016 (UTC)

Is there any basic difference between this article and rep-tile?
It seems like this article is about a method, while rep-tile is about a type of shape. However, they seem to be intimately connected. Is there any essential difference between these two articles, or should we merge this article into the rep-tile article (which to me seems to be the more fundamental of the two)? —Kri (talk) 09:44, 20 January 2022 (UTC)