. SG ] 1 1 N ov 2 00 7 The Symplectic Geometry of Penrose Rhombus Tilings

The purpose of this article is to view Penrose rhombus tilings from the perspective of symplectic geometry. We show that each thick rhombus in such a tiling can be naturally associated to a highly singular 4–dimensional compact symplectic space MR, while each thin rhombus can be associated to another such space Mr; both spaces are invariant under the Hamiltonian action of a 2–dimensional quasitorus, and the images of the corresponding moment mappings give the rhombuses back. The spaces MR and Mr are diffeomorphic but not symplectomorphic. Mathematics Subject Classification 2000. Primary: 53D20 Secondary: 52C23 Introduction We start by considering a Penrose tiling by thick and thin rhombuses (cf. [10]). Rhombuses are very special examples of simple convex polytopes. Because of the Atiyah, Guillemin–Sternberg convexity theorem [1, 8], convex polytopes can arise as images of the moment mapping for Hamiltonian torus actions on compact symplectic manifolds. For example, simple convex polytopes that are rational with respect to a lattice L and satisfy an additional integrality condition, correspond to symplectic toric manifolds. More precisely, the Delzant theorem [6] tells us that to each such polytope in (R)∗, there corresponds a compact symplectic 2n–dimensional manifoldM , endowed with the effective Hamiltonian action of a torus of dimension n. As it turns out, the polytope is exactly the image of the corresponding moment mapping. One of the striking features of Delzant’s theorem is that it gives an explicit procedure to obtain the manifold corresponding to each given polytope as a symplectic reduced space. This correspondence may be applied to each of the rhombuses in a Penrose tiling separately. However, the rhombuses in a Penrose tiling, though simple and convex, are not simultaneously rational with respect to the same lattice. Therefore we cannot apply the Delzant procedure simultaneously to all rhombuses in the tiling. However, if we replace the lattice with a quasilattice (the Z span of a set of R–spanning vectors) and the manifold with a suitably singular space, then it is possible to apply a generalization of the Delzant procedure to arbitrary simple convex polytopes that was given by the second named author in [13]. According to this result, to each simple convex polytope in (R)∗, and to each suitably chosen quasilattice Q, one can associate a family of compact symplectic 2n–dimensional quasifolds M , each endowed with the effective Hamiltonian action