12 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



So we determine the following vertices of the auxiliary poly- 

 tope ce 3 C' 24 : 



1°. 



2°. 



^24 







2 , 



, 



L 24 



2 , 



—2 , 



, Ü 



re C' 



CC S l/ 24 



■4 , 



, 



0,0; 



c 24 



°24 



9 



* > 



, 



2 



Ü , 



, 



2 , 2 



ce 3 C' M 



2 , 



2 , 



2,2 . 



The vertices of the two octahedra used here are: 



1°. (2,2,0, 0) , (2 , -2 , , 0) , (2, , 2 , 0) , (2 , , — 2 , 0) , 

 (2, 0,0, 2), (2, 0,0,— 2); 



2°. (2 ,2,0, 0) , (0 ,0,2, 2) , (2 , , 2 , 0) , (0 , 2 , , 2) , 

 (2 , , , 2) , (0 ,2,2, 0). 



They have the face (2,2,0,0) (2 , , 2 , 0) (2 , , , 2) in 

 common. Hence they are adjacent limiting bodies of 6' 24 , and the 

 vertices deduced from them are adjacent vertices of ce 3 C' 24 . 



As their distance is found to be = 4, the coordinates must be- 

 multiplied by \V2 to obtain those of ce, s C' 24 . 



Two coordinate-symbols are found : 



1°. ce 3 C' 2i 4,0,0, 



ce 3 C 24 [2V 2 , , , ] 8 vertices; 



2°. ce 3 C 24 2,2,2,2 



ce 



h 6 24 [ V2 , V2 , \'2 , \ 7 2] 16 



Total ... 24 vertices. 



133. We now proceed to the further deduction of the coordinate- 

 symbols of the polytopes of the C 24 family. 



The polytope e 1 C 24 is obtained by composition of ce l C 24 and 

 C 24 (one symbol) : 



ce, w>4 4 , z , Ai , u 



C 24 2,2,0,0 



e 1 6' 24 [6 , 4 , 2 , 0J 192 vertices. 



