  We study the cohomology and hence $K$-theory of the aperiodic tilings formed
by the so called 'cut and project' method, i.e., patterns in $d$ dimensional
Euclidean space which arise as sections of higher dimensional, periodic
structures. They form one of the key families of patterns used in quasicrystal
physics, where their topological invariants carry quantum mechanical
information. Our work develops both a theoretical framework and a practical
toolkit for the discussion and calculation of their integral cohomology, and
extends previous work that only successfully addressed rational cohomological
invariants. Our framework unifies the several previous methods used to study
the cohomology of these patterns. We discuss explicit calculations for the main
examples of icosahedral patterns in $R^3$ -- the Danzer tiling, the
Ammann-Kramer tiling and the Canonical and Dual Canonical $D_6$ tilings,
including complete computations for the first of these, as well as results for
many of the better known 2 dimensional examples.
