Geometriae Dedicata

, Volume 162, Issue 1, pp 271–282

Perfect colourings of cyclotomic integers

  • E. P. Bugarin
  • M. L. A. N. de las Peñas
  • D. Frettlöh
Original Paper

DOI: 10.1007/s10711-012-9726-0

Cite this article as:
Bugarin, E.P., de las Peñas, M.L.A.N. & Frettlöh, D. Geom Dedicata (2013) 162: 271. doi:10.1007/s10711-012-9726-0


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.


Colour symmetry Cyclotomic fields Tilings Quasicrystals 

Mathematics Subject Classification

05C25 52C23 11R18 

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  • E. P. Bugarin
    • 1
  • M. L. A. N. de las Peñas
    • 2
  • D. Frettlöh
    • 3
  1. 1.Fakultät für MathematikUniversität BielefeldBielefeldGermany
  2. 2.Mathematics DepartmentAteneo de Manila UniversityQuezon CityPhilippines
  3. 3.Institut für MathematikBerlinGermany

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