Symmetry, Integrability and Geometry: Methods and Applications Ammann Tilings in Symplectic Geometry

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 but not symplectomorphic. These spaces inherit from the tiling its very interesting symmetries.


Introduction
Our aim is to explore the connection between Ammann tilings and symplectic geometry. Ammann tilings are 3D analogues of Penrose rhombus tilings, made of two kinds of tiles: an oblate and a prolate rhombohedron having same edge lengths. The basic idea underlying this connection is that we are able to associate to each rhombohedron in the tiling a compact connected symplectic quasifold in a natural way. Quasifolds generalize manifolds and orbifolds and were first introduced by the second-named author in [6]; locally they are quotients of a manifold modulo the smooth action of a discrete group. The way that we associate a quasifold to a given rhombohedron is by applying a generalization to simple nonrational polytopes [6] of the Delzant construction for simple rational polytopes [3]. We remark that, while each separate rhombohedron in the tiling is a simple rational polytope and could be viewed as a standard symplectic manifold using the Delzant construction, all of the rhombohedra in the tiling are not simultaneously rational with respect to the same lattice. Since we are interested in obtaining a global symplectic view of the tiling we replace the notion of lattice with the more general notion of quasilattice: rationality gets then replaced by quasirationality. Given an Ammann tiling, we introduce a quasilattice Q having the property that each rhombohedron of the tiling is quasirational with respect to Q. We then apply the generalized Delzant procedure simultaneously to each rhombohedron and we show that there is only one symplectic quasifold, M b , associated to each of the oblate rhombohedra of the tiling and one symplectic quasifold, M r , associated to each of the prolate rhombohedra (Theorem 3.1). Both quasifolds are globally the quotient of a manifold, actually the product of three 2-spheres, modulo the action of a discrete group. Furthermore, each quasifold is endowed with the effective Hamiltonian action of the quasitorus R 3 /Q; the oblate and prolate rhombohedra are then obtained as images of the respective moment mappings. As it turns out, the two quasifolds M b and M r are diffeomorphic but not symplectomorphic (Theorem 4.1).
The initial motivation of our work on nonperiodic tilings was the realization that applying the generalized Delzant procedure to the tiles provided a very nice way to obtain new low dimensional examples of symplectic quasifolds with symmetries. We refer the reader to [1,2] for our previous work on Penrose rhombus and kite and dart tilings; the kite, for example, yields an interesting 4-dimensional quasifold that is not a global quotient of a manifold modulo the action of a discrete group.
We remark that Ammann tilings, which were introduced in the 70's [7], later turned out to be crucial in the study of quasicrystals with icosahedral symmetry. Quasicrystals are special alloys having discrete nonperiodic diffraction patterns that were discovered by Shechtman et al. [8] in 1982. They have atomic arrangements with symmetries that are not allowed in ordinary crystals. For a comprehensive review of this fascinating subject we refer the reader to the recent book by Steurer and Douady [9].
The paper is structured as follows: in Section 1 we recall the generalized Delzant procedure; in Section 2 we introduce the quasilattice Q and we discuss its connection with the tiling; in Section 3 we construct the symplectic quasifolds M b and M r ; finally, in Section 4 we show that M b and M r are diffeomorphic but not symplectomorphic.
All pictures were drawn using the ZomeCAD software.

The Generalized Delzant Construction
We now recall from [6] the generalized Delzant construction. For the notion of quasifold and of related geometrical objects we refer the reader to the original article [6] and to [2], where some of the definitions were reformulated.
Let us recall what a simple convex polytope is.
Definition 1.1 (Simple polytope) A dimension n convex polytope ∆ ⊂ (R n ) * is said to be simple if there are exactly n edges stemming from each vertex.
Let us next define the notion of quasilattice, introduced in [5]: Notice that Span Z {Y 1 , . . . , Y d } is a lattice if and only if it admits a set of generators which is a basis of E. 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 (1) Definition 1.3 (Quasirational polytope) Let Q be a quasilattice in R n . A convex polytope ∆ ⊂ (R n ) * is said to be quasirational with respect to Q if the vectors X 1 , . . . , X d in (1) can be 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 by the elements X 1 , . . . , X d in (1). Notice that if X 1 , . . . , X d can be chosen in such a way that they belong to a lattice, then the polytope is rational in the usual sense. Before we go on to describing the generalized Delzant construction we recall what a quasitorus is.

Definition 1.4 (Quasitorus)
Let Q ⊂ R n be 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 [6].
For the purposes of this article we will restrict our attention to the special case n = 3. Theorem 1.5 (Generalized Delzant construction [6]) Let Q be a quasilattice in R 3 and let ∆ ⊂ (R 3 ) * be 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 2πi d j=1 dz j ∧ dz j and the action of the torus T d = R d /Z d given by This is an effective Hamiltonian action with moment mapping given by The mapping J is proper and its image is given by the cone C λ = λ + 0, where 0 denotes the positive orthant of (R d ) * . Take now vectors X 1 , . . . , X d ∈ Q and real numbers λ 1 , . . . , λ d as in (1). Consider the surjective linear mapping 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 be the kernel of the mapping Π and choose λ = d j=1 λ j e * j . Denote by i the Lie algebra inclusion Lie(N ) → R d 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 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, If we want to apply this construction to any simple convex polytope in (R 3 ) * , then there are two arbitrary 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).

Tilings and quasilattices
The purpose of this section is to introduce two quasilattices, R and Q, in R 3 and to study how they relate to an Ammann tiling T with fixed edge length σ = 1 + 1 being the golden section. We will be often using the following fundamental identity Ammann tilings are nonperiodic tilings of three dimensional space by so-called golden rhombohedra; rhombohedra are called golden when their facets are given by golden rhombuses, namely rhombuses with diagonals that are in the ratio of φ. There are two types of such rhombohedra which are called oblate and prolate (see Figures 1 and 2). For a review of Ammann tilings we refer the reader to [7,9]. Let R be the quasilattice in (R 3 ) * that is generated by the vectors These six vectors and their opposites point to the twelve vertices of an icosahedron that is inscribed in the sphere of radius σ (see Figures 3 and 4); they are the only vectors of the quasilattice that have norm σ. One can check that these vectors form a minimal set of generators of R and that R is dense in (R 3 ) * . Consider now the two following golden rhombohedra: the oblate rhombohedron ∆ o b , having nonparallel edges V 4 , V 5 , V 6 , and the prolate rhombohedron ∆ o r , having nonparallel edges The following proposition fully describes how the tiling and the quasilattice R relate. Denote by 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 V 1 with the same orientation.
Proposition 2.1 Let T be an Ammann tiling with edges of length σ. Each vertex of the tiling lies in the quasilattice R. Moreover, for each oblate rhombohedron ∆ b in T (respectively prolate rhombohedron ∆ r in T ) there is a rigid motion ρ, given by the composition of a translation with a transformation of the icosahedral group, such that Proof. Consider first the icosahedron with its twenty pairwise parallel facets. To each pair of parallel facets there correspond two oblate rhombohedra, one the translate of the other, and two prolate rhombohedra, also one the translate of the other. Pick one representative for each such couple. This gives a total of ten oblate rhombohedra and ten prolate rhombohedra. Each of the ten oblate rhombohedra can be mapped to ∆ o b via a transformation of the icosahedral group, and in the same way each of the ten prolate rhombohedra can be mapped to ∆ o r . Now, let C be a vertex of the tiling that is different from 0 and the above vertex B. We can join B to C with a broken line made of subsequent edges of the tiling. We denote the vertices of the broken line thus obtained by T 0 = A, T 1 = B, . . . , T j , . . . , T m = C. Since the tiles are oblate and prolate rhombohedra, each vector Y j = T j − T j−1 is one of the vectors ±V 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 R, that each oblate rhombohedron having C as vertex is the translate of one of the ten oblate rhombohedra described above and that each prolate rhombohedron having C as vertex is the translate of one of the ten prolate rhombohedra described above. We can therefore conclude that, for each oblate rhombohedron ∆ b having C as vertex, there exists a rigid motion ρ, given by the composition of a translation with a transformation of the icosahedral group, such that The same is true for the prolate rhombohedra. ⊓ ⊔ Recall that the rhombic triacontahedron is a convex polyhedron with thirty facets, each given by a golden rhombus. It is first found in Kepler's writings [4]. The long diagonals of its rhombic facets are the edges of an icosahedron while the short diagonals are the edges of a dodecahedron. Consider now the triacontahedron S that has ±V i , i = 1, . . . , 6, among its vertices. Proposition 2.1 implies that all of the facets of the Ammann tiling are parallel to the thirty, pairwise parallel, facets of the triacontahedron S. Therefore, a quasilattice with respect to which all of the rhombohedra of the tiling are quasirational must contain vectors that are normal to the thirty facets of the triacontahedron. We choose as generators of our quasilattice Q the thirty vertices of the icosidodecahedron dual to S (see figure 6 and figure 7); this is an icosidodecahedron inscribed in the sphere of radius 1. Among these thirty unit vectors we select the It can be seen that Q is dense in R 3 and that the thirty unit vectors pointing to the vertices of the icosidodecahedron are the only unit vectors in the quasilattice Q. The quasilattice Q is invariant under icosahedral symmetries (as well as the quasilattice R) and, by construction, all of the rhombohedra of the tiling are quasirational with respect to Q. We will see in Theorem 3.1 that icosahedral symmetry is essential for the uniqueness, from the differentiable viewpoint, of the quasifold corresponding to the tiling.

The Tiling from a Symplectic Viewpoint
In this section we perform the Delzant construction to obtain symplectic quasifolds that can be associated to the oblate and prolate rhombohedra of an Ammann tiling having edge length σ.
Let us consider the quasilattice Q that we introduced in Section 2. As we have seen, all of the rhombohedra of our tiling are quasirational with respect to Q.
We begin by considering the oblate rhombohedron ∆ o b which has one of its vertices at the origin and is determined by the three non-parallel vectors V 4 , V 5 , V 6 . This simple polytope has 6 facets. For our construction we choose the 6 vectors given by X 1 = U 1 , X 2 = U 2 , X 3 = U 3 , X 4 = −U 1 , X 5 = −U 2 and X 6 = −U 3 . Then the corresponding Its kernel, n, is the 3-dimensional subspace of R 6 that is spanned by e 1 + e 4 , e 2 + e 5 and e 3 + e 6 . It is the Lie algebra of N = { exp(X) ∈ T 6 | X ∈ R 6 , π(X) ∈ Q }. If Ψ b is the moment mapping of the induced N -action, then is the sphere in R 4 centered at the origin with radius b = 1 √ φ . In order to compute the group N we need the following linear relations between the generators of the quasilattice Q: 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 as being naturally embedded in N . The quotient group is discrete. In conclusion, the symplectic quotient M b is given by where S 2 b is the sphere in R 3 centered at the origin with radius b. The quasitorus D 3 = R 3 /Q acts on M b in a Hamiltonian fashion, with image of the corresponding moment mapping given exactly by the oblate rhombohedron ∆ o b . Consider now the prolate rhombohedron ∆ o r that has one vertex in the origin and is determined by the three nonparallel vectors V 1 , V 2 , V 3 . We now choose the vectors given by X 1 = U 4 , X 2 = 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 by λ 1 = λ 2 = λ 3 = 0 and λ 4 = λ 5 = λ 6 = −1. It is immediate to check that we obtain the same Lie algebra n as in the case of the oblate rhombohedron. In order to see what happens to the corresponding group we need here the inverse relations: To write the relations in this form we used the fundamental identity 2. This identity also implies that we obtain the same group N as in the case of the oblate rhombohedron.
The moment mapping Ψ r is given by: Therefore where S 3 and S 2 are the unit spheres centered at the origin, of dimension 3 and 2 respectively. The quasifold M r is acted on by the same quasitorus D 3 = R 3 /Q that we obtained for the oblate rhombohedron. This action is Hamiltonian and the image of the corresponding moment mapping is exactly the prolate rhombohedron ∆ o r . Let us remark that M b and M r are both global quotients and that this defines their quasifold structures.
Remark now that, by Proposition 2.1, each of the oblate and prolate rhombohedra in the tiling can be obtained from ∆ o b and ∆ o r respectively by a transformation of the icosahedral group composed with a translation. We can then prove the following Theorem 3.1 Consider an Ammann tiling having edge length σ. Then the compact connected symplectic quasifold corresponding to each oblate rhombohedron in the tiling is given by M b , while the compact connected symplectic quasifold corresponding to each prolate rhombohedron is given by M r .
Proof. Observe that, for each oblate rhombohedron, there exists a transformation P in the icosahedral group that leaves the quasilattice Q invariant, that sends the orthogonal vectors relative to the chosen oblate rhombohedron to the orthogonal vectors relative to ∆ o b , and such that the dual transformation P * sends ∆ o b to a translate of the chosen oblate rhombohedron. The same reasoning applies to the prolate rhombohedra of the tiling. This implies that the reduced space corresponding to each oblate rhombohedron of the tiling, with the choice of orthogonal vectors and quasilattice specified above, is exactly M b . This yields a unique symplectic quasifold, M b , for all the oblate rhomboehedra in the tiling. In the same way we prove that we obtain a unique symplectic quasifold, M r , for all the prolate rhombohedra in the tiling. ⊓ ⊔

Symplectotype and Diffeotype of the Tiles
The purpose of this section is to prove the following Before we proceed with the proof let us describe a special atlas for the quasifold M b . The charts of this atlas are indexed by the vertices of the polytope: in our case we find an atlas given by eight charts, each of which corresponds to a vertex of the oblate rhombohedron. Consider for example the origin: it is given by the intersection of the facets whose orthogonal vectors are X 1 , X 2 and X 3 . We will label the corresponding chart by the index 1. Let B b be the ball in C of radius b, namely Consider the following mapping, which gives a slice of Ψ −1 (0) transversal to the Norbits This induces the homeomorphism where the open subset U 1 of M b 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 by The triple ( Analogously, we can construct seven other charts, corresponding to the remaining vertices of the oblate rhombohedron, each of which is characterized by a different combination of the variables. One can show that these eight charts are compatible and give an atlas of M b . Now let us denote by p b and p r the projections Denote by V n the open subset of S 2 b given by S 2 b minus the south pole and by V s the open subset of S 2 b given by S 2 b minus the north pole. Then, on Ψ −1 (0), consider the action of S 1 × S 1 × S 1 given by (4). We obtain and We have the following commutative diagram: The mappingτ 1 is induced by the diagram and can be written as τ n × τ n × τ n , with τ n : B b → V n . Observe that the mapping is just the stereographic projection from the north pole. We denote by τ s the analogous mapping τ s : B b −→ V s . The two charts (B b , τ n , V n ) and (B b , τ s , V s ) give a symplectic atlas of S 2 b , whose standard symplectic structure is induced by the standard symplectic structure on B b . Analogously, at a local level, the symplectic structure of the quotient M b is induced by the standard symplectic structure on We have already seen that the quasifold M b is a global quotient of a product of three 2-spheres by the discrete group Γ. We remark that the atlas above is the quotient by h 1 is a diffeomorphism of the universal covering models of the induced models. We find the following diagram: Consider the restriction of h 1 to τ −1 1 (W 0 ∩ W 1 ). This restriction admits a lift, given by the restriction of h ♯ 1 to (π 1 • p 1 ) −1 (τ −1 1 (W 0 ∩ W 1 )). Furthermore, by Step 1, the restriction of h 1 admits another lift, defined on p −1 1 (τ −1 1 (W 0 ∩ W 1 )), which is the restriction of h 0 . Therefore, by [2, Lemma 6.3], the restriction of ρ 1 • h ♯ 1 to (π 1 • p 1 ) −1 (τ −1 1 (W 0 ∩ W 1 )) descends to a diffeomorphism defined on p −1 1 (τ −1 1 (W 0 ∩ W 1 )).
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 i=1 W i )/Γ 1 with the model induced by h(∪ k i=1 W i ) ⊂ M r . Remark now that a slight modification of the above argument applies to any choice of point z 1 ∈ B b × B b × B b , z 1 = 0.
Let ǫ > 0 be arbitrarily small. Consider the product of closed balls B b−ǫ × B b−ǫ × B b−ǫ . This, by Step 4, can be covered by a finite number of connected open subsets of the kind (τ 1 • p 1 ) −1 (∪ k i=1 W i )/Γ 1 , whose intersection is a product of three balls centered at the origin. Now [2,Lemma A.3], which guarantees the uniqueness of the lift up to the action of Γ, implies that the homeomorphism h admits a lift toτ 1 (B b−ǫ × B b−ǫ × B b−ǫ ). 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 subset where the action of the quasitorus D b is free. By the uniqueness of the lift, ([2, Lemma A.3]), we obtain a global lifth : r × S 2 r × S 2 r . Moreover, since diagram (7) preserves the symplectic structures, we have thath is a symplectomorphism between S 2 b × S 2 b × S 2 b to S 2 r × S 2 r × S 2 r , which is impossible. ⊓ ⊔ In conclusion, 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 oblate and the prolate rhombohedra.