60 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



By considering the three groups of cases 



a t = 0,(i= 1, 2,3, 4) — , a i9 a. 2 , a*, a k = ( — 1, — 1,0,0,) — , 



«, = -,1,(1=1,2,3,4), 



and adding to the immovable parts the permutable ones taken in 

 any order, generally affected by the sign which tends to decrease 

 the absolute value of the coordinate but — ■ in connection with the 

 negative sign before the lower half of the symbol which exacts 

 an odd number of negative permutable digits — with exception of the 

 smallest of these digits V 2 the sign of which is to be chosen inversely 

 so as to increase the absolute value of the coordinate, we get by the 

 upper half the 96 new vertices [4 + 2\/2, 2 -f- 2\/2, 2 + 2\/2, 0] 

 and by the lower the 96 vertices l[4+3 V2, 2+N/2, 2+Y/2, \/2], 

 obtained above. So the result is a poly tope with 288 vertices repre- 

 sented by the combination of the symbols 



[4 + 3^2,2+^2, 2+V / 2,V / 2], [4 + 2^2,2 + 2^2, 2 + 2\/2,0]. 



As we will prove in section V this polytope with the characteristic 

 numbers (288, 576, 336, 48) limited by 48 tC is ce l e 2 C 2!i . 



Case ce l e s J¥(C 16 ). Besides [I'l'l \~]\/2 = ce 1 e 3 C i6 we have tö 

 look out for the face gap filling and the polytope of vertex import, 

 the edge gap filling being reduced by contraction to the base poly- 

 hedron of the prism occurring in the case of e 1 e s N(C 16 ). 



Face gap prismotope. Here we get for the new origin O' the 

 coordinates § (2 + 2V2), f (2 -{- 2V2), f (2 + 2 \/2), 0, as 2 + 2 V2 

 is the extension number. 



So the first and the second net symbol are 



. 4 



[2- j/2, 2 — j/2, J/2, J/2], (4 — 4j/2) a 1 — i a 2 — i, a^—h «j, , S«i even, 



l 



[2 + 2j/2, 2 ,0,0] 



_, L [2 + yi 2 + K2 ; K2 ; y$ & + W *«i+h *%+*• ** + *• ^ + 1, » odd. 



By taking in the first symbol a { = 0,(^ = 1, 2, 3,4), we find 

 the vertices ( : ^ — , - ~^ , ~ ~~^ J [2] lying at minimum dis- 

 tance fV3 from O', i.e. a P s ; by substituting in the upper half 

 of the second symbol a i = }% (i=l,2, 3), a i = — 1 we get moreover 



/2_+2j/2 2 + 2J/2 -4 + 2|/2\ n ., ., . , , . , f ., 

 I - -y -' - — ö » - -~- - ) 0, the third triangle ot the pris- 

 motope [3; 3] to be found. 



