The article ”A regular pentagonal tiling of the plane” by Philip L. Bowers and Kenneth Stephenson defines a conformal pentagonal tiling. This is a tiling of the plane with remarkable combinatorial and geometric properties.However, it doesn’t have finite local complexity in any usual sense, and therefore we cannot study it with the usual tiling theory. The appeal of the tiling is that all the tiles are conformally regular pentagons. But conformal maps are not allowable under finite local complexity. On the other hand, the tiling can be described completely by its combinatorial data, which rather automatically has finite local complexity. In this thesis we give a construction of the continuous and discrete hull just from the combinatorial data.
For the discrete hull we construct a C-algebra and a measure. Since this tiling possesses no natural R2 action by translation, there is no a priori reason to expect that the K-theory of the C-algebra of the tiling is the same as the K-theory or cohomology of the hull. So it would be very interesting to know the outcome. For the continuous hull, we compute its K-theory and an absolute continuous invariant measure