Abstract
Abstract We consider tilings of the plane with twelve-fold symmetry obtained by the cut-and-projection method. We compute their cohomology groups using the techniques introduced in [ 9]. To do this, we completely describe the window, the orbits of lines under the group action, and the orbits of 0-singularities. The complete family of generalized twelve-fold tilings can be described using two-parameters and it presents a surprisingly rich cohomological structure. To put this finding into perspective, one should compare our results with the cohomology of the generalized five-fold tilings (more commonly known as generalized Penrose tilings). In this case, the tilings form a one-parameter family, which fits in simply one of the two types of cohomology.
Highlights
This paper deals with tilings of the plane R2
Cohomology has been an essential tool of algebraic topology
Worth mentioning are the class of substitution tilings and the class of cut-and-projection tilings
Summary
This paper deals with tilings of the plane R2. Given a tiling, the group of translations of the plane acts on it, allowing us to associate with each tiling a space, the hull of the tiling, defined as the closure of the orbit of the tiling under the group action. Worth mentioning are the class of substitution tilings and the class of cut-and-projection tilings For the former one, there is a well-developed technique that allows to compute the cohomology groups of these tilings [1, 11]. For the latter ones, in spite of the general methods described by Gähler et al [9] to compute these groups, very few examples have been treated in detail. We are going to focus on a two-parameter family of cut-andproject tiling spaces: the generalized twelve-fold tilings The hull of these tilings will be denoted by E1γ2 with γ ∈ R2 (the notation and construction methods are explained in Sections 1.2 and 1.3).
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