DERIVED FROM THE REGULAR POLYTOPES. 87 



in the direction of the line joining O to its centre M, for which 



X\ = X 2 = . . . = 00 n _ k = 1 , 00 n _ k _|_ i = 0C n _ k _|_ 2 = . . • = #? n = U , 



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

 HMpV^) characterized by 



#1 == #2 == • • • == ^w— A; == ^' ^fir-fc + i» ^nr-fc + 2» * * • > ^n " = 2" l_ ' * J' 



n- -A: k 



in which it is a limit HM^V^) of the new polytope \ [AA . . A 11 . . 1]. 

 According to theorem LVI this new polytope has edges of the same 

 length if and only if we put A = 3. This proves the theorem 

 and leads moreover to the result: 



Theorem LX. "In the expansion e k the limits HM^V%) of 

 HMJ^Vfy are moved away from the centre to a distance always 

 three times the original distance." 



This result is also an immediate consequence of the fact that 

 the largest digit 3 of the symbol of the new polytope is the 

 extension number. 



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



n 



the symbol -J- [11. . . 1] by saying that it creates an interval 2 be- 

 tween the n — k th and the n — k -j~ 1 st digit. This is in accordance 

 with the remark inserted at the end of art. 58. In moving out 

 the limits M k of M n the distance to be described in order to give 

 the new edges a length 2\/2 is V^2 times the distance to be 

 described in order to give these edges a length 2 ; so the interval 

 created which was \/2 in the case of M^ must be \/2 times V'Z, 

 i. e. 2 in the case of HM n @W>. 



Theorem LXI. "The influence of any number of expansions 



e k> e i> e »i>'- °f HM^WZ) on its symbol ^[11 . . .1] is found by adding 

 together the influences of each of the expansions taken separately. 



The proof of this theorem can be copied from art. 59. It leads 

 immediately to: 



Theorem LXII. "The operation e k can still be applied to any 

 expansion form deduced from HM(%V%) in the symbol of coordinates 

 of which the n — Jc th and the n — h -f- 1 st digits, i. e. the tt h and 

 the k -J- 1 st digits counted from the end, are equal." 



So in the case ^[9775533311] we have an e 2 e b e 1 e$ HM\. 



99. We have to come back to the face expansion of the hmpd. 

 and to their contraction. 



The faces of the polytope -J- [_a ± a 2 . . .a. n _ 2 1 1] replacing the faces 



