171 



the twinning-process may be con-nl'-i- <1 also as a way 

 it apjuM -ntly increasing tin- existent degree of symnn-tiy "I tin- 

 :rystal under consideration (feldspars). Finally the twinning plain- 

 ly In j>erpendicular to a symmetry-plane of the composing indi- 

 iduals; the same subdivision of the different twins as in the pn \ 

 may be made here also. 



general the classification of twins into such as are produced 

 >y juxtaposition or by penetration, may have certain advantages 

 )m a practical standpoint ; from a theoretical point of view however 

 may be considered as somewhat too limiting, as for instance the 

 individuals of a twin by juxtaposition may at least partially penetrate 



other to some extent. 



Moreover it may be remarked, that in a crystal-aggregate several 

 laws of twinning are often expressed simultaneously, so that very 

 complicated relations may be produced in such compound twins, 

 the same kind of twinning be several times repeated in the forma- 

 tion of a crystal-aggregate, polysynthetic twins are said to be produced 

 when the twinning-plane remains parallel to itself, so that the alter- 

 late individuals of the whole complex are in parallel position. If 

 this twinning-plane 'however changes its direction in the successive 

 repetitions of the twinning-process, so-called cyclic twins will be 

 produced. Of both kinds of twins the mineral aragonite presents 

 well-known examples. 



3. It cannot be our purpose here to go into the details of 

 twinning-phenomena in general, as this is a special chapter of pure 

 crystallography. We have only in the following to consider certain 

 cases of repeated twinning, - - more particularly of penetration- 

 twins, - - between individuals of the same crystal-species, which 

 show approximate, or pseudo-symmetry. 



There are a number of substances, the crystal-forms of which 

 show a more or less close approximation to forms of higher symmetry. 

 Thus if a tetragonal crystal, like chalcopyrite: CuFeS* has an 

 axial ratio a : c very near to unity (here: 1 : 0,9857), the tetragonal 

 crystal has evidently a space-lattice which closely approaches to 

 that of a cubic crystal. Chalcopyrite shows sphenoidal hemihedrism, 

 but the interfacial angle (111) : (if!) is here 10842', while for 

 a regular octahedron it would be 10928'. This mineral has therefore 

 a tetragonal, but clearly pseudo-cubic space-lattice. 



The same is the case if a rhombic crystal has a prism-angle of 

 nearly 60 or 120; in this case the vertical binary axis of the crystal 



