This, however, is not the whole story, for the sixty-five Sohnckian regular point-systems only account for twenty-one of the thirty-two crystal classes, the remaining eleven being those of lower than full holohedral systematic symmetry, and which are characterised by showing complementary right and left-handed forms. In other words, they exhibit two varieties, on one of which faces of low symmetry are developed on the right, while on the other symmetrically complementary faces are developed on the left; that is, these little faces modify on the right and left respectively the solid angles formed by those faces of the crystal which are common to both the holohedral class of the system and to the lower symmetry class in question. In some cases, moreover, these two complementary forms are known to exist alone, without the presence of faces common to both the holohedral class and the class of lower symmetry. The two varieties of the crystals are the mirror images of each other, being related as a right-hand glove is to a left-hand one. Further, the crystals of these eleven classes very frequently exhibit the power of rotating the plane of polarised light to the right or to the left, and complementarily in the cases of the two varieties of any one substance, corresponding to the complementariness of the two crystal forms. The converse is even more absolute, for no optically active crystal has yet been discovered which does not belong to one or other of these eleven classes of lower than holohedral symmetry. The final step of accounting for the structure of these highly interesting eleven classes of crystals was taken simultaneously by a German, Schönflies, a Russian, von Fedorow, and an Englishman, Barlow, who quite independently and by totally different lines of reasoning and of geometrical illustration showed that they were entirely accounted for by the introduction of a new element of symmetry, that of mirror-image repetition, or “enantiomorphous similarity” as distinguished from “identical similarity.” These three investigators all united in finally concluding that when the definition of symmetrical repetition is thus broadened to include enantiomorphous similarity, 165 further point-systems are admitted, and the whole 230 point-systems then account for the whole of the thirty-two classes of crystals.