Ammann Tilings in Symplectic Geometry

Transcription

1 Symmetry, Integraility and Geometry: Methods and Applications Ammann Tilings in Symplectic Geometry Fiammetta BATTAGLIA and Elisa PRATO SIGMA 9 (013), 01, 13 pages Dipartimento di Matematica e Informatica U. Dini, Via S. Marta 3, Firenze, Italy fiammetta.attaglia@unifi.it URL: Dipartimento di Matematica e Informatica U. Dini, Piazza Ghierti 7, 501 Firenze, Italy elisa.prato@unifi.it URL: Received Novemer 09, 01, in final form Feruary 7, 013; Pulished online March 06, Astract. In this article we study Ammann tilings from the perspective of symplectic geometry. Ammann tilings are nonperiodic tilings that are related to quasicrystals with icosahedral symmetry. We associate to each Ammann tiling two explicitly constructed highly singular symplectic spaces and we show that they are diffeomorphic ut not symplectomorphic. These spaces inherit from the tiling its very interesting symmetries. Key words: symplectic quasifold; nonperiodic tiling; quasilattice 010 Mathematics Suject Classification: 53D0; 5C3 1 Introduction Our general aim is to study the connection etween symplectic geometry and the nonperiodic tilings that are related to the geometry of quasicrystals 1. Considering these tilings from the symplectic viewpoint provides a concrete way of otaining new examples of highly singular symplectic spaces that are endowed with very rich symmetries. These examples effectively contriute to understanding the theoretical aspects of the geometry of this type of singular spaces. Moreover, we expect that symplectic geometry may e used to shed light on the study of the tilings. We started this program with the study of two tilings of the plane: Penrose rhomus [] and kite and dart [3] tilings. In this article we focus our attention for the first time on three-dimensional tilings. We consider Ammann tilings, which are the three-dimensional analogues of Penrose rhomus tilings. They were introduced y Ammann in the 70 s [10] and turned out to e related to quasicrystals with icosahedral symmetry [6]. As we will see later, the third dimension yields an initially unexpected richness and complexity. The main idea underlying the connection etween symplectic geometry and tilings is a generalization [8] of the Delzant construction [5], which we use to associate to each tile an explicitly constructed symplectic space. We recall that the Delzant construction associates a symplectic toric n-manifold to each simple convex polytope in ( R n), which is rational with respect to a lattice and satisfies an additional integrality condition. The prolem, in the setting of nonperiodic tilings related to quasicrystals, is that either the tiles are not individually rational, or they are not simultaneously rational with respect to the same lattice. In the generalized 1 Quasicrystals are special materials having discrete nonperiodic diffraction patterns that were experimentally discovered y Shechtman et al. [11] in 198. They have atomic arrangements with symmetries that are not allowed in ordinary crystals. For a comprehensive review of this fascinating suject we refer the reader to the recent ook y Steurer and Deloudi [1].

2 F. Battaglia and E. Prato construction, however, the lattice is replaced y a quasilattice and rationality is replaced y the notion of quasirationality. The resulting space is a n-dimensional quasifold, a generalization of manifolds and orifolds that was first introduced y the second-named author in [8]; the group acting is no longer a torus ut a quasitorus [8]. Ammann tilings are made of two kinds of tiles: an olate rhomohedron and a prolate rhomohedron having same edge lengths. These rhomohedra, although separately rational, are not simultaneously rational with respect to the same lattice. However, the geometry of the tiling ensures that it is possile to choose a quasilattice F having the property that each rhomohedron of the tiling is quasirational with respect to F (Proposition 3.1). We then apply the generalized Delzant construction simultaneously to each rhomohedron and we show that there is one symplectic quasifold, M, associated to each of the olate rhomohedra of the tiling and one symplectic quasifold, M r, associated to each of the prolate rhomohedra (Theorem 4.1). Both quasifolds are gloally the quotient of a manifold (the product of three -spheres) modulo the action of a discrete group. There is a unique diffeotype and two symplectotypes associated to the tiling. In fact, we show that M and M r are diffeomorphic ut not symplectomorphic, consistently with the fact that the two different types of tiles have different volumes (Theorem 6.1). We remark that quasilattices are the fundamental structure underlying oth nonperiodic tilings related to quasicrystals and the corresponding symplectic quasifolds. This is particularly evident for Ammann tilings, and the related physics of icosahedral quasicrystals. The novelty here, with respect to two-dimensional tilings, is that, in this context, there are actually three important quasilattices: the quasilattice F aove, known as face-centered lattice, the odycentered lattice, I, and the simple icosahedral lattice, P. These are the only three quasilattices that have icosahedral symmetry [9]. The quasilattice P, up to a suitale rescaling, has the property of containing all of the vertices of the Ammann tiling. The quasilattices F and 1 I are usually thought of in physics as the dual of each other, as they are projections of two 6- dimensional lattices that are dual to one another in the standard sense. Consistently, we show that, in our symplectic setting, F is the group quasilattice of the quasitorus D 3 = R 3 /F and that 1 I can e thought of as its dual, the weight quasilattice (Section 5). The paper is structured as follows: in Section we recall the generalized Delzant construction; in Section 3 we introduce the quasilattices F, I and P, and we discuss their connection with the tiling; in Section 4 we construct the symplectic quasifolds M and M r ; in Section 5 we study their local geometry; finally, in Section 6 we show that they are diffeomorphic ut not symplectomorphic. The generalized Delzant construction We now recall from [8] the generalized Delzant construction. For the notion of quasifold, of related geometrical ojects and for a numer of examples we refer the reader to the original article [8] and to [3], where some of the definitions were reformulated. Let us recall what a simple convex polytope is. Definition.1 (simple polytope). A dimension n convex polytope ( R n) is said to e simple if there are exactly n edges stemming from each vertex. Let us next define the notion of quasilattice, introduced in [7]: Definition. (quasilattice). Let E e a real vector space. A quasilattice in E is the Z-span of a set of R-spanning vectors, Y 1,..., Y d, of E. Notice that Span Z {Y 1,..., Y d } is a lattice if and only if it admits a set of generators which is a asis of E.

3 Ammann Tilings in Symplectic Geometry 3 Consider now a dimension n convex polytope ( R n) having d facets. Then there exist elements X 1,..., X d in R n and λ 1,..., λ d in R such that = d { ( µ R n ) } µ, Xj λ j. (.1) Definition.3 (quasirational polytope). Let Q e a quasilattice in R n. A convex polytope ( R n) is said to e quasirational with respect to Q if the vectors X1,..., X d in (.1) can e chosen in Q. We remark that each polytope in ( R n) is quasirational with respect to some quasilattice Q: just take the quasilattice that is generated y the elements X 1,..., X d in (.1). Notice that if X 1,..., X d can e chosen in such a way that they elong to a lattice, then the polytope is rational in the usual sense. Before we go on to descriing the generalized Delzant construction we recall what a quasitorus is. Definition.4 (quasitorus). Let Q R n e a quasilattice. We call quasitorus of dimension n the group and quasifold D = R n /Q. For the definition of Hamiltonian action of a quasitorus on a symplectic quasifold we refer the reader to [8]. For the purposes of this article we will restrict our attention to the special case n = 3. Theorem.1 (generalized Delzant construction [8]). Let Q e a quasilattice in R 3 and let ( R 3 ) e a simple convex polytope that is quasirational with respect to Q. Then there exists a 6- dimensional compact connected symplectic quasifold M and an effective Hamiltonian action of the quasitorus D = R 3 /Q on M such that the image of the corresponding moment mapping is. Proof. Let us consider the space C d endowed with the standard symplectic form ω 0 = 1 πi d dz j d z j and the action of the torus T d = R d /Z d given y τ : T d C d C d (( e πiθ 1,..., e πiθ ) d, z ) ( e πiθ 1 z 1,..., e πiθ ) d z d. This is an effective Hamiltonian action with moment mapping given y J : C d ( R d) z d z j e j + λ, λ = const ( R d). The mapping J is proper and its image is given y the cone C λ = λ + C 0, where C 0 denotes the positive orthant of ( R d). Take now vectors X1,..., X d Q and real numers λ 1,..., λ d as in (.1). Consider the surjective linear mapping π : R d R 3 e j X j.

4 4 F. Battaglia and E. Prato Figure 1. The olate rhomohedron. Figure. The prolate rhomohedron. Consider the dimension 3 quasitorus D = R 3 /Q. Then the linear mapping π induces a quasitorus epimorphism Π: T d D. Define now N to e the kernel of the mapping Π and choose λ = d λ j e j. Denote y i the Lie algera inclusion Lie(N) Rd and notice that Ψ = i J is a moment mapping for the induced action of N on C d. Then the quasitorus T d /N acts in a Hamiltonian fashion on the compact symplectic quasifold M = Ψ 1 (0)/N. If we identify the quasitori D and T d /N via the epimorphism Π, we get a Hamiltonian action of the quasitorus D whose moment mapping Φ has image equal to (π ) 1 (C λ ker i ) = (π ) 1 (C λ im π ) = (π ) 1 (π ( )) which is exactly. This action is effective since the level set Ψ 1 (0) contains points of the form z C d, z j 0, j = 1,..., d, where the T d -action is free. Notice finally that dim M = d dim N = d (d 3) = 6. Remark.1. If we want to apply this construction to any simple convex polytope in ( R 3), then there are two aritrary choices involved. The first is the choice of a quasilattice Q with respect to which the polytope is quasirational, and the second is the choice of vectors X 1,..., X d in Q that are orthogonal to the facets of and inward-pointing as in (.1). 3 Ammann tilings and quasilattices The purpose of this section is to introduce three quasilattices P, F and I, that are relevant for our construction. Let φ = 1+ 5 e the golden ratio. We will e using extensively the following fundamental identity φ = φ. (3.1) Let σ e a positive real numer and let us consider an Ammann tiling T with fixed edge length σ. Ammann tilings are nonperiodic tilings of three-dimensional space y so-called golden rhomohedra; rhomohedra are called golden when their facets are given y golden rhomuses, namely rhomuses with diagonals that are in the ratio of φ. There are two types of such rhomohedra which are called olate and prolate (see Figs. 1 and ). For a review of Ammann tilings we refer the reader to [10, 1]. Consider now the vectors in ( R 3) α 1 = 1 (φ 1, 1, 0), α = 1 (0, φ 1, 1), α 3 = 1 (1, 0, φ 1), α 4 = 1 (1 φ, 1, 0), α 5 = 1 (0, 1 φ, 1), α 6 = 1 (1, 0, 1 φ). These six vectors and their opposites point to the twelve vertices of an icosahedron that is 3 φ inscried in the sphere of radius (see Figs. 3 and 4); they generate a quasilattice P that is known in physics as the simple icosahedral lattice [9]. All pictures were drawn using the ZomeCAD software.

5 Ammann Tilings in Symplectic Geometry 5 Figure 3. The vectors ±α 1,..., ±α 6. Figure 4. The icosahedron. Let δ = 3 φσ and consider the two following golden rhomohedra: the olate rhomohedron o, having nonparallel edges δα 4, δα 5, δα 6, and the prolate rhomohedron o r, having nonparallel edges δα 1, δα, δα 3. Denote y AB one edge of the tiling T. From now on we will choose our coordinates so that A = O and so that B A is parallel to α 1 with the same orientation. Proposition 3.1. Let T e an Ammann tiling with edges of length σ. Each vertex of the tiling lies in the quasilattice δp. Moreover, for each olate rhomohedron in T (respectively prolate rhomohedron r in T ) there is a rigid motion ρ, given y the composition of a translation with a transformation of the icosahedral group, such that ρ( ) is o (respectively ρ( r) is o r). Proof. Consider first the icosahedron with its twenty pairwise parallel facets. To each pair of parallel facets there correspond two olate rhomohedra, one the translate of the other, and two prolate rhomohedra, also one the translate of the other. Pick one representative for each such couple. This gives a total of ten olate rhomohedra and ten prolate rhomohedra. Each of the ten olate rhomohedra can e mapped to o via a transformation of the icosahedral group, and in the same way each of the ten prolate rhomohedra can e mapped to o r. Now, let C e a vertex of the tiling that is different from 0 and the aove vertex B. We can join B to C with a roken line made of susequent edges of the tiling. We denote the vertices of the roken line thus otained y T 0 = A, T 1 = B,..., T j,..., T m = C. Since the tiles are olate and prolate rhomohedra, each vector Y j = T j T j 1 is one of the vectors ±α k, k = 1,..., 6. Therefore we have that C A = T m T 0 = Y m + + Y 1. This implies that the vertex C lies in δp, that each olate rhomohedron having C as vertex is the translate of one of the ten olate rhomohedra descried aove and that each prolate rhomohedron having C as vertex is the translate of one of the ten prolate rhomohedra descried aove. We can therefore conclude that, for each olate rhomohedron having C as vertex, there exists a rigid motion ρ, given y the composition of a translation with a transformation of the icosahedral group, such that ρ( ) = o. The same is true for the prolate rhomohedra. We introduce now a quasilattice F R 3 with respect to which all of the rhomohedra of the tiling are quasirational (cf. Remark.1). This is necessary in order to apply the generalized Delzant procedure simultaneously to all of the rhomohedra in the tiling. We take F to e the quasilattice that is generated y the six vectors U 1 = 1 (1, φ 1, φ), U = 1 (φ, 1, φ 1), U 3 = 1 (φ 1, φ, 1), U 4 = 1 ( 1, φ 1, φ), U 5 = 1 (φ, 1, φ 1), U 6 = 1 (φ 1, φ, 1). The quasilattice F is known in physics as the face-centered lattice [9].

6 6 F. Battaglia and E. Prato Figure 5. The star of norm vectors of the quasilattice F. Figure 6. The icosidodecahedron. The vectors U i have norm equal to. It can e easily seen that there are exactly 30 vectors in F having the same norm. These thirty vectors point to the vertices of an icosidodecahedron inscried in the sphere of radius (see Figs. 5 and 6). Remark 3.1. Proposition 3.1 implies that, for each facet of the Ammann tiling, there is a pair of vectors {α i, α j } such that the given facet is parallel to the plane Π ij generated y {α i, α j }. We have 15 such possile pairs {α i, α j }, with i, j = 1,..., 6, i j. For each one of them, two of the 30 vectors aove are orthogonal to the corresponding plane Π ij. This ensures that all of the rhomohedra of the tiling are quasirational with respect to F. Another quasilattice that will e useful in the sequel is the quasilattice I ( R 3) that is generated y the vectors {α 1, α, α 3, φα 4, φα 5, φα 6 }. The quasilattice I is known in physics as the ody centered lattice [9]. Remark 3.. The quasilattices P, F and I are invariant under icosahedral symmetries and are dense in their respective amient spaces. One can show that, if we identify R 3 with its dual using the standard inner product, we have the following proper inclusions: I F P 1 I. (3.) Using the notation of Conway Sloane [4], the lattices P, F and 1 I can e otained as the respective projections from the following lattices in R 6 : Z 6, 6 6 D 6 = Span n j e j n j is even, and D6 1 = Span 6 n j e j nj n k (mod ), the projection eing given y R 6 R 3 e j α j. Coherently with (3.) we have the following proper inclusions: D6 D 6 Z 6 D6.

7 Ammann Tilings in Symplectic Geometry 7 The lattice Z 6 is self dual, whilst D 6 and D6 are the dual of one another. A notion of duality for the icosahedral quasilattices in dimension 3 is derived from the aove relations of duality in R 6. This is coherent with the symplectic setup. In fact, we will see that the quasilattice F is the group quasilattice (see Remark 4.1) and that the quasilattice 1 I plays the role of its dual, the weight quasilattice (see Section 5). 4 The tiling from a symplectic viewpoint In this section we perform the Delzant construction to otain symplectic quasifolds that can e associated to the olate and prolate rhomohedra of an Ammann tiling having edge length σ. Let us consider the quasilattice F that we introduced in Section 3. As we have seen, all of the rhomohedra of our tiling are quasirational with respect to F. We egin y considering the olate rhomohedron o which has one of its vertices at the origin and is determined y the three non-parallel vectors δα 4, δα 5, δα 6. This simple polytope has 6 facets. For our construction we choose the 6 vectors given y X 1 = U 1, X = U, X 3 = U 3, X 4 = U 1, X 5 = U and X 6 = U 3. Then the corresponding coefficients are given y λ 1 = λ = λ 3 = 0 and λ 4 = λ 5 = λ 6 = δ φ. Take now the surjective linear mapping defined y π : R 6 R 3 e i X i. Its kernel, n, is the 3-dimensional suspace of R 6 that is spanned y e 1 + e 4, e + e 5 and e 3 + e 6. It is the Lie algera of N = {exp(x) T 6 X R 6, π(x) F }. If Ψ is the moment mapping of the induced N-action, then Ψ : C 6 ( R 3) ( ) z z 1 + z 4 δ φ, z + z 5 δ φ, z 3 + z 6 δ φ. Therefore Ψ 1 (0) = S 3 S3 S3, where S3 is the sphere in R4 centered at the origin with radius = δ φ. In order to compute the group N we need the following linear relations etween the generators of the quasilattice F : U 4 1 φ 1 U 5 = 1 1 φ U 6 φ 1 1 U 1 U U 3. Then a straightforward computation gives that N = { exp(x) T 6 X = (r + φh, s + φk, t + φl, r, s, t), r, s, t R, h, k, l Z }. We can think of S 1 S 1 S 1 = { exp(x) T 6 X = (r, s, t, r, s, t), r, s, t R } (4.1) as eing naturally emedded in N. The quotient group Γ = N S 1 S 1 S 1 is discrete. In conclusion, the symplectic quotient M is given y M = Ψ 1 (0) N = S3 S3 S3 N = S S S Γ,

8 8 F. Battaglia and E. Prato where S is the sphere in R3 centered at the origin with radius. The quasitorus D 3 = R 3 /F acts on M in a Hamiltonian fashion, with image of the corresponding moment mapping given exactly y the olate rhomohedron o. Consider now the prolate rhomohedron o r that has one vertex in the origin and is determined y the three nonparallel vectors δα 1, δα, δα 3. We now choose the vectors given y X 1 = U 4, X = U 5, X 3 = U 6, X 4 = U 4, X 5 = U 5 and X 6 = U 6. Then the corresponding coefficients are given y λ 1 = λ = λ 3 = 0 and λ 4 = λ 5 = λ 6 = δ. It is immediate to check that we otain the same Lie algera n as in the case of the olate rhomohedron. In order to see what happens to the corresponding group we need here the inverse relations: U 1 U U 3 1 = φ 1 1 φ 1 1 φ 1 U 4 U 5 U 6. To write the relations in this form we used the fundamental identity (3.1). This identity also implies that we otain the same group N as in the case of the olate rhomohedron. The moment mapping Ψ r is given y Therefore Ψ r : C 6 ( R 3) z ( z 1 + z 4 δ, z + z 5 δ, z 3 + z 6 δ ). M r = Ψ 1 r (0) N = S3 r Sr 3 Sr 3 N = S r S r S r Γ where S r R 3 and S 3 r R 4 are the spheres centered at the origin with radius r =, δ. The quasifold M r is acted on y the same quasitorus D 3 = R 3 /F that we otained for the olate rhomohedron. This action is Hamiltonian and the image of the corresponding moment mapping is exactly the prolate rhomohedron o r. Remark 4.1. Let us remark that M and M r are oth gloal quotients and that this defines their quasifold structures. The quasilattice F can e viewed as the group quasilattice of the quasitorus D 3 acting on oth. Remark now that, y Proposition 3.1, each of the olate and prolate rhomohedra in the tiling can e otained from o and o r respectively y a transformation of the icosahedral group composed with a translation. We can then prove the following Theorem 4.1. Consider an Ammann tiling having edge length σ. Then the compact connected symplectic quasifold corresponding to each olate rhomohedron in the tiling is given y M, while the compact connected symplectic quasifold corresponding to each prolate rhomohedron is given y M r. Proof. Oserve that, for each olate rhomohedron, there exists a transformation T in the icosahedral group that leaves the quasilattice F invariant, that sends the orthogonal vectors relative to the chosen olate rhomohedron to the orthogonal vectors relative to o, and such that the dual transformation T sends o to a translate of the chosen olate rhomohedron. The same reasoning applies to the prolate rhomohedra of the tiling. This implies that the reduced space corresponding to each olate rhomohedron of the tiling, with the choice of orthogonal vectors and quasilattice specified aove, is exactly M. This yields a unique symplectic quasifold, M, for all the olate rhomohedra in the tiling. In the same way we prove that we otain a unique symplectic quasifold, M r, for all the prolate rhomohedra in the tiling. The quasifolds M and M r can also e constructed as complex quotients and are Kähler [1].

9 Ammann Tilings in Symplectic Geometry 9 5 Local geometry of the quasifolds M and M r In this section we study the equivariant geometry of the quasifolds M and M r in a neighorhood of the D 3 -fixed points. Let us egin y descriing an atlas for the quasifold M. The charts of this atlas are indexed y the vertices of the polytope: in our case we find an atlas given y eight charts, each of which corresponds to a vertex of the olate rhomohedron. Consider for example the origin: it is given y the intersection of the facets whose orthogonal vectors are X 1, X and X 3. Let B e the all in C of radius, namely B = {z C z < }. Consider the following mapping, which gives a slice of Ψ 1 (0) transversal to the N-orits t B B B 1 { z Ψ 1 (0) z 4 0, z 5 0, z 6 0 } (z 1, z, z 3 ) (z 1, z, z 3, z 1, z, ) z 3. This induces the homeomorphism τ (B B B )/Γ 1 1 U1 [z] [t 1 (z)], where the open suset U 1 of M is the quotient { z Ψ 1 (0) z 4 0, z 5 0, z 6 0 } /N and the discrete group Γ 1 is given y Γ 1 N (S 1 S 1 S 1 {1} {1} {1}), hence Γ 1 = exp {(φh, φk, φl) h, k, l Z}. (5.1) The triple (U 1, τ 1, (B B B )/Γ 1 ) is a chart of M. Analogously, we can construct seven other charts, corresponding to the remaining vertices of the olate rhomohedron, each of which is characterized y a different comination of the variales. One can show that these eight charts are compatile and give an atlas of M. One can check that the moment map, locally, on the first chart is given y Φ([z 1 : z : z 3 ]) = φα 5 φα 5, U 1 z 1 + φα 6 φα 6, U z + φα 4 φα 4, U 3 z 3 = φα 5 z 1 + φα 6 z + φα 4 z 3, while the isotropy action of D 3 on C 3 /Γ 1 is given y ( D 3, C 3 /Γ 1 ) C 3 /Γ 1 ([X], (z 1, z, z 3 )) ( e πiφα 5(X) z 1, e πiφα 6(X) z, e πiφα 4(X) z 3 ). (5.) To otain the local expression of the moment mapping on the other seven charts it suffices to replace φα 5, φα 6, φα 4 in (5.) with all the possile cominations of ±φα 5, ±φα 6, ±φα 4 respectively. Notice that the vectors U 1, U, U 3 are three of the six generators of F, while φα 4, φα 5, φα 6 are three of the six generators of 1 I. An atlas for the the quasifold M r can e constructed in the same way. It can e shown that the moment mapping for the prolate rhomoohedron, is given, locally on the chart corresponding to the origin, y Φ([z 1 : z : z 3 ]) = α α, U 4 z 1 + α 3 α 3, U 5 z + α 1 α 1, U 6 z 3 = α z 1 + α 3 z + α 1 z 3,

10 10 F. Battaglia and E. Prato while the isotropy action of D 3 on C 3 /Γ 1 is given y ( D 3, C 3 /Γ 1 ) C 3 /Γ 1 ([X], (z 1, z, z 3 )) ( e πiα (X) z 1, e πiα 3(X) z, e πiα 1(X) z 3 ). Again, notice that the vectors U 4, U 5, U 6 are the three remaining generators of F, while α 1, α, α 3 are the three remaining generators of 1 I. In conclusion, the weights of the isotropy action of the quasitorus D 3 on a neighoorhood of the D 3 -fixed points for oth M and M r generate the quasilattice 1 I. Therefore 1 I can e thought of, in this setting, as the weight quasilattice of D 3. This is consistent with the fact that 1 I is dual to the group quasilattice F (cf. Remark 3.). Remark 5.1. Remark that, since α i (X), φα i (X) lie in Z + φz whenever X F, and since the local group in each chart of M and M r is equal to Γ 1, the aove actions are well defined. Remark 5.. If we choose as group lattice tf instead, t R, then the corresponding weight lattice would have to e 1 ti. But this would not e consistent with the inclusion and projection schemes in Remark 3.. This is the main reason underlying our choice of the norm of the vectors α j, j = 1,..., 6. 6 Dif feotype and symplectotype of the tiles The purpose of this section is to prove the following Theorem 6.1. The quasifolds M and M r are diffeomorphic ut not symplectomorphic. Before proceeding with the proof of this theorem we need a few more facts on the local geometry of the quasifold M. Let us denote y p the projection S S S M. Denote y V n the open suset of S given y S minus the south pole and y V s the open suset of S given y S minus the north pole. Then, on Ψ 1 (0), consider the action of S 1 S 1 S 1 given y (4.1). We otain and V n V n V n = { z Ψ 1 (0) z 4 0, z 5 0, z 6 0 } / ( S 1 S 1 S 1) U 1 = (V n V n V n )/Γ. We have the following commutative diagram: B B B t 1 { z Ψ 1 (0) z 4 0, z 5 0, z 6 0 } τ 1 B B B V n V n V n (6.1) p 1 (B B B )/Γ 1 τ 1 U 1. The mapping τ 1 is induced y the diagram and can e written as τ n τ n τ n, with τ n : B V n. Oserve that the mapping C V n w [ τ n ( w/ 1 + w )] p

11 Ammann Tilings in Symplectic Geometry 11 is just the stereographic projection from the north pole. We denote y τ s the analogous mapping τ s : B V s. The two charts (B, τ n, V n ) and (B, τ s, V s ) give a symplectic atlas of S, whose standard symplectic structure is induced y the standard symplectic structure on B. Analogously, at a local level, the symplectic structure of the quotient M is induced y the standard symplectic structure on B B B. We have already seen that the quasifold M is a gloal quotient of a product of three -spheres y the discrete group Γ. We remark that the atlas aove is the quotient y Γ of the atlas of the product of three spheres, given y the eight triples V n V n V n, V n V n V s, V n V s V n, V s V n V n, V n V s V s, V s V n V s, V s V s V n, V s V s V s. We are now ready to prove Theorem 6.1: Proof. Let us egin y showing that M and M r are diffeomorphic. Let us denote y p r the projection S r S r S r M r. The natural Γ-equivariant diffeomorphism f : S S S S r Sr Sr induces a homeomorphism f : M M r ; in general, a homeomorphism etween two gloal quotients that is induced y an equivariant diffeomorphism of the manifolds turns out to e a quasifold diffeomorphism [3, Definition A.]. Let us now show that M and M r are not symplectomorphic. Denote y ω and ω r the symplectic forms of M and M r respectively. Suppose that there is a symplectomorphism h: M M r, namely a diffeomorphism h such that h (ω r ) = ω. We prove that this implies that the homeomorphism h: M M r lifts to a symplectomorphism h: S S S Sr Sr Sr, leading thus to a contradiction: such symplectomorphism cannot exists, since the two manifolds have different symplectic volumes. To start with recall from [3, Remark.9] that, to each point m M, one can associate the groups Γ m and Γ h(m). The definition of diffeomorphism implies that these two groups are isomorphic. Let n S e the north pole and take m 0 = p (n n n ). Then, since Γ m Γ h(m), without loss of generality the point h(m 0 ) can e taken to e p r (n r n r n r ), where n r Sr is the north pole. Consider the chart U 1 that we constructed aove. Then, y definition of quasifold diffeomorphism [3, Definition A.3] and [3, Remark A.4], there exists an open suset U U 1 such that m 0 U and h τ1 1 : τ1 1 (U) h(u) is a diffeomorphism of the universal covering models induced y τ1 1 (U) B B B /Γ 1 and h(u) M r respectively. Moreover, y [3, Proposition A.9], any open suset W U enjoys the same property. We can choose W 0 U 1 such that W 0 = (τ 1 p 1 ) 1 (W 0 )) is a product of three alls. In particular, W0 is simply connected. Denote now y W r,0 = (p r ) 1 (h(w ))); this is an open suset of Sr Sr Sr, which is also connected, due to the action of Γ on Sr Sr Sr. Denote y W r,0 its universal covering. Now consider a point z1 B B B such that z1 1 0, z 1 0 and z1 3 0 and let m = (τ 1 p 1 )(z 1 ). For the sequel it is crucial to remark that, ecause of the action of Γ 1 given in (5.1), any Γ 1 -invariant open suset of B B B that contains the point z 1, contains also the product of circles {(z 1, z, z 3 ) B B B z 1 = z1 1, z = z1, z 3 = z1 3 }. Hence, for each point (τ 1 p 1 )(tz 1 ) with t [0, 1], we can find an open suset W t U 1, containing that point, such that the homeomorphism τ1 1 h, restricted to τ1 1 (W t), is a diffeomorphism, and (τ 1 p 1 ) 1 (W t ) is the product of three open annuli. We can cover the curve y a finite numer of these W t s: W 0, W 1,..., W s, with W j W j+1. Notice that (τ 1 p 1 ) 1 (W j W j+1 ), j = 0,..., s 1, is itself a product of three open annuli. The susets W j = (τ 1 p 1 ) 1 (W j ) and W r,j = (p r ) 1 (h(w j )) are open and connected. We divide the remaining part of the proof in susequent steps:

12 1 F. Battaglia and E. Prato Step 1: consider first W 0. Since the isotropy of Γ 1 at 0 is the whole Γ 1, we can apply [3, Lemma 6.]. We find that W r,0 is itself simply connected and that the homeomorphism h τ 1 lifts to a diffeomorphism h 0 : W0 W r,0. Step : consider the homeomorphism h 1 = h τ 1 defined on τ1 1 (W 1). By construction h 1 is a diffeomorphism of the universal covering models of the induced models. We find the following diagram: W 1 h 1 W r,1 π 1 W 1 ρ 1 W r,1 p 1 τ 1 q 1 1 (W 1) h 1 h(w 1 ). Consider the restriction of h 1 to τ1 1 (W 0 W 1 ). This restriction admits a lift, given y the restriction of h 1 to (π 1 p 1 ) 1 (τ1 1 (W 0 W 1 )). Furthermore, y Step 1, the restriction of h 1 admits another lift, defined on p (τ1 (W 0 W 1 )), which is the restriction of h 0. Therefore, y [3, Lemma 6.3], the restriction of ρ 1 h 1 to (π 1 p 1 ) 1 (τ1 1 (W 0 W 1 )) descends to a diffeomorphism defined on p (τ1 (W 0 W 1 )). Step 3: we consider W 0 W 1 W 1 and we apply [3, Lemma 6.5] to the homeomorphism h τ 1 defined on τ1 1 (W 1). We deduce that h τ 1 is a diffeomorphism of the model (τ 1 p 1 ) 1 (W 1 )/Γ 1 with the model induced y h(w 1 ) M r. Step 4: we apply Step 3 to the other successive intersections. We find that h τ 1 is a diffeomorphism of the model (τ 1 p 1 ) 1 ( k W i )/Γ 1 with the model induced y h( k W i ) M r. i=1 i=1 Remark now that a slight modification of the aove argument applies to any choice of point z 1 B B B, z 1 0. Let ɛ > 0 e aritrarily small. Consider the product of closed alls B ɛ B ɛ B ɛ. This, y Step 4, can e covered y a finite numer of connected open susets of the kind (τ 1 p 1 ) 1 ( k i=1 W i )/Γ 1, whose intersection is a product of three alls centered at the origin. Now [3, Lemma A.3], which guarantees the uniqueness of the lift up to the action of Γ, implies that the homeomorphism h admits a lift to τ 1 (B ɛ B ɛ B ɛ ). This in turn implies that h: U 1 h(u 1 ) admits a lift h 1 : V n V n V n p 1 r (h(u 1 )). We apply the same argument to the other eight charts. These charts intersect on the dense connected open suset where the action of the quasitorus D 3 is free. By the uniqueness of the lift [3, Lemma A.3], we otain a gloal lift h : S S S S r Sr Sr. Moreover, since diagram (6.1) preserves the symplectic structures, we have that h is a symplectomorphism etween S S S to S r Sr Sr, which is impossile. In conclusion, similarly to what happens in dimension two for Penrose rhomus tilings [], there is a unique quasifold structure that is naturally associated to any Ammann tiling with fixed edge length, and two distinct symplectic structures that distinguish the olate and the prolate rhomohedra. Acknowledgements We would like to thank Ron Lifshitz for his help on the theory of quasicrystals.

13 Ammann Tilings in Symplectic Geometry 13 References [1] Battaglia F., Prato E., Generalized toric varieties for simple nonrational convex polytopes, Int. Math. Res. Not. (001), , math.cv/ [] Battaglia F., Prato E., The symplectic geometry of Penrose rhomus tilings, J. Symplectic Geom. 6 (008), , arxiv: [3] Battaglia F., Prato E., The symplectic Penrose kite, Comm. Math. Phys. 99 (010), , arxiv: [4] Conway J.H., Sloane N.J.A., Sphere packings, lattices and groups, Grundlehren der Mathematischen Wissenschaften, Vol. 90, 3rd ed., Springer-Verlag, New York, [5] Delzant T., Hamiltoniens périodiques et images convexes de l application moment, Bull. Soc. Math. France 116 (1988), [6] Levine D., Steinhardt P.J., Quasicrystals: a new class of ordered structures, Phys. Rev. Lett. 53 (1984), [7] Mackay A.L., De nive quinquangula: on the pentagonal snowflake, Sov. Phys. Cryst. (1981), [8] Prato E., Simple non-rational convex polytopes via symplectic geometry, Topology 40 (001), , math.sg/ [9] Rokhsar D.S., Mermin N.D., Wright D.C., Rudimentary quasicrystallography: the icosahedral and decagonal reciprocal lattices, Phys. Rev. B 35 (1987), [10] Senechal M., The mysterious Mr. Ammann, Math. Intelligencer 6 (004), [11] Shechtman D., Blech I., Gratias D., Cahn J.W., Metallic phase with long-range orientational order and no translational symmetry, Phys. Rev. Lett. 53 (1984), [1] Steurer W., Deloudi S., Crystallography of quasicrystals: concepts, methods and structures, Springer Series in Materials Science, Vol. 16, Springer-Verlag, Berlin, 009.