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Perfect precise colorings of plane semiregular tilings
Acta Crystallographica Section A: Foundations and Advances
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This paper studies colorings of patterns with multiple orbits, particularly those colorings where the orbits share colors. The main problem is determining when such colorings become perfect. This problem is attacked by characterizing all perfect colorings of patterns through the construction of sufficient and necessary conditions for a coloring to be perfect. These results are then applied on symmetrical objects to construct both perfect and non-perfect colorings.
We consider the problem of colouring the regular triangular tiling {3,n} with n colours, in such a way that one tile of each colour occurs at each vertex. Such a tiling will be called precise, and the most interesting precise colourings are those that are also perfect. We shall find a method of enumerating a large class of perfect precise colourings, and shall also consider various related problems. Other aspects of the subject are discussed in N. Yaz [Perfect Precise colourings of regular triangular tilings, M. Phil. Dissertation, Cardiff: University of Wales, Cardiff, 117 pp. (1997)] and J. F. Rigby [Math. Intell. 20, No. 1, 4-11 (1998; Zbl 0917.52017)], which were written after the present article was first submitted.
Let (p(q)) denote the tessellation of the plane by regular p-gons meeting q at a vertex. In this paper, we consider the problem of coloring (p(q)) using q colors such that all the q colors appear at every vertex, and that every symmetry (or every direct symmetry) of the tiling (p(q)) induces a permutation on the set of q colors. Such a coloring is called a perfect precise coloring.
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 important class of colourings of regular tilings (and some Laves tilings) of the Euclidean or hyperbolic plane, this mainly combinatorial question is addressed here using group theoretical methods.