Perfect colourings of cyclotomic integers

TL;DR

This paper characterizes perfect colourings of cyclotomic integer rings with class number one, establishing conditions based on ideal factorization, and explores their symmetry groups with applications to quasicrystals.

Contribution

It provides a complete characterization of perfect colourings induced by ideals in cyclotomic integer rings and determines their symmetry groups.

Findings

All colourings from ideals are chirally perfect.

A necessary and sufficient condition for perfect colourings is derived.

The colour symmetry group H and the colour preserving group K are explicitly determined.

Abstract

Perfect colourings of the rings of cyclotomic integers with 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.