Colourings of cyclotomic integers with class number one

Abstract

This paper continues the study of colourings of the sets of cyclotomic integers ℳ _n  = ℤ[ξ _n ] (ξ _n  = e ^2πi/n , a primitive nth root of unity) with class number one. We present results for the colour symmetry group and colour preserving group for a given ideal colouring of ℳ _n , with φ(n) = 8 and 10, thus completing the characterisation of the colour preserving group for the cases φ(n) ≤ 10, where φ is Euler's totient function.

Keywords:

• colour symmetry
• ideal colourings
• Bravais colourings
• cyclotomic integers
• non-periodic structures

Acknowledgements

The authors thank Peter Zeiner and Christian Huck for helpful discussions. Ma. Louise de Las Peñas would like to acknowledge funding support of the Ateneo de Manila University through the Loyola Schools Scholarly Work Faculty Grant. Enrico Paolo Bugarin and Dirk Frettlöh are grateful to the CRC 701 of the German Research Council (DFG).