DERIVED FROM THE REGULAR POLYTOPES. 61 



Feriex yap poly tope. The new origin is 2 (2 -|— 2 \/2), 0, 0, 

 and the first and second net symbol become, in the shortest form 

 possible , 



4 



[2-f- j/2, 2 + J/2, y 2, yi\ (4-J-4J/2) a Y , a% , a s , a_ v , 2«i ot R 



1 



YfXVi Wk U. IÀ' (2 + 2K2) 8«, + l.^ + l. »«, + !. ^ + T. f < -en. 



Putting into the first symbol «,, # 2 , # 3 , # 4 = [1, 0, 0, 0] and com- 

 bining with the a t differing from zero one of the two digits 2 -\- V2 

 taken with the sign tending to decrease the absolute value of the 

 coordinate we get the 192 vertices [2 -f 3 V2, 2 -f V2, V2, V2\ 

 Putting into the upper half of the second symbol a t = 0,(^=1,2,3,4), 

 we find moreover the 96 vertices [2 -f 2 \/2, 2 -f- 2 \/2, 2 \/2, 0]. 

 So the result is a poly tope with 288 vertices which will prove later 

 on to admit the characteristic numbers (288, 864, 720, 1 44) and 

 to be e 2 C 2i . 



83. Case e 1 e^x(C 16 ). Here the extension number is 3 -|- V 2. So 

 we have to reduce the three net symbols 



4 



[4,2,0,0], (6-j- 2 V2) a ± , a 2 , a 3 , # 4 ,2^ even, 



1 



£[3,3,1, 1],(3+ V / 2)2fl 1 +l,2fl 2 +l,2fl 3 + l,2a 4 +l,S« i odd, 



— £[3,3,1, l],(3-f \ / 2)2a i -^ r 1 ,2« 2 -f l v 2fl3 + l;2a 4 + 1,2a, even 



1 



for the constituents of body, face, edge, vertex import to the new 

 origins (3 + V2) ^ £,£,£, (3 + \/2) fTfTfTÖ, (3 + V2) 1,1,0,0, 

 (3 ~\- \/2) 2, 0, 0, respectively, the constituent of polytope import 

 being [4, 2, 0, 0] == e 1 C iQ . 



Body yap prism. The three net symbols become 



[4,2,0, 0],(3 + V / 2)2fl 1 — £,2a 2 — £,2a 3 — £,2^ 4 - -£, Eleven, 



£[3,3,l,l],(3-+-V/2)2« 1 +£,2« 2 +£,2%4-£,2« 4 4-£,Za,odd, 



£[3,3,1, l],(3 + V/2)2fl 1 + £,2fl 2 + £,2flr 3 + £,2fl 4 -f£, Steven. 



