Elementary crystallography is an ideal context for introducing students to mathematical geology. Students meet crystallography early because rocks are made of crystalline minerals. Moreover, morphological crystallography is largely the study of lines and planes in real three-dimensional space, and visualizing the relationships is excellent training for other aspects of geology; many algorithms learned in crystallography (e.g., rotation of arrays) apply also to structural geology and plate tectonics. Sets of lines and planes should be treated as entities, and crystallography is an ideal environment for introducing what Sylvester (1884) called "Universal Algebra or the Algebra of multiple quantity." In modern terminology, we need SIMD (Single Instruction, Multiple Data) or even MIMD. This approach, initiated by W.H. Bond in 1946, dispels the mysticism unnecessarily associated with Miller indices and the reciprocal lattice; edges and face-normals are vectors in the same space. The growth of mathematical notation has been haphazard, new symbols often being introduced before the full significance of the functions they represent had been understood (Cajori, 1951; Mclntyre, 1991b). Iverson introduced a consistent notation in 1960 (e.g., Iverson 1960, 1962, 1980). His language, greatly extended in the executable form called J (Iverson, 1993), is used here. For information on its availability as shareware, see the Appendix. Publications suitable as tutorials in , J are available (e.g., Iverson. 1991; Mclntyre, 1991 a, b; 1992a,b,c; 1993). Crystals are periodic structures consisting of unit cells (parallelepipeds) repeated by translation along axes parallel to the cell edges. These edges define the crystallographic axes. In a crystal of cubic symmetry they are orthogonal and equal in length (Cartesian). Those of a triclinic crystal, on the other hand, are unequal in length and not at right angles. The triclinic system is the general case; others are special cases. The formal description of a crystal gives prominent place to the lengths of the axes (a, b, and c) and the interaxial angles ( α, β, and γ). A canonical form groups these values into a 2 x 3 table (matrix), the first row being the lengths and the second the angles.