Local rules for quasi-periodic tilings

The tilings of any dimension $d$ and codimension $d'$ are considered. Such tilings are obtained as sections of a periodic hyper-tiling $\subset\mathbb{R}^D$ by $d$-dimensional subspace $E$ of the hyperspace $\mathbb{R}^{D}$ of dimension $D=d+d'$. By using the projection of the unit $D$-dimensional cube to the space $E'$ orthogonal to $E$, local matching rules are established that determine the local structure of the tiling. In general, the tilings considered may contain ramificated vertices. In the multi-faceted stars of such vertices the polyhedra can overlap each other. A regularization algorithm is given that allows the selection of one of the polyhedral stars of the package.