  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.
