N } of closed bounded subsets of Rd homeomorphic to the unit ball. These are called the prototiles. One usually assumes that the prototiles are polytopes in Rd with a single d-dimensional cell which is the interior of the prototile, but this assumption can be relaxed. e. a translate of one of the prototiles. Given a tiling T of Rd one can form its orbit closure under translations. g. [4], [20]). Tilings can be periodic or aperiodic. There are many familiar examples of periodic tilings, while the best known examples of aperiodic tilings are the Penrose tilings [175].

The inverse of the Dirac operator, and r the scalar curvature. We obtain: − ds2 = −1 48π 2 r dv . 11) M4 In general, one obtains the scalar curvature of an n-dimensional manifold from the integral −dsn−2 . 6. However, there are significant cases where more refined properties of manifolds carry over to the noncommutative case, such as the presence of a real structure (which makes it possible to distinguish between K-homology and KO-homology) and the “order one condition” for the Dirac operator. These properties are described as follows (cf.

In fact, as we discussed in Section 2, one of the fundamental construction of noncommutative geometry (cf. [58]) is that of homotopy quotients. These are commutative spaces which provide, up to homotopy, geometric models for the corresponding noncommutative spaces. The noncommutative spaces themselves, as we are going to show in our case, appear as quotient spaces of foliations on the homotopy quotients with contractible leaves. 5), T ST = S ×Z R . 8) whose generic leaf is contractible (a copy of R).