  The purpose of this paper is to show the relationship in all dimensions
between the structural (diffraction pattern) aspect of tilings (described by
\v{C}ech cohomology of the tiling space) and the spectral properties (of
Hamiltonians defined on such tilings) defined by $K$-theory, and to show their
equivalence in dimensions $\leq 3$. A theorem makes precise the conditions for
this relationship to hold. It can be viewed as an extension of the "Bloch
Theorem" to a large class of aperiodic tilings. The idea underlying this result
is based on the relationship between cohomology and $K$-theory traces and their
equivalence in low dimensions.
