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symmetry may also have the function of twinning-elements ; in this 

 case, therefore, the twinning-process may also be considered as a way 

 of apparently increasing the existent degree of symmetry of the 

 crystal under consideration (feldspars). Finally, the twinning plane 

 may be perpendicular to a symmetry-plane of the composing indi- 

 viduals ; the same subdivision of the different twins as in the previous 

 cases may be also made here. 



In general the classification of twins into such as are produced 

 by juxtaposition or by penetration, may have certain advantages 

 from a practical standpoint; from a theoretical point of view, 

 however, it may be considered as somewhat too limiting, as for 

 instance the individuals of a twin by juxtaposition may at least 

 partially penetrate each 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. 

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

 tion of a crystal-aggregate, poly synthetic twins are said to be produ- 

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

 alternate 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. The mineral aragonite presents wellknown 

 examples of both kinds of twins. 



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 to consider in the following certain 

 cases of repeated twinning, more particularly of penetration- 

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

 show approximate, or pseudosymmetry . 



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 2 , 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) : (111) 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 



