It was noted that complete dislocations in quasicrystals were the intersections of dislocations in a high-dimensional lattice with an irrational cut that represented physical space. Predictions were made of the resultant properties by using Volterra and topological methods. The restriction of the Volterra method to quasicrystals introduced very unusual geometrical properties which could be described in terms of phason deformations and their mismatch. Thus, the motion of a defect was generically non-commutative. The pattern of mismatches that was created by complete moving dislocations or disclinations depended upon the path which was traced out by the defect in going from one position to another, when 2 such paths surrounded a defect. The same sort of reasoning was applicable to the intersection of 2 defects. Other properties resulted from the use of the Volterra method, such as the existence of stacking faults that were bounded by incomplete dislocations and the relationship between mismatches and the re-shuffling of atoms. It was suggested that the natural method to use was the topological one. In particular, it was shown that the group which classified the dislocations was non-Abelian. This property was directly related to non-commutativity. Complete dislocations were termed disvections, because of their relationship to Cartan’s transvections. The latter involved translations of non-Abelian nature in a hyperbolic space.

M.Kléman: Journal of Physics - Condensed Matter, 1996, 8[49], 10263-77