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Perfect colourings of cyclotomic integers

Abstract

Perfect colourings of the rings of cyclotomic integers of class number one are studied. It is shown that all colourings induced by ideals (q) are chirally perfect, and vice versa. A necessary and sufficient condition for a colouring to be perfect is obtained, depending on the factorisation of q. This result yields the colour symmetry group H in general. Furthermore, the colour preserving group K is determined in all but finitely many cases. An application to colourings of quasicrystals is given.

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Correspondence to D. Frettlöh.

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Bugarin, E.P., de las Peñas, M.L.A.N. & Frettlöh, D. Perfect colourings of cyclotomic integers. Geom Dedicata 162, 271–282 (2013). https://doi.org/10.1007/s10711-012-9726-0

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  • DOI: https://doi.org/10.1007/s10711-012-9726-0

Keywords

  • Colour symmetry
  • Cyclotomic fields
  • Tilings
  • Quasicrystals

Mathematics Subject Classification

  • 05C25
  • 52C23
  • 11R18