Červeň I. - CRYSTALLOGRAPHIC PLANE GROUPS, Types of symmetry of planar periodic structures, their geometric and mathemathical construction

CRYSTALLOGRAPHIC PLANE GROUPS
Types of symmetry of planar periodic structures, their geometric and mathemathical construction

Ivan Červeň
cervenivan@gmail.com
Slovak University of Technology 
Faculty of Electrical Engineering and Information Technology,
Department of Physics

Abstract:

This text describes both geometric and mathematical methods for the construction of 17 types of symmetries of planar periodic structures. Such symmetries are characteristic, for example, for printing on wallpaper, but they are also applied in crystals in which the atoms are regularly - periodically arranged in parallel planes.  In terms of the symmetry arrangement  of the atoms in crystals there are up to 230 types of symmetry. Their exact derivation is very demanding in scope and content, and so the exact procedure is better documented on two-dimensional periodic structures, where there are only 17 such types. The first part of the text contains an unpretentious geometric derivation, the third already uses the mathematical theory of groups and the algebra of tensors. The content of the second part is the mathematical expression (representation) of the symmetry operations of such structures using matrices and tensors.

DOI: 10.61544/OFDN3064

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