We present a complete formulation of the two-dimensional and three-dimensional crystallographic space groups in the conformal geometric algebra of Euclidean space. This enables a simple new representation of translational and orthogonal symmetries in a multiplicative group of versors. The generators of each group are constructed directly from a basis of lattice vectors that define its crystal class. A new system of space group symbols enables one to unambiguously write down all generators of a given space group directly from its symbol.

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Most of the current work on conformal GA can be found at the website in Ref. 3 and links therein.

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Equation (27) corrects a mistake in Eq. (68) of Ref. 1. It is also worth mentioning that the point at infinity was defined with opposite sign in that paper—probably not a good idea!

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, SPACE GROUP VISUALIZER, open source software freely available at http://www.spacegroup.info
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