DERIVED FROM THE REGULAR POLYTOPES. 59 



the two sets of permutable digits having to be combined with the 

 same set of immovable ones. Here we find only vertices lying at a 

 greater distance from 0' , unless we take a ± = a 2 = 0. So we get 

 for a 3 , « 4 = (0 , — ■ 1) by means of the upper half of the symbol 

 the 16 new vertices [2, 0] [2 (1 -f- V/2), 0], by means of the lower 

 the 1 6 vertices x, , x, = \ [2 -f- V 2, V 2] , x s , x, = — | [2 -f V% V/2] 

 already contained in the set £ [2 + V% V2] [2 -f- V/2, V2\ 

 deduced from the first symbol. From this may be deduced that 

 the two halves of the third symbol will furnish the two sets 

 [2,0] [2 (I -f V/2), 0] and x u x 2 = £ [2 + V/2, V/2], x 3 , x„ = 

 -i[2 + V/2,V / 2]. 



So the result is a poly tope with 48 vertices represented by the 

 combination of the two symbols -|[2 -j- V/2, V/2] [2 -f V/2, V2] 

 and [2, 0] [2 (1 -\- V2), 0J. It proves to be a P lC . For, by applying 

 on the /<7 represented by the symbol [V2] [2 + V2, 2 -f V2, V/2] 

 the transformation 



x x -jr x 2 = y x V2 \ x 3 -f- x\ =y 3 V2 i 



X \ X -l = j/'2 ^ ^ ! ' ^3 t ^4 = j/ 4 V 7 2 ) 



we get -J- [2 -f- V/2, V/2] [2 -j- V/2, V"2] for the 32 vertices 

 [V/2] [2 + V 7 2] [2 -f V/2, V/2] and [2, 0] [2 (1 +■ V/2), 0] for 

 the remaing 16 vertices [V/2] [V/2] [2 -f V2, 2 + V'2]. 



Vertex gap poly tope. By extension the vertex 2,0,0,0 of [2, 0,0,0] 

 gives 4 (2 -j- V/2), 0, 0, for O' . With respect to this origin the first 

 net symbol is 



[4 + V/2, 2 + V/2, 2 + V / 2,V / 2],(8 + 4V / 2)^ 1 — l,tf 2 ,0 3 ,0 4 , Steven, 



l 



which can be reduced to 



[4 + V/2, 2 + Y/2, 2 + V/2, V/2], (8 + 4 V/2) a,, a 2) a 3 ,a„ E a t odd. 



i 



By taking in this last symbol a it a. 2 , a 3 , a k = [1,0, 0, 0] and putting 

 the digit 4 — |— V^2 always Avhere the 1 stands with the opposite sign 

 of it, we get the 192 vertices [4 -f- 3 V 2, 2 + V/2, 2 + V/2, V/2] 

 lying at the minimum distance 4 (1 -|- V 2) from O' . 



With respect to the same origin O' the second net symbol is 



[4+2J/2, 2 , 2 , ]j 2«, — 1, 2*0+1, 2« 3 +l, 2a.+ l, Yfli odd, 



i[4+ j/2, 2+^2, 2+^2, V'lY l 



the immovable part of which can be reduced to 



(4 + 2V/2) 2 *! + 1, 2 a 2 -)- 1, 2 a 3 -f- 1, 2 a k + 1, Zeeven. 



l 



