46 ANALYTICAL TREATMENT OE THE POLYTOPES BEGULAELY 



as the amount of truncation at (/) , (l) i , (/) 2 , (/) 3 . As these 

 numbers 



_|_ (4 4 + BV/2), î |y(13 + 4V/?), î iy(8 + 13V/2), î ^(9\/2-5) 



are rather impractical, we only put on record in Table IV the 

 results relating to the cross poly tope forms proper, where the 



MP 



denominator and the numerator of the fraction — -— are both 



MO 



integer multiples of Y/2. Here the result 9 6,3,1 correspon- 

 ding to [3321 0] V/2 expresses that the amount of truncation at 



2 11 



OOo, 00i> OO2 is respectively -, -, -. 



I). Expansion and contraction symbols. 



75. What we have to prove here is: 



Theorem XLVI „The expansion e k , (fc=I, 2, 3, . . , n — 2), applied 

 to the cross polytope GyP^ of S n changes the symbol of coordinates 



n — 1 



[100. . .0]\/2 of that regular polytope by addition of \/2 to the 



ƒ■ n — k — 1 



first /;+ 1 digits into [211. .1 ÖÖT7Ö]V / 2, whilst in the case of 

 e n _i where application of this rule would give a symbol without 

 zero we have to add unity instead of \/2 to all the digits, giving 



n — 1 



[I'll. . .1]". 



Proof. We treat the cases k<^n — 1 and k = n — 1 separately. 



Case k<in — 1 . The operation e h acts upon the limits (l) h = S{h -\- 1 ) 

 of the cross polytope. Now the centre M of the limit (/),,. represen- 

 ted by 



x x , x 2 , . . . , œ,. + ! = (100 . .0) Y 7 '2, x k + 2 = os k + 3 = . . . = x t = 

 has the coordinates 



_ , _ _ _ V2 __ _ _ _ n 



X i ■ - X 2 — . . . M,. + ] i , i %k+2 cT k+Z • • ■ °°n ^ > 



If we move this limit (l) k parallel to itself in the direction 

 OM to a position (l)' k for which the centre M' satisfies the rela- 

 tion OM' = a. OM, where A is to be determined, we find for the 

 coordinates of M' 



A\/2 

 k-\-\ 



af i — x 2 — . . . = x,, +i = — — , x k+2 = w = . . . = Wn = 



