Cohomology groups for spaces of 12-fold tilings

Nicolas Bedaride (1) , Franz Gähler , Ana G Lecuona

1 I2M - Institut de Mathématiques de Marseille

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.

Domaines: Mathématiques [math]; Systèmes dynamiques [math.DS]; Combinatoire [math.CO]; Mathématique discrète [cs.DM]

Nicolas Bedaride, Franz Gähler, Ana G Lecuona. Cohomology groups for spaces of 12-fold tilings. International Mathematics Research Notices, In press, ⟨10.1093/imrn/rnab117/6291057⟩. ⟨hal-03171546⟩