Journal of Symplectic Geometry

Volume 6 (2008)

Number 2

The Symplectic Geometry of Penrose Rhombus Tilings

Pages: 139 – 158

DOI: http://dx.doi.org/10.4310/JSG.2008.v6.n2.a2

Authors

Fiammetta Battaglia

Elisa Prato

Abstract

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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