2 00 8 The Symplectic Geometry of Penrose Rhombus

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. Primary: 53D20. Secondary: 52C23