ON THE STKIJCT[TIIE OF CRYSTALS. 331 



and are called ' simple ' ; the remainder may be regarded as consisting of 

 two ' analogous ' systems, the one the mirror image of the other, and are 

 called ' double systems.' 



The systems, as a whole, are divided into three groups : (1) Symmor- 

 phous, whose elementary figures possess the same class of symmetry as the 

 system itself ; (2) hemisymmorphous, consisting of two analogous 

 symmorphous simjDle systems, which together make up a ' double system,' 

 the latter itself not being symmorphous ; (3) asymmorphous. In the first 

 class all the elements of symmetry meet in a point within each figure ; in 

 the second class only the symmetry axes meet in a point ; in the third class 

 none of the elements of symmetry meet in a point ; here, consequently, 

 adjacent figures are differently orientated. 



Fedorow first proves that the classes of symmetry of the regular sys- 

 tems of figures are only special cases of the classes of symmetry of the 

 finite figures, and that it is possible to have several regular systems 

 belonging to the same class of symmetry. 



The classes of symmetry are, of course, thirty-two in number ; they 

 are limited by virtue of the fact that the axes, whether symmetry axes, 

 screw axes, or axes of composite (alternating) symmetry, can only be two- 

 fold, three-fold, four-fold, or six-fold. 



The definition of the regular partitioning of space given by Schonflies ^ 

 is practically identical with that given by Fedorow in his ' Elements of 

 Figures' in 1885 : 'A division of space into absolutely similar cells in 

 which each cell is surrounded in the same way by the remainder.' If in 

 a regular system of figures all the figures dilate uniformly until they 

 come into contact, the system is converted into one of cells regularly 

 partitioning space. A noteworthy property of the ' elementary ' or 

 minimum cells is that the axes and planes of symmetry of the system 

 cannot pass through them,''^ but must lie in their surfaces ; in other 

 respects unless bounded on all sides by planes of symmetry their actual 

 form is quite arbitrary. 



Here, again, Fedorow introduces the classification mentioned above : 

 (1) Symmorphous systems have the same symmetry as their cells ; here 

 the cells are parallelohedra,^ and therefore arranged in parallel positions ; 

 e.g., an arrangement of parallel cubes ; (2) hemisymmorphous systems, 

 which only have the elements of simple translation and rotation in 

 common with the constituent cells ; e.g., the triangular prisms of fig. 1 ; 

 here the parallelohedron (rhombic prism of 60°) is composed of two 

 ' analogous ' stei'eohedra (two triangular prisms) ; (3) asymmorphous 

 systems in which the parallelohedra are indeterminate and not neces- 

 sarily closed polyhedra. 



Now a parallelohedron possesses a centre of symmetry (centre of 

 inversion), and if it is a convex figure this centre lies within it ; if, on 

 the other hand, it is concave, the centx'e lies without it. Further, in 

 every convex parallelohedron the faces are only parallel and equal in 

 pairs ; and there are only four sorts of parallelohedra, namely, those 



' Kachr. d. It. Ges. d. Wiss., Gottingen, 1888, ix. p. 223. = See above, p. 326. 



' Like Haiiy's molecules sonstraotives Fedorow's parallelohedra are mere geo- 

 metrical entities, and in many cases the grouping of the stereohedra which produces 

 them has to be very arbitrary. The same stereohedra can in all cases be grouped 

 to form parallelohedra in an infinite number of ways. The parallelohedra will often 

 have re-entrant angles even if the angles of the stereohedra of which they are com- 

 posed are all salient. 



