  We calculate the growth rate of the complexity function for polytopal cut and
project sets. This generalises work of Julien where the almost canonical
condition is assumed. The analysis of polytopal cut and project sets has often
relied on being able to replace acceptance domains of patterns by so-called cut
regions. Our results correct mistakes in the literature where these two notions
are incorrectly identified. One may only relate acceptance domains and cut
regions when additional conditions on the cut and project set hold. We find a
natural condition, called the quasicanonical condition, guaranteeing this
property and demonstrate via counterexample that the almost canonical condition
is not sufficient for this. We also discuss the relevance of this condition for
the current techniques used to study the algebraic topology of polytopal cut
and project sets.
