C. Radin, The pinwheel tilings of the plane, Ann. of Math. 139 (1994) 661-702.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....the skeleton of any normal partition of the plane without 3 , 4 and 5 gons, is embeddable. The Robinson subdivision of above Penrose zototopal tiling and another tiling by golden triangles (see both on Figure 2 [BaSc95] are not 5 gonal, as well as a pinwheel tiling by 1 2 p 5 right triangles([Ra94]) and ( We91] pages 104, 89) a D urer tiling (by regular pentagones and rhombi) and the tiling by special pentaminos, Greek crosses (combinatorially equivalent to (4 4 ) with two new vertices on each edge) Above pinwheel tiling is not edge to edge, but we consider it as edge to edge tiling by ....

C.Radin, The pinwheel tiling of the plane, Annals of Math. (2) 139 (1994) 661-702.

....The techniques used in this paper do not seem to apply to irrational repsets. This raises a question: are there any irrational reptiles Our numerical calculations indicate that there are no irrational 2 reptiles. The answer is less clear for n reptiles where n 2. Conway s Pinwheel Tiling (see [R]) involves rotations by some irrational multiples of #, but it also involves reflections, which are prohibited in our setting. We conjecture: Conjecture 1.5 All reptiles are rational. The rest of this paper is organized as follows: In 2, we establish several preliminary results on self a#ne ....

C. Radin, The pinwheel tiling of the plane, Ann. of Math. 139 (1994), 661-702.

....2) this contradicts (7) 6 ) i.e. d o u b l y periodic. 7 ) It can be shown, that for inflation species satisfying some rather general conditions there is always such a radius. 5Colouring does help (Theorem 2) 10 5 Colouring does help 5. 1 A coloured inflation species After the papers [7] by Ch. Radin and [12] by Ch. Goodman Strauss the next to do is, to find a colouring of S(F; infl) that does permit a local matching rule. In principle it would be enough to colour every member of our species and then find a radius ae such that the atlas A of all coloured ae clusters (used as ....

Charles Radin. The pinwheel tiling of the plane. Annals of Math., 139:661--702, 1994.

....) has Sym(X) f1g. If we choose ae so that NX (ae) k, then (4.11) implies that ae grows at least proportionally to p k as k 1. For other properties of Penrose tilings, see [4] 5] 13] 21] 22] and [25] Example 4.3. Pinwheel Tilings) The pinwheel tilings of the plane studied in Radin [20] have the locally finite atlas property under isometries, but do not have the locally finite atlas property under translations. We obtain a Delone set X from the Conway tesselation of the plane described in x2 of [20] and [23, Sect. 7.4] by choosing a fixed point in each prototile. It can be ....

....4.3. Pinwheel Tilings) The pinwheel tilings of the plane studied in Radin [20] have the locally finite atlas property under isometries, but do not have the locally finite atlas property under translations. We obtain a Delone set X from the Conway tesselation of the plane described in x2 of [20] and [23, Sect. 7.4] by choosing a fixed point in each prototile. It can be proved that NX (ae) is finite, with NX (ae) O(ae 2 ) Radin s results imply that N X (ae) 1 for ae 4R. Acknowledgments. We are grateful to the Fields Institute of the University of Toronto for its hospitality ....

C. Radin, The pinwheel tilings of the plane, Ann. Math. 139 (1994), 661--702.

....equal to inflated prototiles. However, it easily extends to e.g. the kite and dart Penrose tiling, since it is mutually locally derivable with a self similar Penrose tiling having triangular tiles. 5. Since we assume translational finiteness, tilings such as the Conway Radin pinwheel tiling [Ra1] are excluded from consideration; in fact, our methods do not seem to work in that case. The statement (non periodicity implies unique composition) might still be true though. Next we discuss the more general case, when the tiling may have non trivial periods. Definition. Let K(T ) fx 2 R d : ....

C. Radin, The pinwheel tiling of the plane, Annals of Math. 139 (1994), 661-702.

....[24] which proves that perfect local rules under isometries exist for nearly all substitution tilings. There do exist Delone sets which have perfect local rules under isometries, but do not have perfect local rules under translations. One such example comes from the pinwheel tilings of Radin [52]. Another example comes from the vertices of the SCD tiles in Danzer [12] for SCD(m=n; s; h) with n odd, using fact (13) in Danzer [12] The quaquaversal tilings of Conway and Radin [9] can be suitably decorated to yield a Delone set with perfect local rules under isometries, using the result of ....

C. Radin, The Pinwheel Tilings of the Plane, Ann. Math. 139 (1994), 661--702.

....tiling dynamical system. Our goal is to study ergodic properties of such systems. Tiling dynamical systems in a different setting were considered by Rudolph [Rud] The investigation of tiling dynamical systems as actions of various groups of rigid motions, including R 2 , was initiated by Radin [RW, Rad1, Rad2, Rad3, BR] and Robinson, Jr. Ro1] There is an interesting aspect which we don t touch upon in this paper: connection with finite type systems in two dimensions. For this development see [Moz, Rad1] Methods of the theory of word substitutions play an important role in the papers of Radin [Rad1, Rad2] ....

....4.1. If T is a self affine tiling, then the R d action (X T ; Gamma g ) is not mixing. Remark. This is analogous to the theorem of Dekking and Keane [DK] from substitution dynamics. It is an open problem to find a mixing tiling dynamical system. Radin conjectured that the pinwheel tiling [Rad3] gives rise to such a system. The pinwheel tiling has many common features with our self similar tilings, but there is a crucial distinction: the set of tiles is finite up to translations and rotations but not up to translations only. Proof. Consider the set of translation vectors between T ....

C. Radin, The pinwheel tiling of the plane, Annals of Math. 139 (1994), 661-702.

....and the number of orientations is infinite. A piece of a tiling with two sizes and an infinite number of orientations is shown in Figure 1. Figure 1. Part of the tiling T il(1=2) These tilings all arise from a substitution scheme that is quite similar to the pinwheel tiling of Conway and Radin [R1]. In all cases the tilings have the sibling edge to edge property, which is to say that two daughter tiles of a single parent tile can only meet full edge to full edge. However, in general the tiling need not be globally edge to edge. Tiles that are not siblings generally do meet in ways that ....

.... b=2. Finally, if b = 2a, then all five triangles in T 1 have the same size. All five are subdivided in the next stage, yielding T 2 with 25 congruent triangles, all of which are then subdivided to give T 3 with 125 congruent triangles, and so on. This is the pinwheel tiling of Conway and Radin [R1]. We will show that, for any angle , this subdivision scheme generates nonperiodic tilings of the plane. We first need three technical lemmas: Lemma 1: In T n , the ratio of the hypotenuse of the largest tile to the hypotenuse of the smallest tile is strictly less than max(c=a; 2c=b) Proof: ....

C. Radin, The pinwheel tilings of the plane, Annals of Math. 139 (1994), 661--702.

No context found.

C. Radin, The pinwheel tilings of the plane, Ann. of Math. 139 (1994) 661-702.

No context found.

C. Radin, The pinwheel tilings of the plane, Annals of Math. 139 (1994), 661-702.

No context found.

C. Radin, The pinwheel tilings of the plane, Annals of Math. 139 (1994), 661-702.

No context found.

C. Radin, The pinwheel tilings of the plane, Annals of Math. 139 (1994), 661-702.

....symmetry and statistical 10 fold symmetry of the kite dart tilings) Consider again the pinwheel tiling of Fig. 13. The substitution version of the pinwheel is made from a 1; 2 ; p 5 right triangle by the substitution function of Fig. 30. There is also a finite type version of the pinwheel [Ra2]. We want to focus first on the rotational symmetries of these tilings, and this requires overcoming a technical problem. For the kite dart tilings X k d we used the simplification that X k d decomposes into the subsets X ff k d . Technically this was advantageous since the sets S(P) could ....

C. Radin, The pinwheel tilings of the plane, Annals of Math. 139 (1994), 661-702.

No context found.

C. Radin, The pinwheel tilings of the plane, Ann. of Math. 139 (1994), 661-702.

....1 2 3 4 3 1 4 2 2 3 4 1 Figure 5. The Penrose substitution 7 Figure 6. A pinwheel tiling 8 Figure 7. The substitution for pinwheel tilings Figure 8. The substitution for (2,3) pinwheel tilings 9 stretching factor is # 2 = 3 # 5) 2. A pinwheel tiling of the plane, Fig. 6 [Ra1], has tiles that appear in an infinite number of orientations. The two basic tiles, a 1 2 # 5 right triangle and its mirror image, are shown in Fig. 7, with their substitution rule. Notice that at the center of a tile of level 1 there is a tile of level 0 similar to the level 1 tile but rotated ....

C. Radin, The pinwheel tilings of the plane, Annals of Math. 139 (1994), 661-702.

....d actions. Mozes proof can probably be extended without major difficulty to subshifts in higher dimensions and to tiling systems in which each polyhedron only appears, in any single tiling, in only finitely many orientations. However, more general tiling systems, such as the pinwheel in R 2 [Ra1] and quaquaversal tilings in R 3 [CoR] require a significant extension of the proof in [Moz] This was done for the pinwheel in [Ra1] and then in general in [GoS] In each case a measureable conjugacy is constructed, but the map is only continuous after exclusion of certain sets of measure ....

....which each polyhedron only appears, in any single tiling, in only finitely many orientations. However, more general tiling systems, such as the pinwheel in R 2 [Ra1] and quaquaversal tilings in R 3 [CoR] require a significant extension of the proof in [Moz] This was done for the pinwheel in [Ra1] and then in general in [GoS] In each case a measureable conjugacy is constructed, but the map is only continuous after exclusion of certain sets of measure zero. We demonstrate, as with subshifts, that these theorems cannot be strengthened to the topological category: there exist substitution ....

C. Radin, The pinwheel tilings of the plane, Ann. of Math. 139 (1994) 661-702.

No context found.

C. Radin, The pinwheel tilings of the plane, Ann. of Math. 139 (1994), 661-702.

....continuous spectrum. We do not solve this problem, but we do make a significant generalization of classical substitution dynamics and give evidence of its usefulness in our problem. Our basic problem has origins in several directions. For a review see [7] and for more recent developments see [8,9,10,11,1]. One historical root of our problem was the search for tiling dynamical systems which have no closed orbits. The original examples were obtained by use of some form of hierarchical structure; we will follow this path. So given a tiling dynamical system, we assume further that we are given a ....

....radius of A[m] is strictly less than that of A[0] and so 7) holds. Getting back to 6) we therefore define Z j = ffl=2 : And from the above analysis it is easy to see that 6) holds for large N , which completes the proof. Example. As an example we consider John Conway s pinwheel system [9] defined as follows. There are two prototiles, a right triangle with legs 1; 2; 5 1=2 and its reflection; two iterations of the substitution rule are given in Figure 1, and a portion of a tiling is given in Figure 2. Fig. 1. Two iterations of the substitution rule. Figure 4 Fig. 2. Part of a ....

[Article contains additional citation context not shown here]

Radin, C., The pinwheel tilings of the plane, Ann. of Math. (2), to appear.

....in fact with orientations which are uniformly distributed in SO(3) Furthermore, the number of such orientations which occur in a sphere of volume N in a tiling grows polynomially in N . As we shall see, only logarithmic growth is possible in analogous 2 dimensional tilings, such as the pinwheel [Ra1, Ra2], due to the commutativity of rotations in 2 dimensions. We also discuss two local properties of the tilings, concerned with the neighborhoods of tiles. II. The quaquaversal tilings Consider the triangular prism ( tile ) made from a 1; p 3; 2 right triangle, with depth 1. We want to define a ....

....red tile is made similarly, but with the other choice. We then make sandwich tilings by repeated deflation expansion, starting with some tile. We note that any such tiling will consist of parallel red and black layers. By definition the colors appear in a Morse sequence [Que] Now it is known [Ra1] that in each red layer the tiles appear in infinitely many orientations. It is easy to see that in each black layer the tiles appear in only four orientations. Therefore wherever black and red layers meet the tiles must abut in infinitely many ways. This proves the following. Proposition 2. In a ....

Radin, C.: The pinwheel tilings of the plane. Annals of Math. 139, 661-702 (1994)

....not conjugate (or c equivalent) then x and x 0 cannot belong to equivalent substitution tiling systems. Before defining substitution tiling systems in general, we present an example. Hopefully, the general definitions will be clearer with this example in mind. The pinwheel tiling of the plane [Ra1] is made as follows. Consider the triangles of Fig. 1. Divide one of them into five small triangles as in Fig. 2 and expand the figure about the origin by a linear factor of p 5, producing 5 triangles congruent to the originals. Figure 1. Two pinwheel tiles Figure 2. The substitution for ....

C. Radin, The pinwheel tilings of the plane, Annals of Math. 139 (1994), 661-702.

....of different tiles finite we allow rotation of the tiles when making the tilings. This was unnecessary for the Berger or Penrose tilings, since we were careful to include in B each tile in each orientation needed. We now digress to examine this detail. The pinwheel tilings came about as follows [Ra1]. In 1990 I noticed that in all the known aperiodic tiling examples the tiles only appeared in finitely many orientations. In fact, most of the examples were produced using a technique due to N.G. de Bruijn [Bru] based on projection of a lattice in a high dimensional Euclidean space, which ....

....(1=2) is irrational with respect to , which implies the number of orientations. But in order to be useful as a model for matter it was necessary to have an example which could only tile that way, a property certainly not shared by these 2 triangles That was much harder, and was solved in [Ra1]. The point I want to make is that producing matching rules (edge bumps, or edge colors) for this iterative example was quite difficult it takes about 30 pages in [Ra1] even to define them but I felt I had good enough reason to think they existed to justify the effort. This was based on ....

[Article contains additional citation context not shown here]

C. Radin, The pinwheel tilings of the plane, Annals of Math. 139 (1994), 661-702.

.... w: Therefore (ffi n t; r) m1 t 00 = ffi n t; r) m2 t 00 , which implies that t 1 = 0. In Theorem 2 the key assumption is the map D, which is a nontrivial representation on X of a similarity of E d ; the canonical illustrative example is John Conway s pinwheel tilings [10,11], defined as follows. The alphabet A consists of the two triangles in Figure 1. The substitution rule for A consists of the dilation D about the origin of E 2 by the factor ffi = 1= p 5, and a set fC jk : 1 j 2; 1 k 5g of elements of G, such that for each P j 2 A we have: P j = k ....

C. Radin, `The pinwheel tilings of the plane' Annals of Math. (2), to appear.

....pinwheel system conjugate to the pinwheel substitution, although it cannot be obtained directly from words of fixed size by the above method. Furthermore, the technique used by Mozes for square ish tiles has also proved to be too simple. Next we will sketch a method which works for the pinwheel [Rad4]. First an overview. The letters in the new alphabet A 0 come in two families, one for each of the two triangles in the original A. The letters in each family can be thought of as perturbed versions of the original triangle, that is, the original triangle with some pattern of bumps and dents on ....

....as perturbed versions of the original triangle, that is, the original triangle with some pattern of bumps and dents on each edge. But this is not the best way to think of them; it is preferable to view each letter as a triangle with certain information associated with each vertex. It is proven in [Rad4] that the information can transformed to bumps and dents. The finite type system is then defined by requiring that the triangles may abut if and only if the information associated with vertices of abutting triangles is consistent in a certain precise sense. Roughly speaking, the information can ....

[Article contains additional citation context not shown here]

C. Radin, The pinwheel tilings of the plane, Annals of Math. 139 (1994), 661-702.

....a tiling dynamical system . Figure 4. The alphabet for pinwheel tilings: 2 tiles Figure 5. Part of a pinwheel tiling We are interested primarily in certain closed subcollections X OE ae X(A) which we call substitution tiling systems. Two planar examples relevant to this paper are the pinwheel [Ra1] and chair [Sen] systems. The chair tilings were discussed in the introduction; see Figs. 4,5 for the pinwheel alphabet A p and a portion of a pinwheel tiling. To define these systems we first need the notion of patches . A patch is a (finite or infinite) subset of an element x 2 X(A) the set of ....

....substitution tiling systems X OE ae X(A) have been of interest primarily when there are alphabets A 0 such that X OE X(A 0 ) for some notion of isomorphism) as is the case for the pinwheel and chair systems. For a discussion of the alphabets A 0 for the pinwheels and chairs see [Ra1] and [Sen] This aspect of substitution tilings has only indirect relevance to this paper, although it is partial justification for our notion of isomorphism. Figure 8. The substitution for pinwheel variant tilings We are interested in classification of substitution systems. Basically, we view ....

C. Radin, The pinwheel tilings of the plane, Annals of Math. 139 (1994), 661-702.

No context found.

Charles Radin. The pinwheel tilings of the plane. The Annals of Mathematics, 139(3):661--702, 1994.

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC