Journal of Symplectic Geometry

Volume 6 (2008)

Number 2

The Symplectic Geometry of Penrose Rhombus Tilings

Pages: 139 – 158



Fiammetta Battaglia

Elisa Prato


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 $M_R$, while each thin rhombus can be associated to another such space $M_r$; 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 $M_R$ and $M_r$ are diffeomorphic but not symplectomorphic.

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