DERIVED FROM THE REGULAR POLYTOPES. 

 and therefore 



"4-1 



PP ' = S (ft - ijJS 



i = i 



But here PP' is expressed in OA i as unit. By taking the edge 

 A L A k = O A, ]/2 as unit we find as above 



i = i 



The formula 1) enables us to find an answer to the following 

 question, now forming our starting point: 



"Under what circumstances will the series of points obtained by 

 giving to the set of barycentric coordinates a? 4 , x 2 ,..., x n + i a 

 determinate set of values taken in all possible permutations form 

 the vertices of a poly tope all the edges of which have the same 

 length, say the length of the edge of the simplex of coordinates?'' 



The very simple answer is given by the theorem : 



Theorem I. "If the n -f- 1 values a l9 a 2 ,. . ., a n +\ satisfying the 



n 4-1 



relation 2 a t = 1 are arranged in decreasing order, so that we have 



i = l 



a ± *^>a 2 >.... >a k ^>a k + i > .... >a n + 1 , 



the difference a k — % + 1 of any two adjacent values must be either 

 one or zero." 



Proof. According to 1) the square of the distance PQ between 

 any two vertices P, Q of the set is a sum of squares; from this 

 it is evident that in order to make the distance a minimum we 

 have to select two points P, Q which are transformed into each 

 other by interchanging only one pair of coordinate values, say a k 

 and a,... But then the square of PQ is | [(«,. — a,,) 2 -j- {a k > — %) 2 ] = 

 (a k — a k ) 2 , and therefore PQ itself is a k — a k >. Now this diffe- 

 rence becomes a minimum, if a k and a,, are unequal adjacent 

 values. As this minimum distance must be an edge, the condition 

 that all the edges are to have the length unity implies that the 

 difference between any two different adjacent values must be one. 



2. As the condition stated in theorem I depends upon the 

 differences of the corresponding barycentric coordinates x t and œ \ 

 we may drop the conditions 2 x { = 1 , X x' { == I by allowing these 

 coordinates either to increase or to diminish all of them by the 

 same amount, so as to make e. g. either the smallest or the greatest 

 of the n -f- 1 values equal to zero. So in order to avoid fractions 



