Bravais colourings of planar modules with N-fold symmetry

Abstract

Abstract. The first step in investigating colour symmetries for periodic and aperiodic systems is the determination of all colouring schemes that are compatible with the symmetry group of the underlying structure, or with a subgroup of it. For an important class of colourings of planar structures, this mainly combinatorial question can be addressed with methods of algebraic number theory. We present the corresponding results for all planar modules with N -fold symmetry that emerge as the rings of integers in cyclotomic fields with class number one. The counting functions are multiplicative and can be encapsulated in Dirichlet series generating functions, which turn out to be the Dedekind zeta functions of the corresponding cyclotomic fields. Key Words:Colourings, Planar Modules, Cyclotomic Fields, Dirichlet Series IntroductionColour symmetries of crystals [23,24,21,22,19] and, more recently, of quasicrystals [16,13,1] continue to attract a lot of attention, simply because so little is known about their classification, see [13] for a recent review. A first step in this analysis consists in answering the question of how many different colourings of an infinite point set exist which are compatible with its underlying symmetry. More precisely, one has to determine the possible numbers of colours, and to count the corresponding possibilities to distribute the colours over the point set (up to permutations), in line with all compatible symmetry constraints. In this generality, the problem has not even been solved for simple lattices. One common restriction is to demand that one colour occupies a subset which is of the same Bravais type as the original set, while the other colours encode the cosets. Of particular interest are the cases where the point symmetry is irreducible. In this situation, to which we shall also restrict ourselves, several results are known and can be given in closed form [1,5,6,13,14,19].Particularly interesting are planar cases because, on the one hand, they show up in quasicrystalline T -phases, and, on the other hand, they are linked to the rather interesting * Correspondence author (e-mail: mbaake@math.uni-bielefeld.de) classification of planar Bravais classes with n-fold symmetry [15], which is based on a connection to algebraic number theory in general, and to cyclotomic fields in particular. The Bravais types correspond to ideal classes and are unique for the following 29 choices of n, n ∈ {3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 24, 25, 27, 28, 32, 33, 35, 36, 40, 44, 45, 48, 60, 84}. ( 1) The canonical representatives are the sets of cyclotomic integers M n = Z[ξ n ], the ring of polynomials in ξ n , a primitive nth root of unity. To be explicit (which is not necessary), we choose ξ n = exp(2πi/n). Apart from n = 1 (where, the values of n in (1) correspond to all cases where Z[ξ n ] is a principal ideal domain and thus has class number one, see [26,6] for details. If n is odd, we have M n = M 2n . Consequently, M n has N -fold symmetry whereTo avoid duplication of resu...

Multiple planar coincidences with N-fold symmetry

“…Referring back to [6,22], and to the theory of Dirichlet characters summarized there, one can express the zeta functions in terms of L-series as follows,…”

“…As in [6,22], G n denotes the set of primitive Dirichlet characters for the field extension K n /Q, and thus contains φ(n) elements. The principal character in this formulation, χ 0 ≡ 1, always leads to L(s, χ 0 ) = ζ(s), i.e., to Riemann's zeta function itself.…”

“…II.7, Thm. 15] for the general formulation, and [6,Prop. 4] for the reduction to the situation at hand) leads to the following result.…”

“…In [6], we additionally used the class number formula to transform the calculation of res s=1 ζ Kn (s) into an algebraic problem. However, given the fact that ζ(s, a) and Γ(s) are accessible by means of algebraic program packages to arbitrary numerical precision, it seems easier to directly use formulas (25) and (26) together with (31) for these calculations.…”

“…Some values for α n and γ n , including exact expressions (except for γ 7 , see a previous footnote), are also contained in [6] and [17], respectively.…”