Cohomology Groups for Spaces of Twelve-Fold Tilings

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.