54 ANALYTICAL TREATMENT OF THE POL1TOPES REGULARLY 



The first part of this lemma is evident by itself. As to the second 

 part related to a face we state that the angle formed by the two 

 spaces of adjacent tetrahedra ABCD , ABCD' of C i6 at the common 

 face ABC is 120° (see my paper: "On the angles of the regular 

 polvtopes, etc.", Amer. Joiirn. of Math., vol. XXXI, p. 307), 

 from which it ensues that any face is common to three (7 16 ; as any 

 two of these three C iQ have a limiting tetrahedron in common they 

 belong to different groups, etc. 



The lemma just proved immediately shows the truth of the 

 theorem. If, after having driven asunder the celJs C^ 1 ^' 1 ) of the 

 net iV(C lu ) so as to create room for the extension recorded above, 

 the extended C i(i receive the shape exacted by the character of the 

 net under consideration by means of a regular truncation, the 

 contact of the cells — belonging to different groups — by faces will 

 remain uninfluenced if the operations e 3 , e k do not yet present 

 themselves, the truncations being then restricted either to the ver- 

 tices alone or to vertices and edges; so, as any vertex of the net 

 belongs at least to one face and each face belongs to three poly- 

 topes of the set without e 3i e ki one of each group, each vertex of 

 the net must be contained in each of the three net symbols of 

 any case of that set. 



So in this case the net itself can be represented by any of the 

 three symbols, which includes that the constituents furnished by 

 one symbol are identical with those furnished by each of the two 

 others, though constituents of polytope and body import of one 

 symbol may become under certain circumstances constituents respec- 

 tively of vertex and edge import of an other. 



Now the state of affairs changes as soon as <? 3 makes its appea- 

 rance. This operation still preserves the contact by limiting bodies 

 of body import between cells belonging to different groups, but it 

 annihilates at the same time face contact between limiting bodies 

 of body import of the same cell. So here the limiting bodies of 

 body import of any constituent have been split up into two sets 

 B and Q dividing the vertices equally between them, in such a 

 way that any two of these limits which were in face contact 

 before belong to different sets. So here the arrangement of the 

 three groups A, B, C of constituents is such that any constituent 

 of group A is in body contact by its set of limits B with consti- 

 tuents of group B, by its set of limits Q with constituents of 

 group C. So each of the three net symbols contains all the verti- 

 ces of one group and only half the number of vertices of each of 

 the two other groups, i.e. -| of the total amount. 



