Analytical treatment of the polytopes regularly derived 



From the regular polytopes. 



Section II : Polytopes and nets derived prom the measure polytope. 



A. The symbol of coordinates. 



46. The distance r between two points P, P', the ordinary 

 rectangular coordinates of which are fa, fa, . . . , f/, n and f/,\, f/ 2 . . . , y! n 

 is represented by the formula 



n 



^=E(^— [z'if 2). 



i = 1 



Now we repeat here the question of art. 1 : 



"Under what circumstances will the series of points obtained 

 by giving to the set of coordinates fa,^,. . . t fZ n a determinate 

 set of values taken in all possible permutations form the vertices 

 of a polytope all the edges of which have the same length, say 

 unity?" 



The answer is nearly the same as that given in art. 1: 



"If the n values a ± , a. 2 ,. . ., a n are arranged in decreasing order, 

 so that we have 



the difference a,. — a k + i of any two adjacent values must be either 

 Y V2 or zero." 



The proof runs on the same lines as that given in art. 1. The 

 geometrical result can be stated in the following general form: 



"Under the conditions stated, the polytope the vertices of which 

 are represented by the symbol 



is the same as that obtained in the first section for n — 1 and 

 a k — a,. + 1 either one or zero. It is a derivative of the regular 

 simplex the vertices of which determine on the n axes OX i of 

 coordinates positive segments OA L , (i = 1 , 2, . , . , n), of the same 



• n 



length b = 2 a-\ 

 1 



L* 



