From: Chris Hillman Subject: Re:Quasicrystal, Penrose Tilings and A_4 Date: Tue, 2 May 2000 03:55:36 GMT Newsgroups: sci.physics.research Summary: [missing] Sigh... since there have been several questions about this topic (stupid of me to have raised the subject, really), I suppose I am honor-bound to attempt to sketch some answers, even though I'd much rather not. Kwok Man Hui asked if there are higher dimensional analogues of the Penrose tilings. Indeed, de Bruijn gave two completely equivalent constructions for the natural common generalization of Sturmian sequences and Penrose tilings to any dimension and codimension in a classic 1981 paper (the general construction is clearly implicit in this paper, and made explicit in a 1986 paper). In 1988, Oguey, Duneau and Katz gave a third completely equivalent construction. I won't attempt to describe any of these constructions in detail; I'll just try to give some intuition for how one of de Bruijn's constructions works. Start with any p-dimensional subspace W in R^d, where d = p+q, p,q positive integers. Recall that the quotient vector space R^d/W consists of cosets x+W, where each x+W is said to be a "p-flat" (p-dimensional affine subspace) which is "parallel" to W. Consider the p-skeleton of the integer lattice Z^d (that is, the p-dimensional cubes with vertices in Z^d). There is a natural concept of a "digital approximation" D(x+W) of x+W by p-dimensional cubes in the p-skeleton. Suppose P is the orthoprojection of R^d onto W. Then if we identify W with R^p, P(D(x+W)) = T(x+W) is a Sturmian tiling of R^p. The set of all tilings obtained in this way is S(W). In the generic case, P(Z^d) will be a countable dense subset of W, and the vertices of each tiling in S(W) will be drawn from this set, so S(W) is obviously not invariant under translations in R^p. However, there is a natural way of defining two tilings in S(W) to be "close" if, after a "small" translation, they agree perfectly on a "large" ball around the origin of R^p. Then, we can take the translation orbit of each tiling in S(W), take the topological closure of that orbit, and take the union of these to the (p,q) Sturmian tiling dynamical system T(W). This is a "topological dynamical system" because the translation action by R^p is continuous on T(W). There is also a very natural way to place invariant probability measure on T(W) and then it becomes a measure-theoretic dynamical system. The basic question I asked in my thesis was this: For fixed positive integers p,q, as we vary W over the Grassmannian G(p,q), how do the combinatorial, statistical, (*) geometrical, spectral, and dynamical properties of T(W) change? Examples of combinatorial properties, or rather questions: what protopatches can occur? (A patch is a compact simply connected configuration of nonoverlapping tiles; a protopatch is an orbit of a patch under translation.) If patch A occurs at location x, at what other locations can this patch occur? What protopatches [B] can "extend" the protopatch [A]? (A useful analogy: protopatches are like words, and the rules on how they can occur with other protopatches are the "grammar" of the Sturmian "language".) Examples of statistical questions: what are the frequencies with which the various protopatches occur in a given tiling? What are the frequencies with which various recurrence alternatives of a patch are chosen? Do these properties change from one tiling to another in T(W)? Examples of geometrical questions: what are the geometric shapes of the prototiles? Are there any tilings in T(W) which have a global geometric symmetry (e.g. under a 1/4 turn about the origin)? Examples of spectral questions: what is the nature of the diffraction spectrum of a tiling in S(W)? How does this relate to the "dynamical spectrum" of T(W)? Examples of dynamical questions: does T(W) contain only one orbit closure? Do the tilings in T(W) have any periods? Of course, it's not always easy to say if a particular property is "combinatorial" or "dynamical"! Example: in the generic case, every tiling in T(W) will be aperiodic (but quasiperiodic, that is, as close as possible to being periodic without actually having any periods, in a precise sense). When T(W) does have one or more periods, all the tilings in T(W) share each period. In this situation, typically T(W) will be periodic in a few "special directions" and quasiperiodic in all the other directions. The periods, if any, can be considered as algebraic properties of W, or dynamical properties of T(W). Example: every tiling T(0+W) contains a "singular patch" at the origin; that is, T(0+W) defines not one but several genuine tilings. By moving off this coset in just the right directions, you can systematically "unfold" this singular patch to obtain singular tilings with "simpler" singular patches, until eventually you have no singular patches at all. In the generic case, there will be at most one singular patch in any tiling in T(W) (and "most" tilings contain not even one), but for some choices of W, singular tilings feature a "large singularity" plus "smaller" singular patches which recur in quasiperiodic fashion along certain "directions" or even in certain "planes" (if p > 2). These recurrence properties can be considered as algebraic properties of W or dynamical properties of T(W), and they are closely connected to geometric symmetry properties of the tiling T(0+W) and indeed to "local geometric symmetries" in any other tiling in T(W). Example: can T(W) be defined by matching rules? This can be considered either a combinatorial or dynamical question, but the answer turns to be number-theoretic. (The generic answer is "no"). Example: does T(W) admit a composition arising from a d by d integer matrix acting on Z^d? This can be considered a combinatorial or dynamical question, but the answer is again number-theoretic. (The generic answer is "no".) The general answer to question (*) turns out to be, roughly speaking, that -all- these properties are completely determined by various algebraic/number-theoretical characterizations of how W is "situated" in R^d wrt Z^d. All properties undergo -simultaneous- "bifurcations" whenever W sweeps over a rational point W', or more generally, a point W'' which "satisfies one or more rational relations". These "special" points for an uncountable dense set in G(p,q), and the in particular G(1,q) is organized into a hierarchy of "nets" describing the various "levels of detail" in the "bifurcations". This should not seem surprising if you recall the construction of T(x+W) via a "digital approximation": by their very nature, Sturmian systems are in essence concrete geometrical realizations of problems in Diophantine approximation. In particular, when p = 1, G(1,q) is just real projective space RP^q and then the digital approximations D(x+W) are closely related to the notion of rational approximations of the line W, or in more conventional language, to the "simultaneous rational approximation of q real numbers". Example: when p = q = 1, G(1,1) = RP^1 (the real projective line) and the tilings in each S(W) are combinatorially the same as the classical Sturmian sequences from the corresponding "Sturmian shift". Here the "best rational approximations" may be enumerated using the simple continued fraction expansion of the slope of W. Example: when p = 1, q > 1, we have Sturmian tilings of R by d types of tiles, which can be identified with generalizations of Sturmian sequences to alphabets larger than {0,1}. Example: when p = 2, q = 3, consider the five cycle (permutation matrix) R acting on the standard unit basis vectors of R^5. This has a fixed line and two real invariant subspaces, in which it effects a 1/5 and 3/5 turn, respectively. Let W be the invariant subspace in which it effects a 1/5 turn. Then T(W) is a Sturmian tiling dynamical system which turns out to have a one real parameter family or orbit closures, one of which is precisely the space of Penrose tilings. Example: when p = q = 3, one can find a representation of the icosahedral group and choose W to be a three dimensional invariant subspace. This gives a (3,3) Sturmian system which may be regarded as the natural three-dimensional generalization of the Penrose tilings. There are two fundamental observations about Sturmian systems. First, Sturmian tilings are hierarchical: every (p,q) Sturmian tiling can be decomposed into a "layer" of (p-1,q) tilings, and these can be further decomposed into (p-2,q) tilings, and so forth, until you reach (1,q) tilings, the "ribbons" of the tiling. Second, there is a duality between properties of T(W) and properties of T(W*), where W* is the orthogonal complement of W. For example, the integer vectors (if any) in W* correspond to the rational relations (if any) of W, where n in Z^d is an integer relation on W if for all w in W, (n,w) = 0. It is easy to see that the integer vectors in W* correspond to the periods (common to any and all tilings in the space) of T(W*); it is a bit harder to see that the rational relations of W (if any) determine the nature of the recurrence of a singular patches in the singular tilings of T(W). I apologize if I gave John Baez, Kwok Man Hui, or Greg Egan the impression that the connection between root lattices and Sturmian tilings is deep: it isn't. The connection is pretty much what Greg Egan guessed: you can factor the orthoprojection P of R^d onto W through the orthoprojection on that rational closure rat(W), where rat(W) is the smallest rational subspace (one spanned by integer vectors) which contains W. In the generic case, rat(W) will be all of R^d but sometimes it is smaller, and in this situation, the orthoprojection of Z^d onto rat(W) will give a periodic lattice in rat(W). In particular, consider the (2,3) system T(W) for which one orbit closure is the space of Penrose tilings. W contains no integer vectors, so every tiling in T(W) is aperiodic. However, rat(W) is a four dimensional subspace of R^5, and the orthoprojection of Z^5 onto rat(W) is precisely the root lattice A_4. As I said earlier, this means that T(W) consists of a one parameter family of orbit closures (they all have distinct combinatorial and statistical properties from one another). The dual system T(W*) contains the period (1,1,1,1,1) noticed by Greg Egan, so the tilings in T(W*) are tilings of R^3 which are periodic in one direction and quasiperiodic in the "orthogonal layers". The single rational line in W* corresponds to the single rational relation in W: for every w = (w1, w2, w3, w4, w5) in W, we have w1 + w2 + w3 + w4 + w5 = 0 This means that the singular tilings in T(W) have recurrent singular patches. Indeed, the generic singular tiling contains (roughly speaking) a "line" of recurrent singular fat hexes, or else singular thin hexes. In the Penrose orbit closure, these two types are superimposed to give the "worms" of Conway. The tiling T(0+W) contains a central singular decagon with five worms piercing the decagon and arranged in a symmetrical way. Much of what I have said above should become much clearer if one first carefully studies the "translation action by W" on the torus T^d. Here, the orbit closures are precisely the cosets in T^d of "rat(W) mod Z^d". Gerard Westendorp asked about "Aman rhombi". I think he meant the five fat and five thin rhombic prototiles serve as the prototiles for each of the tilings in T(W), the (2,3) system one of whose orbit closures is the space of Penrose tilings. These are not to be confused with the prototiles (two squares and four rhombs) for the "Ammann octagonal tiling", aka "Ammann-Beenker tiling". (The "Ammann bars" are decorations of the -Penrose prototiles- which give an alternative set of matching rules for the Penrose orbit closure). The -small- interior angles for the thin and fat Penrose rhombs are pi/5 and 2pi/5 radians respectively (36 and 72 degrees, as Gardner stated). The "thin" angle for the Ammann tiling rhombic prototile is pi/4 (45 degrees). From this, you can figure out what the -big- interior angles are! I suspect that the "art shapes" created by Westendorp's friend do not represent patches of true quasiperiodic tilings, but rather aperiodic tilings more allied to what mathematicians would call "random tilings", which the artist input amounting to the choice of an "aesthetically pleasant" example of such a random tiling. To see why, note that, on the one hand, unmarked rhombs can always be used to tile the plane -periodically-, and on the other, most Sturmian tilings cannot be constructed using matching rules. I have already pointed out that physical quasicrystals (this term is usually taken to mean a substance whose diffraction spectrum shows Bragg peaks, but cannot be the the diffraction spectrum of any crystal) do not, for the most part, correspond to quasiperiodic tilings of E^3, although some alloys do correspond to aperiodic quasiperiodic tilings of E^3 (the "icosahedral" tilings) or to tilings with one period (stacks of aperiodic quasiperiodic tilings, including the Penrose tilings). Most physical quasicrystals seem to involve distributions of "crystal dislocations" which render their diffraction spectra similar to those of true quasiperiodic structures. I should stress the articles by Baake et al. found by Greg Egan (there are many more in the journals) and the article by Kellendonk do not discuss only Sturmian tilings. Indeed, only a few papers discuss -only- Sturmian tilings! On the other hand, for obvious reasons, most papers by mathematical physicists consider -only- tilings of R^2 or R^3. But as I have already observed, this is enough to already include mathematical structure capable of modeling anything which can even be discussed in "ordinary mathematics". I could go on and yak about how tiling theory fits into the grand scheme of things, but I've done that on other occasions, so I won't repeat myself. I'll just offer a brief summary of how -I- see the above results on -Sturmian systems- fit into the grand scheme of things. On the one hand, Sturmian systems clearly are a basic example (which can and do appear in many textbooks, although never in full generality) where rigorous long-range order coexists with the absence (in general) of any periods (periods are the "trivial enforcement mechanism" for such long range order!). This makes them -highly atypical- as dynamical systems, but also provides them with a surprisingly rich mathematical structure, and one could argue that any class of mathematical creatures with a rich mathematical structure is worthy of study. On the other hand, since the "major pedagogical points" already emerge by the (2,3) case, and since the Sturmian systems form a tiny subset of the class of "interesting" tilings of R^2 or R^3, the study of Sturmian tilings -in general- is clearly not of great importance for tiling theory or dynamical systems theory. However, the very same Diophantine approximation phenomena which arise naturally in the study of general Sturmian systems also arise naturally in many other mathematical contexts, including ones which are very "important" indeed, to judge by the list of Fields Medals citations over the past 40 years. For this reason, because Sturmian tilings can be regarded as "geometric realizations" of these Diophantine approximation problems, one could argue that they are important, but only because they provide a geometric realization of these Diophantine approximation problems, which are "the real object of interest". Chris Hillman Home Page: http://www.math.washington.edu/~hillman/personal.html