The Symplectic Penrose Kite

The purpose of this article is to view the Penrose kite from the perspective of symplectic geometry. Mathematics Subject Classification 2000. Primary: 53D20 Secondary: 52C23 Introduction The kite in a Penrose aperiodic tiling by kite and darts [8, 9] is an example of a simple convex polytope. By the Atiyah, Guillemin–Sternberg convexity theorem [1, 6], convex polytopes that are rational can be obtained as images of the moment mapping for Hamiltonian torus actions on compact symplectic manifolds. Moreover, the Delzant theorem [5] provides an exact correspondence between symplectic toric manifolds and simple rational polytopes that satisfy a special integrality condition; a crucial feature of this theorem is that it gives an explicit construction of the manifold that is associated to each polytope. The Penrose kite however is the most elementary and beautiful example of a simple convex polytope that is not rational. The purpose of this article is to apply to the kite a generalization of the Delzant construction for non–rational polytopes, which was introduced by the second–named author in [10]. We recall that this generalized construction allows to associate to any simple convex polytope ∆ in (R)∗ a 2k–dimensional compact symplectic quasifold. Quasifolds are a natural generalization of manifolds and orbifolds: a local n–dimensional model is given by the quotient of an open connected subset of R by the action of a finitely generated group. In the generalized construction the lattice of the rational case is replaced by a quasilattice Q, which is the Z–span of a set of generators of R. The torus is replaced accordingly by a quasitorus, which is the quotient of R modulo Q. The action of the quasitorus on the quasifold is smooth, effective and Hamiltonian, and exactly as in the Delzant case, the image of the corresponding moment mapping is the polytope ∆. In order to apply the generalized Delzant construction to the kite we need to choose a suitable quasilattice Q, and a set of four vectors in Q that are orthogonal to the edges of the kite and that point toward the interior of the polytope. The most natural choice is to consider, among the various inward–pointing orthogonal vectors, those four which have the same length as the longest edge of the kite, and then to choose Q to be the quasilattice that they generate. We remark that these choices are justified by the geometry of the kite, and, more globally, by the geometry of any kite and dart tiling, in the following sense. Let us consider the quasilattice R which is generated by the