Towards a more comprehensive crystallography

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Acta Crystallographica Section B, vol. 51, issue 4 (1995) pp. 386-401

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Despite the intrinsic difference from the point of view of
structure of various types of crystals, such as
commensurate, incommensurately modulated, intergrowth
crystals and quasicrystals, a common approach to their
symmetry seems to be possible which eventually will lead to
a more comprehensive crystallography. The unifying elements
become apparent through treatments which, at first, seem to
be contradictory with the geometry of the crystal
structures involved. Examples are the description of
aperiodic crystals in terms of lattice-periodic structures
(going beyond three-dimensionality), the investigation of
scaling symmetry in quasicrystals by means of a Z-module of
translations generating a dense set of translationally
equivalent atomic positions (going beyond discreteness) and
finally the characterization of Euclidean properties of
normal crystals through non-Euclidean symmetries (going
beyond Euclidean metric). These changing approaches do not
modify, however, the fundamental nature of crystals to be
three-dimensional, discrete and Euclidean. They only allow
implicit symmetry groups like the superspace groups
(unifying the crystallography of incommensurate and
commensurate crystals) and the multimetrical space groups
(unifying the possible symmetries of quasicrystals and
normal crystals) to be made explicit. Aspects of crystal
diffraction, morphology and crystal structure are presented
from this unifying point of view, without intending to
cover the whole crystallography.

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