  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, while each thin rhombus can be associated to another such
space; 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. These two spaces are diffeomorphic but not
symplectomorphic.
