10 GEOMETRICAL DEDUCTION OP SEMIREGULAB ETC. 



Generalization. 



13. The foregoing result may be generalized thus. If any set 

 of limits e r be the subject of expansion in a regular poly tope P n 

 in a space of n dimensions the polytope P n ' defined by the new 

 positions of the members of the subject has for its limits /' n _i : 



1 st : a group consisting of the limits l n _ i of P n transformed by 

 the e r expansion (<?,. l n _^ 9 



2 nd : a group of vertex import, each member of the group being 

 determined by its vertices, the number of which is equal to the 

 number of limits l r meeting in a vertex of P n and having one 

 kind of edge. This polytope is regular in the e x and the e n _ i expan- 

 sions. These two groups are the principle ones. 



3 rd : there are besides various kinds of prisms. Those of edge 

 import (1 -import) are determined by the new positions of an edge 

 of P n and the number of these positions is equal to the number 

 of limits /,. meeting in an edge of P n . The prisms of face import 

 (2-import) are determined each by the new positions of a face of 

 P n , and the number of these is equal to the number of limits l r 

 meeting in a face of P n and so on. The whole series of prisms 

 is as follows: 1 -import, 2-import, ....r — 1 -import. 



Combination of operations. 



14. The expansions described above have been applied to regular 

 bodies according to the definition given on page 5, transforming 

 them into bodies possessing a particular kind of semiregularity. 



The question now arises: can these semiregular bodies be trans- 

 formed by the application of any further expansion without having 

 lost the kind of semiregularity defined above? 



It is evident in the first place that a movement of all the edges 

 or of all the faces would produce bodies with edges of different 

 lenghts. But an inspection of the transformed bodies in three- 

 dimensional space (Figs. 1 b, 2 a and 2 b) shows that in each of 

 the polyhedra IC, tT and tO there are two groups of faces, each 

 of which taken alone defines the polyhedron: one group corres- 

 ponds to the faces (expanded), the other to the vertices (expanded) 

 of the original polyhedron, and these two groups differ as to a 

 particular characteristic. 



The members of the first group are in contact with members of 

 the same group; the members of the second are separated by at 

 least the length of an edge from members of their own group. As 



