THE CRYSTALLOGRAPHY OF APERIODIC CRYSTALS

Abstract
It is shown that a complete crystallographic classification of aperiodic crystals is possible starting from the known point groups. We first construct the [MATH]-space Bravais sets consistent with the point group. This leads to a hyperspace representation in the space of incommensurate hydrodynamic translations. A unique hyperlattice dual to the Bravais set describes the discrete hydrodynamic symmetry of the aperiodic crystal. From the transformation properties of the phases an algorithm for the construction of the relevant hyperspace groups is constructed in which the hydrodynamic representations of the group play a special role. Results are only discussed briefly. It is shown that there are six cubic sets and four tetragonal sets. There are four icosahedral sets only two of which give a six dimensional hyperlattice. There is only one pentagonal set.