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K. Stróz
Plane groups — from basic to advanced crystallographic concepts
The plane groups are rarely discussed in the crystallography courses and the didactic role of the plane groups for teaching symmetry is rather underestimated. Most crystallographic concepts known from 3-dimensional space group descriptions concern also these 2-dimensional groups and can be easier illustrated. Symmetry of atomic layers as well as symmetry of mosaics, lattice designs, symmetry of electron diffraction patterns or any pattern with two-dimensional periodicity can be characterised on a unique basis. Such a pattern can be generated by decorating the points of a crystallographic orbit by different graphical objects. The orbits are visualised as “crystallographic mosaics” (by connection of the closest points of the orbit), they can be characterised by Shubnikov or Laves nets and there always exists a number (circle packing density) that is related to a given orbit. The paper and a didactic computer program presented in it give some ideas and framework for the plane symmetry experiments: from finding the isometries on the generated patterns to identifying the non-characteristic orbits. It is also shown that the contour map of a circle packing density is a good frame for locating symmetry elements, asymmetric units or points with higher plane symmetry into a unit cell. The applied complex approach to the orbit characterisation will narrow the gap existing between concepts used by practising and theoretical crystallographers and mathematicians interested in tessellations.
Zeitschrift für Kristallographie, Oldenbourg Wissenschaftsverlag
Print ISSN: 0044-2968
Volume: 218, 09/2003
Pages: 642