Cohomology groups for spaces of 12-fold tilings

Abstract

We consider tilings of the plane with 12-fold symmetry obtained by the cut and projection method. We compute their cohomology groups using the techniques introduced in [5]. 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 12-fold tilings can be described using 2-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 5-fold tilings (more commonly known as generalized Penrose tilings). In this case the tilings form a 1-parameter family, which fits in simply one of two types of cohomology.