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Perfect colourings of cyclotomic integers

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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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  1. Baake, M.: Combinatorial aspects of colour symmetries. J. Phys. A: Math. Gen. 30 (1997) 2687–2698, mp_arc/02-323

    Google Scholar 

  2. Baake, M., Grimm, U.: Bravais colourings of planar modules with N-fold symmetry. Z. Krist. 219 (2004) 72–80, math.CO/0301021

    Google Scholar 

  3. Baake, M., Grimm, U., Scheffer, M.: Colourings of planar quasicrystals. J. Alloys Compounds 342 (2002) 195–197, cond-mat/0110654

    Google Scholar 

  4. Bugarin E.P., de las Peñas M.L.A.N., Evidente I., Felix R.P., Frettlöh D.: On color groups of Bravais colorings of planar modules with quasicrystallographic symmetry. Z. Krist. 223, 785–790 (2008)

    Article  Google Scholar 

  5. Conway J.H., Burgiel H., Goodman-Strauss C.: The Symmetries of Things. AK Peters, Wellesley (2008)

    MATH  Google Scholar 

  6. de van Peñas M.L.A.N., Felix R.P., Laigo G.R.: Colorings of hyperbolic plane crystallographic patterns. Z. Krist. 221, 665–672 (2006)

    Article  MathSciNet  Google Scholar 

  7. Dräger J., Mermin N.D.: Superspace groups without the embedding: The link between superspace and Fourier-space crystallography. Phys. Rev. Lett. 76, 1489–1492 (1996)

    Article  Google Scholar 

  8. Grünbaum B., Shephard G.C.: Tilings and Patterns. Freeman, New York (1987)

    MATH  Google Scholar 

  9. Lifshitz R.: Theory of color symmetry for periodic and quasiperiodic crystals. Rev. Mod. Phys. 69, 1181–1218 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  10. Lück R.: Colour symmetry of 25 colours in quasiperiodic patterns. Phil. Mag. 88, 2049–2058 (2008)

    Article  Google Scholar 

  11. Mermin N.D.: Copernican crystallography. Phys. Rev. Lett. 68, 1172–1175 (1992)

    Article  MathSciNet  Google Scholar 

  12. Moody R.V., Patera J.: Colourings of quasicrystals. Can. J. Phys. 72, 442–452 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  13. Schwarzenberger R.L.E.: Colour symmetry. Bull. Lond. Math. Soc. 16, 209–240 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  14. Senechal M.: Color groups. Discrete Appl. Math. 1, 51–73 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  15. Senechal M.: Quasicrystals and Geometry. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

  16. van der Waerden B.L., Burckhardt J.J.: Farbgruppen. Z. Krist. 115, 231–234 (1961)

    Article  Google Scholar 

  17. Washington L.C.: Introduction to Cyclotomic Fields. Springer, New York (1996)

    Google Scholar 

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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).

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