The space of Penrose tilings and the noncommutative curve with homogeneous coordinate ring $k⟨x,y⟩/(y^2)$

Abstract

It is shown that the noncommutative algebraic curve with homogeneous coordinate ring $\mathbb{C}⟨x,y⟩/(y^2)$ is a noncommutative algebraic-geometric analogue of the space of Penrose tilings of the plane. Individual tilings determine “points” on the noncommutative curve and the tilings coincide under isometry if and only if the skyscraper sheaves of the corresponding points are isomorphic. The category of quasi-coherent sheaves on the curve is equivalent to the category of modules over a von Neumann regular ring that is a direct limit of finite dimensional semisimple algebras. The norm closure of this von Neumann regular ring is the AF-algebra that Connes associates to the space of Penrose tilings. There is an algebraic analogue of the fact that every isometry-invariant subset of tilings is dense in the set of all Penrose tilings.