50 ANALYTICAL TREATMENT Oh' THE POLYTOPES REGULARLY 



x x = 1 -f- a V x 2, œ 2i œ s , . . . , œ n = (1 -j- a \/2, 1 -\- b V // 2, . . . , 1), 

 œ 2 = 1 -\-aV2, œ x ,œ> d ,. . .,x n = (l -]~a\ / 2, 1 -\-bV% t . . ., 1), 



which are in contact with each other by the n — 2-dimensional 

 polytope 



a\ = 1 -j- a \/2, a? 2 = 1 -\-aX / 2, x 3 , <z? 4 , . . . , œ n ={\ -\-b V/2, . . . , 1). 



Cte [a-j- 1, #, #, . . . , 0] \/2. — Here we have to consider the 

 influence of the replacing of a -j- 1 by a. The proof runs exactly 

 in the same lines. 



Remark. By combining the theorems XLVIII and XLIX we 

 can find the symbols in c and e k of any form deduced from C 2 { ^\ 

 But this process can be simplified by introducing the operation 

 e which transforms the centre O of C 2 n } considered as an 

 infinitesimal cross poly tope OJ$ into C 2 ^ ] . Then the contraction 

 symbol c can be shunted out by substituting e k e t . . ,e m C 2 ® } for 

 ce k e t . . ,e m Cf\ but this implies that we replace e k e\. . .e m C 2 ^ 



by e o e k e i • • • e ,n ^2'? ) - ^ n ^ s remar k — corresponding literally to that 

 of art. 60 — will also be useful in part F of this section. 



Meanwhile we have shown now that any coordinate symbol be- 

 tween square brackets satisfying the laws of the first part of theorem 

 XXVIII (art. 47) can be interpreted both ways, either as a form 

 deduced from the measure polytope or as a descendent from the 

 cross polytope. So we have proved the following theorem already 

 stated implicitly in art. 48 : 



Theorem LI. "The families of polytopes deduced from the two 

 patriarchs, measure polytope and cross polytope, are identical.' 5 



E. Nets of polytopes. 



78. In accordance with the last theorem the net of measure 

 polytopes N(M\f) can also be considered as a net iV(ce n _ 4 C 2 \f) of 

 polytopes ce n _ i C 2 ^\ So the nets put on record for n = 3, 4, 5 

 can be transcribed as nets of cross polytope descent. 



But instead of doing this we point out a particularity of 

 the case n = 4. For n = 4 both the half measure polytopes 

 4- -o [I. 1, 1, 1J are cells (7 16 and in relation with this fact we find 

 a new fourdimensional net of regular polytopes, i. e. /S 4 possesses 

 besides the measure polytope net exceptionally a cross polytope net 

 too. If we suppose that the net N{M l ^ 1) ) be composed of alternate 

 white and black polytopes, so that two J/ 4 (2) with a common M.J 2) 

 differ in colour, and that each white J/ 4 (2; is truncated at one 

 set of eight vertices, so as to retain a -{- ± [1, 1, 1, 1], whilst 



