GEOMETRICAL DEDUCTION OF SEMIREGULA.Il ETC. 1 5 



Partial operations. 



23. It has been seen that in both expansion and contraction 

 it is a necessary condition that the subject of operation shall define 

 the polytope both before and after the movement. 



In expansion, each member of the subject must be in contact 

 with other members. In contraction, each member must be separated 

 from the other members by at least the length of an edge. 



It sometimes happens that one of these conditions is satisfied by a 

 group consisting of the alternate members of a set of limits. Such 

 a group may then be made the subject of expansion or contraction. 

 If the members be in contact, they may be made the subject 

 of expansion; if they be not in contact, they may be made the 

 subject of contraction. 



24. Thus, an octahedron is defined by a group of four alter- 

 nate triangles, but each of these triangles is in contact with the 

 other three, so that these four may be made the subject of expan- 

 sion. This partial operation, which changes the octahedron into a 

 truncated tetrahedron , is denoted by the symbol J e 2 0. So ^ e 2 O = tT. 



Again, a CO whose symbol is c e i C is defined by a group of 

 four alternate triangles. Each of these is separated from the 

 others by the length of an edge. This group may therefore 

 be made the subject of the c operation, which changes the CO 

 into a T. So ^ c c e i C = T. 



It may be remarked that the partial, contraction hc can never 

 take place without a previous complete contraction c . 



25. The corresponding case in fourdimensional space is expressed 

 by the symbol -J- c c e ± C 8 . This indicates that first, the edges of 

 the C s are made the subject of expansion; second, the sixteen 

 tetrahedra of vertex import are made the subject of contraction; 

 third, a group of eight alternate tetrahedra are made the subject 

 of still further contraction. This last partial movement changes 

 the cubes of the C s into tetrahedra and annihilates eight of the 

 tetrahedra of vertex import, thus changing the C 8 into a C i6 , eight 

 of whose limiting tetrahedra are derived from the limiting cubes 

 of the C s , the remaining eight being of vertex import. So 



These examples suffice to show in what manner and under what 

 conditions the partial operations may be applied. 



