Perfect colourings of cyclotomic integers

Abstract

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.

Perfect precise colorings of plane semiregular tilings

“…If there is only one orbit of tiles under the symmetry group of T , then the following theorem (with X ¼ T and H ¼ G) characterizes perfect colorings of T as partitions of T obtained from the orbit of a tile in T by some subgroup of the symmetry group of T (cf. De Las Pen ˜as et al, 2006;Bugarin et al, 2013). The theorem is applicable not only to perfect colorings but also to chirally perfect colorings.…”

Perfect precise colorings of plane regular tilings

“…If there is only one orbit of tiles under the symmetry group of T , then the following theorem (with X = T and H = G) characterizes perfect colorings of T as partitions of T obtained from the orbit of a tile in T by some subgroup of the symmetry group of T (cf. [4,1]). The theorem is applicable not only to perfect colorings but also to chirally perfect colorings.…”

“…Observe from the proof of Theorem 2.3 that the subgroup J in (1) consists of elements of H that fix the color of tile t, or equivalently, J is the stabilizer of the set Jt under the action of H on the elements of the partition of X in (1).…”

“…Of particular interest are perfect colorings of tilings, that is, colorings where every symmetry of the uncolored tiling sends all tiles of a given color to tiles of the same color [7,9]. In addition, colorings of patterns in the hyperbolic plane have garnered attention because of their connection with quasicrystals and structural chemistry [4,1,10,16].…”

Symmetries and color symmetries of a family of tilings with a singular point

“…4 ),(6 3 ) and(3 6 ) tilings, respectively. The following proposition states that if α is an admissible value, then replacing α by any of its associates in (1) yields the same tiling S α .…”