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tilling up that space completely. The distribution of matter within such 

 an t K nu-ntary volume, which Von Fedorow calls a sterohedron, 

 while Schoenflies prefers the name of fundamental domain for it, - 

 is supposed to be completely arbitrary and free from all symmetry. 

 It> volume is constant and equal to that of the elementary paralh !<- 

 piprdon of the space-lattice, or a multiple of it. When some of these 

 identical or enantiomorphous "fundamental domains" in symmetrical 

 space-lattices or structures, are eventually combined into greater 

 units exhibiting a certain symmetry, these symmetrical "complex 

 domains" which by similar repetition are also filling up the whole 

 space, are discriminated by Von Fedorow as parallelohedra 1 ). 

 Corresponding points of such parallelohedra are also corresponding 

 (homologous) points of the regular system, and they are always 

 arranged in a space-lattice characterised by a definite group of 

 translations, etc. 



It is easy to demonstrate, moreover, that no existing symmetry- 

 elements can ever lie within the fundamental domain of a regular 

 structure, but that they are always situated on its surface. This 

 follows immediately from the fact that each symmetrical operation 

 must always bring a fundamental domain into coincidence with 

 another one present in the whole complex. From this it is clear that 

 the existence of symmetry-axes and of symmetry-planes in the 

 structure will then of course be in some way determinative for the 

 shape of the fundamental domain, as e. g. symmetry-planes must 

 be always limiting parts of the surface of such fundamental cells 

 (fig. 1 1 8). In the latter cases it also becomes clear that in general 

 to every fundamental domain A, a second one A', being the mirror- 



1) If a point of a space-lattice be joined with all nearest points situated 

 round it, and planes perpendicular to the midst of these lines be constructed, 

 a volume of space is separated which is limited by fourteen planes which 

 are pairwise parallel to each other. In a cubic space-lattice for instance these 

 planes are perpendicular to the edges of the cubic cell and to the four cube- 

 diagonals. The ,, fundamental domain" thus determined, a hepta-parallelo- 

 hedron, is in the latter case a cube, the corners of which are truncated by 

 planes of the octahedron. With elements of this shape space can be filled 

 without any room remaining between the composing cells. These hepta-parrallelo- 

 hedra, already used by Lord Kelvin, have an important share in the 

 deductions of Von Fedorow. However, it may be remarked here that it is 

 not necessary to determine the special shape of the fundamental domain. This 

 form can be quite arbitrary; but its volume is always constant and equal to 

 that of the elementary cell of the space-lattice, or in regular systems in 

 general a multiple of this. 



