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Projected Wallpaper Patterns

  • Isabel S. Labouriau
  • Eliana M. Pinho
Conference paper
  • 515 Downloads
Part of the Trends in Mathematics book series (TM)

Abstract

Consider a periodic function f of two variables with symmetry Γ and let ⊂ Γ be the subgroup of translations. The Fourier expansion of a periodic function is a sum over *, the dual of the set of all the periods of f. After projecting f, some of its original symmetry remains. We describe the symmetries of the projected function, starting from Γ and from the structure of *.

Keywords

Tilings in two dimensions crystallographic groups periodic solutions group-invariant bifurcations 

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References

  1. [1]
    Armstrong, M.A., Groups and Symmetry, Springer-Verlag, 1988.Google Scholar
  2. [2]
    Bosch Vivancos, I., Chossat, P. and Melbourne, I., New planforms in systems of partial differential equations with Euclidean symmetry, Arch. Rat. Mech. Anal. 131 (1995) 199–224.CrossRefMathSciNetGoogle Scholar
  3. [3]
    Chillingworth, D. and Golubitsky, M., Symmetry and pattern formation for a planar layer of nematic liquid crystal, J. Math. Phys., Vol. 44, No. 9 (2003) 4201–4219.CrossRefMathSciNetGoogle Scholar
  4. [4]
    Dionne, B. and Golubitsky, M., Planforms in two and three dimensions, Z. Angew. Math. Phys. 43 (1992) 36–62.CrossRefMathSciNetGoogle Scholar
  5. [5]
    Dionne, B., Silber, M. and Skeldon, A.C. Stability results for steady, spatially periodic planforms, Nonlinearity 10 (1997) 321–353.CrossRefMathSciNetGoogle Scholar
  6. [6]
    Golubitsky, M. and Stewart, I., The Symmetry Perspective, Prog. Math. 200, Birkhäuser Verlag, 2002Google Scholar
  7. [7]
    Gomes, M.G.M., Black-eye patterns: A representation of three-dimensional symmetries in thin domains, Phys. Rev. E 60 (1999) 3741–3747.CrossRefMathSciNetGoogle Scholar
  8. [8]
    Melbourne, I., Steady-state bifurcation with Euclidean symmetry, Trans. Amer. Math. Soc. 351 (1999) 1575–1603.CrossRefMathSciNetGoogle Scholar
  9. [9]
    Pinho, E.M., Symmetries of Projected Symmetric Patterns, Thesis, University of Porto — in preparation.Google Scholar
  10. [10]
    Rucklidge, A.M., Silber, M. and Fineberg, J., Secondary instabilities of hexagons: a bifurcation analysis of experimentally observed Faraday wave patterns, in Bifurcation, Symmetry and Patterns (Buescu, J., Castro, S., Dias, A.P. and Labouriau, I. eds.), Trends in Mathematics, Birkhäuser Verlag (2003) 101–114.Google Scholar
  11. [11]
    Senechal, M., Quasicrystals and Geometry, Cambridge University Press, 1995.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2006

Authors and Affiliations

  • Isabel S. Labouriau
    • 1
  • Eliana M. Pinho
    • 2
  1. 1.Centro de Matemática da Universidade do PortoPortoPortugal
  2. 2.Departamento de Matemática AplicadaFaculdade de Ciências da Universidade do PortoPortoPortugal

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