Penrose tilings as coverings of congruent decagons
The Penrose tilings show that there is a set of two prototiles, with which the Euclidean plane can be tiled, but only in a nonperiodic way. Is there a set of first one prototile with the same property? This problem remains open; but there the author solves the analogous problem for covering: A regular decagon can serve as a single proto``tile''. To cover the Euclidean plane some decagons will of course overlap. These overlaps are however governed by matching rules: The proto``tile'' consists of white and dark regions, and we require that all dark regions are overlapped, and that colours match. Since the colouring of the regions is based on Penrose tilings, the ``tiling'' is necessarily nonperiodic.
scientific article; zbMATH DE number 929904