Crystallography, geometry and physics in higher dimensions. II. Point symmetry of holohedries of the two hypercubic crystal systems in four-dimensional space

Abstract

The interpretation of physical properties of incommensurate modulated crystals leads to the use of their point groups and their total character tables in their superspaces. Examples are chosen of point groups of holohedries of the two hypercubic crystal systems - primitive and body-centred- in four-dimensional space. A geometrical presentation is given of the point group - including its character table - of the primitive hypercubic crystal system, as it is useful for the prediction and simplification of tensorial physical properties of the corresponding crystals. Through geometrical considerations, the exceptional splitting of the hypercubic family of \bb E4 into two crystal systems is easily proved. Finally, the two different relations - according to the parity of n - existing between the point group of the primitive hypercube of \bb En and its subgroup of rotations are explained.