88 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



(11—1) of IIJ/,y 2 y 2 ) are represented by (31 — 1) for a n _ 2 = 3 

 and by (11 — 1) for a n _. 1 = 1. So we treat these two cases together 

 by considering the face. 



x\ = a x (i = 1, 2, . . ., n — 3) , œ n _ 2i œ n _ { , œ n — (a n _ 2 1 — 1) 



with the centre 



%. = a t (i = 1, 2, . . . , # — 3) , 3 x n _ 2 = 3 x n _ 1 = 3 x n = # w _ 2 . 



By moving this face away from the centre O to a distance A 

 times as large its centre is transported to the point 



x ( = hdi (i = 1, 2, . . . , n — 3) , 3 w n _ 2 = 3 x n _ 1 = 3 x n = Xa n _ 2 . 



So the new position of the face leads to a new poly tope 

 ![/«!, Xa.,,. . .Xa n _ 3 , . . .]. As the length 2\/2 of the sides of this 

 face is maintained and the length of the edge (Xa k) Aa,. + 1 ) is 2A\/2 

 if a k and % + 1 are unequal, we only can arrive for A^ 1 at a 

 polytope all the edges of which have the same length 2V 7 2 if all 

 the digits a it a 2 ,. . . , fl n _ 3 are equal, i. e. in the four cases 



n n — 3 n — 2 n — 3 



\ [ÏÏTTT], \ [337731 1 1], \ [3377311], \ [557733 1 1]. 

 In these cases the face becomes 

 _ _ À +.2 "A -f- 2. A — 4' 



X t — A, \l — L,Z,...,n o),X n _ 2 ,X n _ ] ,X u — I — w ~, o" ' <T 



x t = oA, ,, „ = 5J 



^=3A, „ „ =(A-f-2, A ,A — 2) 



a?^ = 5A, „ 9t = J} 



furnishing for the edge (# n _ 3 , a n _ 2 ) of the new polytope the four symbols 



A 4- 2\ / A + 2\ 

 A, -J— J, (3A,— ±^-J, (3A,A+2), (5A,A + 2). 



So, it' s represents either 2 or we have in these four cases 

 A = f* + 1, SA = 3f + 2, 2A = 5 + 2, 4A = g -f 2; 

 so the values of A different from unity are respectively 

 4 1 2 J- 



of which the integer values are the only available ones. So the 



n n— 3 



face expansion can be applied to 1 [11 .. 1] giving [44. .4220] 



n—2 n — 3 



and to £[88.. 311] giving [00776420], i. e. in both available 

 cases measure polytope forms deducible from J/ n (4) . Therefore we can 

 disregard altogether the expansion of the fonpd. according to their own 



