Quasicrystallographic tilings (Q580723)

The main result of the paper says that for every finite group G of symmetries of n-space there exist tilings of Penrose type (remember the ``kites and darts'' tiling by \textit{R. Penrose} in Bull., Inst. Math. Appl. 10, 266-271 (1974)) that have all the symmetries in G. The ``Penrose type'' means that 1) the shapes used are a finite set of polytopes which are fitted together by matching congruent faces; 2) the tiling is quasicrystallographic; 3) it is not translation invariant); 4) it is generated by an inflation operation that consists of magnifying each shape equally and then subdividing it into copies of the original shape.