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In Matching Rules and Substitution Tilings Sketch of the techniques in \Matching rules and substitution tilings" Chaim Go o dman-Strauss The fol lowing orginal ly appearedaspart of the introduction to \Matching rules and substitution tilings", soon to appear in the Annals of Mathemat- ics. Because of space limitations, this section was cut from the nal draft. Nontheless, it may be helpful in understanding the paper. In \Matching rules and substitution tilings", we set out to prove: d Theorem Every substitution tiling of E , d > 1,can be enforced with nite matching rules, subject to a mild condition: the tiles arerequired to admit a set of \hereditary edges" such that the sub- stitution tiling is \sibling-edge-to- edge". As an immediate corollary, in nite families of ap erio dic sets of tiles are constructed. The fundamental idea b ehind the construction is rather simple: in essence, we wish the tiles to organize themselves into larger and larger images of the in ated tiles these images are \sup ertiles". Each of these sup ertiles is to b e asso ciated with a small amount of information| its alleged p osition with resp ect to its parent sup ertile. This information resides on a \skeleton" of edges; these skeletons are designed so that every edge in the tiling b elongs to only nitely manyskeletons, and the skeletons of one generation of sup ertile connect to the skeletons of the previous generation. The pro of that the con- struction succeeds is by induction: if every structure is well-formed for one generation, we showevery structure must b e well-formed for the previous generation as well. Before continuing, we give a more detailed summary.We will de ne our terms precisely in Section 1, and in \Addressing and Substitution Tilings" In preparation. d \Tilings" are to b e coverings of d-dimensional Euclidean space E by d congruences of a nite set of \prototiles"| marked compact subsets of E . The images of the prototiles under congruences are to b e called \tiles"; we require the tiles in a tiling to have disjointinteriors. 1 0 0 A \matching rule tiling" M, T is the set of all tilings by prototiles T , that satisfy some lo cal rules M that sp ecify allowed b ounded con gurations. Given a set of prototiles T , a \substitution acting on the prototiles" is d an expanding linear map called an \in ation" or \similarity" on E , + such that for each prototile A, A is the union of a set A of \daughter" tiles with disjointinteriors. Thus may also b e thought of as an \in ate and sub divide" op eration on con gurations of tiles. We take care to require k that can b e iterated; any con guration congruent to some A, for some prototile A,we call a \sup ertile". The sp eci c substitutions are enco ded in a set S of images of the prototiles in the in ated prototiles. A \substitution tiling" T , ,S is the set of tilings by \p olyhedral" pro- totiles T such that any b ounded subset of the tiling app ears in some sup ertile given by the substitution de ned through and S . To de ne the \enforcement of substitution tiling by matching rules" we must de ne a \lab eling" of a substitution tiling; essentially, this formally allows one to mark information concerning the hierarchy on the sup ertiles. Then a matching rule tiling \enforces" a substitution tiling if and only if it repro duces this lab eling. We b egin with a given substitution tiling T , ,S . Weintend for the tiles to organize themselves into larger and larger sup ertiles |in ations of the original tiles| further and further up the hierarchy. Each n-level sup er- n1 tile congruent to, say, A, A 2T is to lie in a n + 1-level sup ertile n + congruentto A , where A = fB 2T j A 2 B g. The essential informa- tion asso ciated with each sup ertile is its own shap e and p osition in the next level of the hierarchy. Much of the construction given here is foreshadowed in \Tilings, substitu- tion systems and dynamical systems generated by them", in which S. Mozes gives matching rules enforcing substitution tilings in which the tiles are all rectangular blo cks. Mozes uses twokey observations: each sup ertile needs only to know its ancestry only a nite numb er of generations back, and each sup ertile should b e combinatorially active at only a few key sites. This information needs to b e consistent across the sup ertile, needs to b e manifest at a few key p oints on the b oundary of the sup ertile, and any 2 neighb orho o d in the tiling must contain only a nite amount of information. We can use the combinatorial structure of our addressing scheme as the basis for a \lab eling" of T , , S : this lab eling will enco de \skeletons" and \wires" to compare and transp ort information across sup ertiles. We will de- ne nite classes of \lab els"| combinatorial enco dings of these mechanisms; the lab eling will consist of marking the tiling with these lab els. We will de ne the elements of our lab eling in in Section 2. In Section 3 we derive tiles and matching rules from the lo cal structure of the lab eling, and in Section 4 show these force the hierarchy to organize. Because we de ne our structures on in ated prototiles they will b e avail- able, scaled up, on every sup ertile. Note we are selecting these structures. We do not assume the \nice" choices are b eing made. There is thus still ample ro om to nd elegant con- structions in more sp eci c cases. Each n-level sup ertile will consist of n 1-level sup ertiles held together byan n-level \skeleton", de ned in Section 2.1, of edges for the parent su- p ertile. The essential information de ning this sup ertile is conveyed in a \packet" of lab els along this skeleton. That is, our matching rules will as- sure that wehave identi ed each sup ertile's intended p osition with resp ect to its parent, and p erhaps with resp ect to a few recent ancestral sup ertiles. Askeleton will b e lo ose and oppy, a lo cally de ned top ological ob ject, combinatorial in nature, on top of which is enco ded information concern- ing the role of the sup ertile in the hierarchy. Sup ertiles are geometrically rigid, and combinatorially inert. Skeletons provide combinatorial cohesion; sup ertiles provide geometrical rigidity.Together they force the hierarchyto emerge. Matching rules at its vertices ensure the skeleton is formed correctly; matching rules at certain \sites" ensure that an n-level skeleton correctly meets its descendantn 1-level skeletons and its parentn + 1-level skele- ton. In the lower left of gure 1, the substitution for the pinwheel tiling \The pinwheel tilings of the plane", C. Radin is shown; ab ove and to the right 3 skeletons for three generations of sup ertile are shown. Note the sites, shown as half circles, connecting the skeletons of child to parent. Figure 1: Skeletons As a technical p oint, to ensure that skeletons are connected, that a su- p ertile's skeleton meets each of the sup ertile's children, and that sites can be chosen, we allowan n-level skeleton to enter lower level sup ertiles cf. gure 7. However, for any substitution tiling we nd a constant so that all n-level skeletons include only edges of level at least n and less than n.In fact, though, in most well known examples, we can take = 1. In this case, esp ecially when d = 2, the construction simpli es enormously. Thus skeletons mightoverlap, but only to a b ounded depth. Each n-level sup ertile, and skeleton, is asso ciated with its \address" X X :::X , relative to its n + 1-level ancestor. These digits are 1 1 in S . Each edge, vertex, and site in a skeleton carries the sup ertile's address and its own lab el relative to the skeleton, in classes de ned throughout Sec- tion 2. As an edge or vertex might b elong to manyskeletons, wemay enco de many such pairs; however the total information at any p oint in a tiling will b e b ounded. We need a sup ertile to \know" where certain vertices{\terminals"{ are; these vertices are endp oints of the sup ertile's parent's edges. That is, if the sup ertile is of level n, the terminals are endp oints of n-level edges in the 4 b oundary of the sup ertile. We can ho ok the terminals into the sup ertile's skeleton, if they meet lower level edges in the interior of the sup ertile such terminals are \endovertices". Alas, this is not often the case and wemust intro duce another device{ we link certain terminals \mesovertices" to the skeleton through a series of lower level sup ertiles. Such a series is a \ver- tex wire". A sup ertile maythus carry, for certain of its vertices, certain information asso ciated with some higher level sup ertile. In gure 2 vertex wires are shown for three vertices on the pinwheel proto- tile the vertex in the middle of the large edge is not really necessary in the actual enforcement of the pinwheel tiling but gives a more interesting vertex wire to illustrate.
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