Geodesic
Ammann Tilings in Symplectic Geometry
Symmetry, integrability and geometry: methods and applications, Tome 9 (2013) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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.
Keywords: symplectic quasifold; nonperiodic tiling; quasilattice. 
@article{SIGMA_2013_9_a20,
     author = {Fiammetta Battaglia and Elisa Prato},
     title = {Ammann {Tilings} in {Symplectic} {Geometry}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2013},
     volume = {9},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a20/}
}
TY  - JOUR
AU  - Fiammetta Battaglia
AU  - Elisa Prato
TI  - Ammann Tilings in Symplectic Geometry
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2013
VL  - 9
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a20/
LA  - en
ID  - SIGMA_2013_9_a20
ER  - 
%0 Journal Article
%A Fiammetta Battaglia
%A Elisa Prato
%T Ammann Tilings in Symplectic Geometry
%J Symmetry, integrability and geometry: methods and applications
%D 2013
%V 9
%U http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a20/
%G en
%F SIGMA_2013_9_a20
Fiammetta Battaglia; Elisa Prato. Ammann Tilings in Symplectic Geometry. Symmetry, integrability and geometry: methods and applications, Tome 9 (2013). http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a20/

[1] Battaglia F., Prato E., “Generalized toric varieties for simple nonrational convex polytopes”, Int. Math. Res. Not., 2001, 1315–1337, arXiv: math.CV/0004066 | DOI | MR

[2] Battaglia F., Prato E., “The symplectic geometry of Penrose rhombus tilings”, J. Symplectic Geom., 6 (2008), 139–158, arXiv: 0711.1642 | MR | Zbl

[3] Battaglia F., Prato E., “The symplectic Penrose kite”, Comm. Math. Phys., 299 (2010), 577–601, arXiv: 0712.1978 | DOI | MR | Zbl

[4] Conway J. H., Sloane N. J. A., Sphere packings, lattices and groups, Grundlehren der Mathematischen Wissenschaften, 290, 3rd ed., Springer-Verlag, New York, 1999 | MR | Zbl

[5] Delzant T., “Hamiltoniens périodiques et images convexes de l'application moment”, Bull. Soc. Math. France, 116 (1988), 315–339 | MR | Zbl

[6] Levine D., Steinhardt P. J., “Quasicrystals: a new class of ordered structures”, Phys. Rev. Lett., 53 (1984), 2477–2480 | DOI

[7] Mackay A. L., “De nive quinquangula: on the pentagonal snowflake”, Sov. Phys. Cryst., 1981, 517–522 | MR

[8] Prato E., “Simple non-rational convex polytopes via symplectic geometry”, Topology, 40 (2001), 961–975, arXiv: math.SG/9904179 | DOI | MR | Zbl

[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 | DOI | MR

[10] Senechal M., “The mysterious Mr. Ammann”, Math. Intelligencer, 26 (2004), 10–21 | DOI | MR | Zbl

[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 | DOI

[12] Steurer W., Deloudi S., Crystallography of quasicrystals: concepts, methods and structures, Springer Series in Materials Science, 126, Springer-Verlag, Berlin, 2009 | DOI