Morphology of modulated crystals and quasicrystals

A survey is given of new types of crystallographic faces and forms occurring on modulated crystals which have to be indexed with four integers (hklm). These faces have so far been found on modulated crystals with an average jS-K2S04 structure which can be grown from an aqueous solution and the mineral crystal calaverite (AuTe2) which can be grown from the melt. Interesting data on crystal growth obtained from in situ observations are presented. Among others, spiral-like patterns can be recognized. The occurrence on quasicrystals of crystal forms to be indexed with six integers (hklmno) is discussed and it is shown that these faces can in principle be explained by a generalization of the Hartman- Perdok theory. Crystal forms (hklm) and (hklmno) occurring on modulated and quasicrystals respectively can be explained by the higher than three-dimensional crystallography of de Wolff, Janner and Janssen where these crystals are considered as threedimensional cuts out of a higher than three-dimensional truly translational invariant structure.