The Symplectic Geometry of Penrose Rhombus Tilings

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Battaglia, Fiammetta
Prato, Elisa
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Abstract
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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.
Comment: 22 pages, 11 figures. Minor improvements. To appear in J. Symplectic Geom
Keywords
Mathematics - Symplectic Geometry, Mathematical Physics, Mathematics - Geometric Topology, 53D20 (Primary) 52C23 (Secondary)
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