Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 9 (2013), 021, 13 pages      arXiv:1004.2471      http://dx.doi.org/10.3842/SIGMA.2013.021

Ammann Tilings in Symplectic Geometry

Fiammetta Battaglia a and Elisa Prato b
a) Dipartimento di Matematica e Informatica ''U. Dini'', Via S. Marta 3, 50139 Firenze, Italy
b) Dipartimento di Matematica e Informatica ''U. Dini'', Piazza Ghiberti 27, 50122 Firenze, Italy

Received November 09, 2012, in final form February 27, 2013; Published online March 06, 2013

Abstract
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.

Key words: symplectic quasifold; nonperiodic tiling; quasilattice.

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References

  1. Battaglia F., Prato E., Generalized toric varieties for simple nonrational convex polytopes, Int. Math. Res. Not. (2001), 1315-1337, math.CV/0004066.
  2. Battaglia F., Prato E., The symplectic geometry of Penrose rhombus tilings, J. Symplectic Geom. 6 (2008), 139-158, arXiv:0711.1642.
  3. Battaglia F., Prato E., The symplectic Penrose kite, Comm. Math. Phys. 299 (2010), 577-601, arXiv:0712.1978.
  4. Conway J.H., Sloane N.J.A., Sphere packings, lattices and groups, Grundlehren der Mathematischen Wissenschaften, Vol. 290, 3rd ed., Springer-Verlag, New York, 1999.
  5. Delzant T., Hamiltoniens périodiques et images convexes de l'application moment, Bull. Soc. Math. France 116 (1988), 315-339.
  6. Levine D., Steinhardt P.J., Quasicrystals: a new class of ordered structures, Phys. Rev. Lett. 53 (1984), 2477-2480.
  7. Mackay A.L., De nive quinquangula: on the pentagonal snowflake, Sov. Phys. Cryst. (1981), 517-522.
  8. Prato E., Simple non-rational convex polytopes via symplectic geometry, Topology 40 (2001), 961-975, math.SG/9904179.
  9. Rokhsar D.S., Mermin N.D., Wright D.C., Rudimentary quasicrystallography: the icosahedral and decagonal reciprocal lattices, Phys. Rev. B 35 (1987), 5487-5495.
  10. Senechal M., The mysterious Mr. Ammann, Math. Intelligencer 26 (2004), 10-21.
  11. Shechtman D., Blech I., Gratias D., Cahn J.W., Metallic phase with long-range orientational order and no translational symmetry, Phys. Rev. Lett. 53 (1984), 1951-1953.
  12. Steurer W., Deloudi S., Crystallography of quasicrystals: concepts, methods and structures, Springer Series in Materials Science, Vol. 126, Springer-Verlag, Berlin, 2009.

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