On color groups of Bravais colorings of planar modules with quasicrystallographic symmetries

Abstract

Abstract. In this work we study the color symmetries pertaining to colorings of M n ¼ Z½x, where x ¼ exp ð2pi=nÞ for n 2 f5; 8; 12g which yield standard symmetries of quasicrystals. The first part of the paper treats M n as a four dimensional lattice L with symmetry group G and a result is provided on sublattices of L which are invariant under the point group of G. The second part of the paper characterizes the color symmetry groups and color fixing groups corresponding to Bravais colorings of M n using an approach involving ideals.

Colourings of cyclotomic integers with class number one

“…Thus, ð1 À 5 15 Þ 2 ðqÞ and so, ' j ½M 15 : ð1 À 5 15 Þ ¼ N n ð1 À 5 15 Þ ¼ N n ð1 À 3 Þ ¼ 3 4 ¼ 81. Hence, if ' B 81, then we have R 3 = 2 K. facilitate the characterisation of its structure in addition to what has already been determined in previous work [10,11,13]. Moreover, the results make the derivation of K more straightforward for a large number of cases.…”

“…Our previous work derived the colour symmetry group for any n and focussed on determining the colour preserving group for the crystallographic cases: n ¼ 3, 4 [1], as well as the non-periodic cases: n ¼ 5, 8, 12 (involving standard quasicrystallographic symmetries) [10] and n ¼ 7, 9 [11]. In this note, we enumerate the colour preserving group for the cases n ¼ 11, 15,16,20,24, thus completing the characterisation of colour symmetries of ideal colourings associated with M n for (n) 10.…”

“…In these tables, many entries under H and K follow immediately from the results we have established (also, see [11]). Entries in brackets require further computations (compare [10,13]), depending on the case. For instance, if n ¼ 16, ' ¼ 4 (see Table 2), one checks that the four cosets are invariant under R 8 but not R 16 .…”

Perfect colourings of cyclotomic integers

“…A further object of interest is the subgroup K of H which fixes the colours. For a given X ⊂ R 2 and a colouring c of X, we consider the colour preserving group K (in [4] called colour fixing group):…”

“…The cases n = 6, 14, 18 are covered implicitly.) This table can be seen as a complement to Table 4 in [4], where values of n for which φ(n) = 4 are considered. The entries in the second and third column follow from [2].…”