2 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



extended over such a distance that the two new positions of any 

 vertex which was common to two adjacent M^ shall be separated 

 by the length 2 of an edge. 



Now if we move the limit M k (2) for which w T e have 



k 



#1 = cV 2 = . . = <K n _ k = 1 , 02 n _ fe _j_ 1? W n _ k _j_ 2 , • • , 0S n = [_ 1 , 1 , . . . , 1 J 



in the manner described in the direction of the line joining O to 

 its centre M, for which 



x \ == ^2 == • • == &n — k === *■ f %n - k + 1 == ^n — fc + 2 == ■ ■ === ^n == ^ > 



to a À times larger distance from O we get a new position of this 

 M k (2) characterized by 



k 



%\ ^^ &2 == • • == <%n — k == A > %n — k + 1 > c ^n — & + 2 > ■ • > ^n ==:: [_ * * ' ' ' * _J> 



w — /c k 



in which it is a limit J^ fc (2) of the new poly tope [ÀÀ. . .A 11 ... 1] 

 and according to the last ten lines of art. 48 this polytope belongs 

 to the progéniture of M n (2) if we have À = 1 -\~ \/2. So the result 



n — k k 



is [1' 1'. . . 1' 1 1 .,. . 1], which proves the theorem, and we find 

 by the way: 



Theorem XXXIII. "In the expansion e k the limits M,P of M n {2) 

 are moved away from the centre to a distance always equal to 

 1 -f \/2 times the original distance." 



This comes true, for 1 -j- V2 is the first digit of the symbol 

 of coordinates of the new polytope and, as we found in art. 56, 

 this first digit represents the extension number. 



As the distance OM was V(n — ft) it becomes (1 -\-V2)V(n — Jc). 



Remark. We may express the influence of the operation e h on 



the symbol [11 ... 1] without interval between the digits by saying 

 that it creates an interval V2 between the n -\~ k th and the 

 n -\- k -\- 1 st digit. 



59. Theorem XXXIV. "The influence of any number of expan- 



ii 



sions e ki e u e m ,. . . of M n (2) on its symbol [11. . .1] is found by 

 adding together the influences of each of the expansions taken 

 separately." 



Proof. We begin by combining two expansions only. 



In the succession of two expansions the subject of the second 

 is to be what its original subject has become under the influence 

 of the first. So in the case e 2 e 1 C of the cube 6' (fig. 13 a ) the 



