DERIVED FROM THE REGULAR POLYTÖPES. 



e { e 2 C 24 ( 576 , 1152 , 720 , 144); 



01*3^24 | ( 57 6 f 1440 f ii04 , 240); 



^2 ^3 24 ' 

 «1*2 «8*54 ( 1152 ' 2304 ' l392 ' 240 >- 



127. Forms ce. 6' 24 , ce 2 C 24 , ce. 6 6 24 . In these polytopes there 

 is coincidence of two vertices of each edge [ce x C^), three vertices 

 of each face (ce 2 C 24 ) or six vertices of each octahedron (ce B C 24 ). 

 So the number of vertices will be j,. 192, - 1 . 288, i \. 144, viz. 

 96, 96, 24. This, in combination with the limiting bodies, 

 (24 CO, 24 C for ce x C 24 ; 24 C, 24 CT> for ce 2 C 24 ■ 24 O for 



<*3 C 24) g iveS : 



«1 <h 



24 

 ^2 24 



(96 , 288 , 240 , 48) 

 ce 3 C 24 (24 , 90 , 90 , 24) 



128. Forms ce x e 2 C 24 , ce 1 <? 3 C 24 , ce 2 e 3 6' 24 , ce 1 e 2 e 3 C 24 . By 

 the application of the principle of contraction we obtain a trans- 

 formation of the bodies of body-import tCO , tO , BCO , tCO into 

 tC, CO, C, W. Consequently 0/, 24 r, 24 r, 48 r are trans- 

 formed into 2>f, 12 r, 8r, 24 r, leading to 288, 288, 192, 

 570 vertices. 



The limiting bodies enable us to complete the symbols (48 tC 

 for ce t e 2 C u ; 24 CO , 96 P 3 , 24 7^0 for c^ e 3 C 24 ; 24 C, 

 24 tO for «? 2 é> 3 C 24 ; 24 tC , 96 P 3 , 24 /CO for ce 1 e 2 e\ C 24 ): 



ce 1 e 2 C 24 (288 , 576 , 336 , 48); 



ce 1 e B C 24 (288 , 864 , 720 , 144); 



ce 2 e. A C 24 (192 , 384 , 240 , 48); 



ce l e 2 e 3 C 24 (576 , 1152 , 720 , 144). 



D. The coordinates of the po^-f amity. 



129. The C 24 from which the other polytopes of the family will 

 be deduced, is supposed to be referred to a rectangular system of 

 coordinates with axes passing through the centres of opposite limiting 

 bodies. The vertices are then represented by a single coordinate- 

 symbol. 



Let the length of the edge be = 2\/2 ; then this symbol is 

 found to be 



[2,2,0,0]. 



