November 6, 2005
The coincidence problem for planar patterns with $N$-fold symmetry is considered. For the N-fold symmetric module with $N<46$, all isometries of the plane are classified that result in coincidences of finite index. This is done by reformulating the problem in terms of algebraic number fields and using prime factorization. The more complicated case $N \ge 46$ is briefly discussed and N=46 is described explicitly. The results of the coincidence problem also solve the problem of colour lattices in two dimensions and its natural generalization to colour modules.
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January 3, 2003
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 th...
September 11, 2016
A submodule of a $\mathbb{Z}$-module determines a coloring of the module where each coset of the submodule is associated to a unique color. Given a submodule coloring of a $\mathbb{Z}$-module, the group formed by the symmetries of the module that induces a permutation of colors is referred to as the color group of the coloring. In this contribution, a method to solve for the color groups of colorings of $N$-planar modules where $N=4$ and $N=6$ are given. Examples of colorings...
November 21, 2008
It is shown that the coincidence isometries of certain modules in Euclidean $n$-space can be decomposed into a product of at most $n$ coincidence reflections defined by their non-zero elements. This generalizes previous results obtained for lattices to situations that are relevant in quasicrystallography.
November 11, 2005
Planar coincidence site lattices and modules with N-fold symmetry are well understood in a formulation based on cyclotomic fields, in particular for the class number one case, where they appear as certain principal ideals in the corresponding ring of integers. We extend this approach to multiple coincidences, which apply to triple or multiple junctions. In particular, we give explicit results for spectral, combinatorial and asymptotic properties in terms of Dirichlet series g...
July 10, 2012
Given a Bravais colouring of planar modules $\M_n:=\Z[\xi_n]$, where $\xi_n$ is a primitive $n$th root of unity, two important colour groups arise: the colour symmetry group $H$, which permutes the colours of a given colouring of $\M_n$, and the colour preserving group $K$, a normal subgroup of $H$ that fixes the colours. This paper gives a complete characterisation of $H$ and $K$ for all $\ell$-colourings of $\M_n$ for values of $n$ for which $\M_n$ has class number one.
October 31, 2001
The investigation of colour symmetries for periodic and aperiodic systems consists of two steps. The first concerns the computation of the possible numbers of colours and is mainly combinatorial in nature. The second is algebraic and determines the actual colour symmetry groups. Continuing previous work, we present the results of the combinatorial part for planar patterns with n-fold symmetry, where n=7,9,15,16,20,24. This completes the cases with values of n such that Euler'...
February 2, 2010
If $G$ is the symmetry group of an uncolored pattern then a coloring of the pattern is semiperfect if the associated color group $H$ is a subgroup of $G$ of index 2. We give results on how to identify and enumerate all inequivalent semiperfect colorings of certain patterns. This is achieved by treating a coloring as a partition $\{hJ_iY_i:i\in I,h\in H\}$ of $G$, where $H$ is a subgroup of index 2 in $G$, $J_i\leq H$ for $i\in I$, and $Y=\cup_{i\in I}{Y_i}$ is a complete set ...
May 10, 2019
A two-dimensional configuration is a coloring of the infinite grid Z^2 with finitely many colors. For a finite subset D of Z^2, the D-patterns of a configuration are the colored patterns of shape D that appear in the configuration. The number of distinct D-patterns of a configuration is a natural measure of its complexity. A configuration is considered having low complexity with respect to shape D if the number of distinct D-patterns is at most |D|, the size of the shape. Thi...
October 19, 2010
In this work, a theory of color symmetry is presented that extends the ideas of traditional theories of color symmetry for periodic crystals to apply to non-periodic crystals. The color symmetries are associated to each of the crystalline sites and may correspond to different chemical species, various orientations of magnetic moments and colorings of a non-periodic tiling. In particular, we study the color symmetries of periodic and non-periodic structures via Bravais colorin...
July 29, 2008
A first step in investigating colour symmetries of periodic and nonperiodic patterns is determining the number of colours which allow perfect colourings of the pattern under consideration. A perfect colouring is one where each symmetry of the uncoloured pattern induces a global permutation of the colours. Two cases are distinguished: Either perfect colourings with respect to all symmetries, or with respect to orientation preserving symmetries only (no reflections). For the im...