E.M.S. Hitzer, C. Perwass

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

[PDF] 1.5Mb

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