4 ANALYTICAL TRE/VTMENT OF THE POLYTOPES UEGÜLA1ILY 



Section I : Polytopes deduced from the simplex. 



A. The symbol of coordinates. 



1. In a preceding paper (Memo Archief voor Wiskunde, vol. TX, 

 p. 133) I found that the distance r between two points P,P', the 

 barycentric coordinates of which — with respect to a regular simplex 

 S(n -\- 1) of space S n — are ft i9 (£ 2 , • . . , f£ n +i ana< /A> ^2» • • • > ft'n + \> 

 is represented , with the length of the edge of the simplex of coor- 

 dinates as unit, by the simple formula 



1 



= |E(^ — ^',) 2 1). 



We insert here a much simpler deduction of this formula; of 

 this deduction fig. 1 gives a geometrical represensation for the 

 particular case n = 2 of the plane. 



n + 1 



Let 2^ = 1 represent the space JS n determining in a space of 



i = l 



operation # w+ i, on the axes of a given system of rectangular coor- 

 dinates with origin O, the points A if A 2 ,..., A n + i at positive 

 distances unity from 0. 



Let P and P' with the orthogonal coordinates m ±i m 2 , . . . , m n + i 

 and m\, m 2 ,. . . , m' n+i be any two points of this S n ; then, accor- 

 ding to the expression for the distance in rectangular coordinates, 



n + l 



we have PP 2 = 2 {m l — m$ and, as the points lie in IS ni the 



i = 1 



n+l n+l 



conditions 2 m i = 1,2 m\ = 1 hold. 



i = 1 i = 1 



Now let us consider the" normal distance coordinates y l and y! { 

 [i=\ j 2,. . ., # -|- 1) of P and P' with respect to the regular 

 simplex S {n -\- 1) with the vertices A l9 A 2 , . . ., A n + i in JS n ; then 

 from similar rectangular triangles we deduce immediately the relation 



Pi = t*i = z- 



m l m i 1 



where h is the height of the regular simplex. But, as the barycentric 

 coordinates are normal distance coordinates measured by the corres- 

 ponding height of the simplex and all these heights are equal in the 

 regular simplex, we find for the barycentric coordinates fz t and f/ { 



fa = , = "h > ft i = - 7 = m i » 

 k k 



