Has anyone produced a quasiperiodic tiling of the hyperbolic plane?
Or is there a reason it cannot be done?
By quasiperiodic I mean that the structure is not strictly periodic (i.e. equal to itsef after translation) but that any arbitrary large neighbourhood of any point can be found identically at an infinity of other locations.
Yes, this question has been somewhat studied, for instance by Chaim Goodman-Strauss. See this paper of his. See also this paper and references in both. Below is an image from the second paper which gives the first step in building a strongly aperiodic set of tiles in the hyperbolic plane, which I think Chaim would be ok with me copying here.
Perhaps one of the most important points brought up in this work is that the notion of aperiodicity or quasiperiodicity in the hyperbolic setting is more subtle than in the Euclidean case, and one should be careful with the definition being used (thus the use of 'weakly aperiodic' and 'strongly aperiodic').