Cohomology groups for spaces of 12-fold tilings

TL;DR

This paper computes the cohomology groups of 12-fold symmetric tilings created by cut and projection, revealing a complex structure that contrasts with the simpler cohomology of 5-fold tilings.

Contribution

It provides a complete description of the cohomology of 12-fold tilings, using detailed analysis of the window and orbit structures, and introduces a 2-parameter family of such tilings.

Findings

Cohomology groups are rich and complex for 12-fold tilings.

The family of 12-fold tilings is parametrized by two parameters.

Compared to 5-fold tilings, 12-fold tilings exhibit more intricate cohomological behavior.

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 by the second author, Hunton and Kellendonk. 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.