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Sobczyk, Clifford Algebra to Geometric Calculus, a Unified Language for Mathematics and Physics, G. Reidel Publ. , Dordrecht jBoston, 1984. [7] International Tables for X-ray Crystallography, Volume A: SpaceGroup Symmetry, Kluwer Academic, Dordrecht, 1992. [8] O'Keefi'e, M. and B. G. Hyde, Orystal Structures 1. Patterns and Symmetry, Mineralogical Society of America, Washington DC, 1996. Chapter 2 The Inner Products of Geometric Algebra Leo Dorst ABSTRACT Making derived products out of the geometric product requires care in consistency.

69) has the same value for every point x. 68) has a conformal split of the form (1. 1). 66): a = a 1\ E = a 1\ eo 1\ e = 2m 1\ n 1\ eo = 2[(m 1\ n)eolE, (1. 71) 1. 65) was used to get the form on the right. The magnitude of the translation is therefore lal = lal = 21(m 1\ n) . eol, (1. 72) which holds for any point eo chosen as origin. Also, using to show that eE = e = -Ee (1. 67) in the form Ta = 1 + ~ae = 1 - ~ae == Ta, (1. 75) where, despite the conformal split of vector a, the right side is independent of the choice of origin.

An oriented reflection in one of these oriented (silvered) planes maintains the physical distinction between an object and its reflected image. So the two oriented reflections specified by ±b, describe the two possible placements of an object on opposite sides of the reflecting plane. 9) makes no distinction between objects and reflected images. The notion of oriented reflection is consistent with the notion of oriented rotation. 2. Thus, each element of 2116 characterizes some oriented symmetry of a hexagon.