E.M.S. Hitzer, C. Perwass
Three Vector Generation of Crystal Space Groups in Geometric Algebra
Bulletin of the Society for Science on Form, 21(1), pp. 55,56 (2006).

Abstract: This paper focuses on the symmetries of crystal space lattices. All two dimensional (2D) and three dimensional (3D) point groups of 2D and 3D crystal cells are exclusively described by vectors (two in 2D, three in 3D for one particular cell) taken from the physical cells. Geometric multiplication of these vectors completely generates all symmetries, including reflections, rotations, inversions, rotary-reflections and rotary-inversions. We then extend this treatment to 3D space groups by including translations, glide reflections and screw rotations. We focus on the monoclinic case as an example. A software demonstration shows the spacegroup visualizer.

Keywords. crystal lattice, space group, geometric algebra, OpenGL, interactive software


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