  The integer Cech cohomology of canonical projection tilings of dimension
three and codimension three is derived. These formulae are then evaluated for
several icosahedral tilings known from the literature. Rather surprisingly, the
cohomologies of all these tilings turn out to have torsion. This is the case
even for the Danzer tiling, which is, in some sense, the simplest of all
icosahedral tilings. This result is in contrast to the case of two-dimensional
canonical projection tilings, where many examples without torsion are known.
