Three Vector Generation of Crystal Space Groups in Geometric Algebra

Authors: Eckhard Hitzer, Christian Perwass

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.

Comments: 2 Pages. 1 figure, 1 table. Bulletin of the Society for Science on Form, 21(1), pp. 55,56 (2006).

Download: PDF

Submission history

[v1] 2013-06-19 03:03:40

Unique-IP document downloads: 340 times is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary. In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution. will not be responsible for any consequences of actions that result from any form of use of any documents on this website.

Add your own feedback and questions here:
You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.

comments powered by Disqus