10 ANALYTICAL TREATMENT OF THE POLYTOPES REGULABLE 



The primitive polytopes of the family, from which the others 

 may he deduced by composition (cf. art. 1 1 6) are 6' 24 , ce x C 24 , 

 ce 2 C 24 and ce s C 24 ; the hitter three being respectively similar to 

 the convex polytopes whose vertices are the middle-points of the 

 edges, the centres of the faces and those of the limiting bodies 



of. a 



24 



As a consequence of the self-reciprocity of the C 24 we know, 

 that ce 3 C 24 is again a 24 , placed however in another position to 

 the system of coordinates than the C 24 from which we started. 



Denoting by C 24 this second C' 24 we have the identities : 



Ç> 4 = 



= -, <2* 



r '\ r 24 = 



= «| ^24 



ce 2 ( 24 = 



= cc x C 2i 



'■", ^24 = 



a 



24 



Similar identities exist between the polytopes deducible from 

 these by composition. 



Three of these polytopes are deduced from the primitive ones 

 by a symmetrical composition. They may be derived either from 

 C 24 or from G' 24 by the same operations and will be obtained but 

 once in the following investigations. They are 



H C 2i 



e S ^24 ' 



ce i e 2 24 == 



rt '\ e 2 24 ' 



e \ e 2 e 3 ^24 == 



e \ H '':', 2 4 ' 



deducible by composition 

 from f', and 



r '\ {[ >i 



C h ^24 

 C H ([ 24 



C 24 , ce l C 24 , ce 2 C 2i and ce 3 C 24 



Each of the other polytopes of the family is obtained in two 

 different positions relative to the system of coordinates : 



<'. ^24 = 



ce 2 e 3 C 24 



, from 



'L>4 



and ce 1 C 24 ; 





2 ^3 24 ' — 



-i Çu 



> )) 



OP (' 



tt 2 ( 24 



,, C6q o 24 ; 





H G 2i = 



ce l e 3 C 24 



, 



c 24 



„ ce 2 C 24 ; 





ce \ e 3 ^24 = 



e c 



6 2 24 



J) 



ce C 



1 1 °24 



„ ce. à C 24 ; 





C 'l C 2 C 24 = 



Ce ] ' J 2 C 3 ^24 



' Î 



24 



, «?, C 24 and 



CC 2 6 24 



Ce \ C 2 e S 24 == 



e \ e 2 2 1 



î J 



n \ C 24 



, ce 2 C 24 „ 



ce 8 .C 2A 



e l e 3 C 24 = 



e 2 & 3^24 



». 



1 '24 



, ce i C 24 „ 



ce 3 ^24 



e 2 e 3 ^24 = 



e i H ^24 



J? 



ü 24 



, ce 2 v 24 ,, 



C h C 2i 



