18 ANALYTICAL TREATMENT OE THE POLYTOPES REGULARLY 



qV% , . 

 fraction ^— - is irrational if the symbol of coordinates of the 



polytope winds up in 1 and rational if the last digit of that symbol 

 is zero. 



57. If we indicate the truncation numbers corresponding succes- 

 sivily to a truncation at a vertex, an edge, a face, ... by r , T x T 2 , . . . 

 and p stands once more for 1 -j-jpV/2 we have: 



Theorem XXXII. "If [m' , m\, m' 2 , . . -, ni n -_i] is the symbol of 

 coordinates of a polytope deduced from the measure polytope M n {2) 

 of S n — where m\ l _ i stands for either 1 or — the truncation 

 numbers r , T ±i T 2 , • . • are 



n — 4 n — 2 n — 3 



T =n w — 2 m i3 T ± = (n — 1) w — S m t , T 2 = (n — 2) m — 2 %, 



i = i = i = 



Proof. Here m is the extension number. Now, if we wish to 

 calculate r k and we take for the vertices P and Q of the ex tented 

 measure polytope [m , m ,. . ., w' ] the points m' , m ,. . ., w' 

 and — m , m' , . . . , w' differing in the sign of ^ only, we have 

 to apply the theorem of page 27 (art. 17) with respect to the equation 



#! -|- #2 ~h • • • ~t~ œ n-k — c t> it — 1 > 2,3), where c ( is determined 

 by the condition that this space is to contain successively the points 

 P, Q and the pattern vertex m , m ± , m' 2 , . . . , vi n _ i of the polytope 

 under consideration. So we find 



n — k — 1 



c d = (n — k) m f , c 2 = (n — /; — 2) m , c 3 = 2 m % 



i=0 



and therefore 



n—k—\ 



PA O — &) m'o — 2 m'i 



PQ = ' 2^ 



But, as 2?n' is PQ, the numerator is P72. As the rational part 



n — k — i 



of (n — k) m'o is equal to that of ^m if viz. n — k for m' n _ 1 = 1 



i=0 



and zero for m' n _ i = 0, this numerator is a multiple of V2 t 

 as we have stated at the end of the preceding article. So we find 



n — k — 1 



Th = (n — k) m — 2 %, as the theorem has it. 



i = 



In the case of the polytope P i0 represented by [&' 4' 4' S' 3' 2' 2' 2' V 1] 

 and in the case of [5443322210] we get 



T = 24, T 4 = 19, T 2 = 15, T 3 = 12, T 4 = 9, T 5 = 6, T 6 = 4, 



T 7 = 2 , T 8 = 1 . 



