E.M.S. Hitzer, C. Perwass
Crystal Cell and Space Lattice Symmetries in Clifford Geometric Algebra
in TE. Simos, G. Sihoyios, C. Tsitouras (eds.), International Conference on Numerical Analysis and Applied Mathematics 2005, Wiley-VCH, Weinheim, 2005, pp. 937-941.

Introduction: The structure of crystal cells in two and three dimensions is fundamental for many material properties. Many elements, including Aluminium, Copper and Iron have e.g. cubic unit cells. The nearest neighbors of diamond structures form tetrahedrons. About 30 elements show hexagonal close-packed structure. Important organic molecules like benzene have hexagonal symmetry. Today some 80\% of crystal structure analysis is carried out on crystallized biomolecules with huge investments from pharmaceutical companies.
In two dimensions atoms (or molecules) often group together in triangles, squares and hexagons (regular polygons). Crystal cells in three dimensions have triclinic, monoclinic, orthorhombic, hexagonal, rhombohedral, tetragonal and cubic shapes (see Fig. 1).
The geometric symmetry of a crystal manifests itself in its physical properties, reducing the number of independent components of a physical property tensor, or forcing some components to zero values. There is therefore an important need to efficiently analyze the crystal cell symmetries.
Mathematics based on geometry itself offers the best descriptions. Especially if elementary concepts like the relative directions of vectors are fully encoded in the geometric multiplication of vectors.

Key words: crystal cell, point symmetry, Clifford geometric algebra, OpenGL, interactive software.
Subject classification: 15A66

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