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.

中文翻译：

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十二折平铺空间的上同调群

我们考虑通过切割和投影方法获得的具有十二重对称性的平面平铺。我们使用 [9] 中介绍的技术计算它们的上同调群。为此，我们完整地描述了窗口、群作用下线的轨道以及 0-奇点的轨道。可以使用两个参数来描述完整的广义十二折平铺系列，并且它呈现出惊人的丰富上同调结构。为了正确看待这一发现，我们应该将我们的结果与广义五重平铺（通常称为广义彭罗斯平铺）的上同调进行比较。在这种情况下，平铺形成一个单参数族，它仅适合两种类型的上同调中的一种。