Abstract
The law of complication, advanced by Goldschmidt, rests on an empirical basis. In this paper, the law is investigated rationally. It is shown that if missing terms are neglected, any zone of any crystal must conform to the law of complication for purely geometrical reasons. A transformation to ordinary Miller indices makes it obvious that the law implies the fundamental property of permutable axes. Strictly permutable, non-identical axes are only encountered in certain zones of certain classes of isometric, tetragonal and hexagonal crystals, because this condition requires a properly located symmetry element through the dominant node, 1. A re-examination of Goldschmidt’s evidence for the reality of the law of complication indicates that this evidence points only to the statistical reality of the law, and that individual zones of a given crystal species do not, in general, conform to the law except in the special symmetry cases given above.
Two factors enter into mechanism of controlling the permutability of axes in a zone: the lattice frame, which limits the slopes of crystal faces to definite values but which does not prohibit any crystal from conforming to the law of complication, and the growth environment of the crystal. In any individual instance, the latter factor supplies the mechanism for making the axes of a zone non-permutable. A statistical study of a given crystal species from all environments, however, gives a mass of data from which the specific effect of any individual environment is, at least partly, eliminated. Under these circumstances, the data may fit the crystal species into the law of complication. Junghann’s addition-rule aspect of the law of complication, although it implicitly includes Goldschmidt’s (essentially) reciprocal term law, is a still poorer approximation because it is, without theoretical justification, still more specific.
