Color groups

Senechal, Marjorie

doi:10.1016/0166-218x(79)90014-3

Cited by 46 publications

(18 citation statements)

References 8 publications

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“…In addition, one of the sets is a similarity submodule of M n . The interest in this kind of partition originates from its rather important grouptheoretical structure, compare [21,22,23,24,13] and references therein. Also, each such partition induces a colouring of tilings that are compatible with the module (usually formulated in terms of local indistinguishability, compare [13]), see [3,5,20] for examples.…”

Section: Number Theoretic Formulationmentioning

confidence: 99%

“…Colour 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.…”

Section: Introductionmentioning

confidence: 99%

“…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}.…”

Section: Introductionmentioning

confidence: 99%

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Bravais colourings of planar modules with N-fold symmetry

Baake¹,

Grimm²

2004

Zeitschrift Für Kristallographie - Crystalline Materials

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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...

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Section: Number Theoretic Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Bravais colourings of planar modules with N-fold symmetry

Baake¹,

Grimm²

2004

Zeitschrift Für Kristallographie - Crystalline Materials

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“…The study of color symmetries of periodic [8,9,11,[12][13][14] and aperiodic [1][2][3]11] structures remains an interesting area of investigation. More recently in [8], color groups associated with colorings arising from sublattices of square and hexagonal lattices have been discussed.…”

Section: Introductionmentioning

confidence: 99%

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

Bugarin¹,

Peñas

Evidente

et al. 2008

Zeitschrift Für Kristallographie

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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.

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“…Rather, a local antiferromagnetic symmetry emerges as a pattern of oriented spins. Symbolizing each local orientation by colour, the symmetry of the spin crystal may be described by the crystal colour group (g, γ) -combinations of translations and rotations and permutations of the colours that leave the coloured crystal invariant [26][27][28][29]. This antiferromagnetic symmetry characterizes antiferromagnetic phases.…”

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confidence: 99%

A uniform approach to antiferromagnetic Heisenberg spins on low-dimensional lattices

Benoit¹,

Dandoloff²

2000

Physics Letters A

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Using group theoretical methods we show for both the triangular and square lattices that in the continuum limit the antiferromagnetic order parameter lives on SO(3) without respect of the initial lattice. For the antiferromagnetic chain we recover the Haldane decomposition. This order parameter interacts with a local gauge field rather than with a global one as implicitly suggested in the literature which in our approach appears in a rather natural manner. In fact this merely corresponds to a novel extension of the spin group by a local gauge field. This analysis based on the real division algebras A d applies to low dimensional lattices (d = 1, 2, 3, 4). c 2000 Elsevier Science B. V. All rights reserved. 75.50.Ee, Since the discovery of the high temperature superconductivity bidimensional antiferromagnetic Heisenberg models and their continuum limit have attracted considerable interest [1][2][3][4][5]. The problem of the continuum limit is directly related to the presence or not of a Hopf term in the Lagrangian of the bidimensional antiferromagnetic Heisenberg model [6][7][8]. This problem is still open [9,10].The continuum limit of the bidimensional antiferromagnetic Heisenberg model is much more difficult to achieve than may be expected in particular for the triangular and hexagonal lattices. This difficulty has led the authors to outline the construction of novel local antiferromagnetic degrees of freedom for low dimensional antiferromagnetic lattices. This new construction naturally reproduces the real division algebra hierarchy [11][12][13] satisfied by the nonlinear σ-model [8,14,15] (which is accepted to correspond to the continuum limit of the Heisenberg Hamiltonian for classical isotropic antiferromagnets in two dimensions [16][17][18][19][20]). Real division algebras are known as powerful tools to describe regular tessellations in low dimension (the complex algebra C on plane and the quaternion algebra H in space [13,21]) and interactions in particle physics [12,14]. This powerfulness is employed to introduce effortlessly new degrees of freedom. This algebraic construction suggests an original microscopic mechanism and reveals the presence of a novel local gauge field which is expected to give rise to a Hopf term in the continuum limit [22]. In the present paper we will focus on the antiferromagnetic chain, the antiferromagnetic triangular and square lattices only.First, let us consider compact regular tessellations partitioning low dimensional Euclidian space (d = 1, 2, 3, 4) with identical regular polytopes [23]. Then each crystal will be characterized by a regular polytope: the crystal will have the dimension d of the regular polytope, the symmetries which leave the regular polytope invariant will leave the crystal invariant, the translations which reproduce the crystal with a single regular polytope will leave the crystal coinciding with itself, and the length of the regular polytope edge will correspond to the crystal step a. Implicitly we have associated to each crystal site (i) a spin operator...

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Color groups

Cited by 46 publications

References 8 publications

Bravais colourings of planar modules with N-fold symmetry

Bravais colourings of planar modules with N-fold symmetry

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

A uniform approach to antiferromagnetic Heisenberg spins on low-dimensional lattices

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