Penrose patterns and related structures. I. Superstructure and generalized Penrose patterns

The section method is applied to derive the Penrose pattern and related patterns with a ten- or fivefold axis. These are derived from a four-dimensional decagonal crystal or from a five-dimensional icosahedral crystal as a two-dimensional section. The two descriptions correspond to the three- and four-dimensional ones in the usual superstructure and the Penrose pattern can be regarded as the superstructure in the four-dimensional space. The diffraction intensities and symmetries of these patterns are discussed. The present study indicates that the point group in the quasicrystals is noncrystallographic but (m + d)-reducible similarly to that in incommensurate structures, where m and d are the dimension of the real space and the orthogonal complement to it.