DERIVED EROM THE REGULAR POLYTOPES. 



65 



Edge gap prism. We get 



[6+ j/2, 4 + ^2, 2 + J/2, 1/2], (6 + 3^2) 2^—1, 2a, 2 — 1, 2« 3 , 2« 4 



4 



, 2ai even, 

 1 



[6 + 2J/2, 4 , 2 ,0 1] 

 |[6 + j/2, 4 + >/2, 2 + J/2, j/2j 



[6 + 2J/2, 4 , 2 , 0] 

 i[6 + j/2, 4 + J/2, 2 + J/2, j/2] 



, (6 -|— 3j/2) 2« 1 , 2« 3 , 2«3--l, 2a£--l, 2te; odd, 



1 



>, (6 + 3J/2) ^öj , 2« 3 , 2# 3 --l, 2«5j, — |— 1 , 2tei even, 



giving by means of the suitable substitutions 



-±-[2 + 2/2, 2/2] [2 + /2, /2] 



[ 2 



iP+ ^2 

 [ 2 



1[2+ /2 



0J -J [2 + 3/2, /2], 

 /2] -J- [2.+ 2/2, 2/2], 



0] -±-[2 + 3/2, V/2], 

 ^2] -J [2 -J- 2/2, 2/2], 



which can be combined into 



\ [2 + 2j/2, 2j/2] [2 + j/2, j/2] —, [2, 0] [2 -f 3j/2, j/2] -, * [2 + j/2, j/2,] [2 + 2j/2, 2j/2], 



representing together the 96 vertices of a P iC0 . For the transformation 



œ x + œ 2 = g i V2 \ œ 3 + a? 4 = y 3 \/2 j 



a? 4 — a?2 — ^2 /2 ).' ^ 3 — # 4 = y 4 V 7 2 I 



gives immediately 



y, = |V2] , ^,^,^ = [4 + ^2,2 + ^2/2]. 

 Vertex gap polytope. Finally we have to deal with 



[6+ j/2,4 + |/2,2 + |/2,j/2], (12 + 6J/2) ^ 



[6 + 2 j/2, 4 , 2 , ]j 

 i[6+ j/2, 4 + j/2, 2+ j/2, j/2]; 



, «o 



a, 



, 1,3 , «e, 4 



a A 



, E«i odd, 

 l 



, ( 6 -- 3j/2) 2« T + 1, 2// 3 + 1, 2« 3 + 1, 2a 4 + 1 , 2«j even, 



IÎT^+V 2 ;2+V^A! (,i+aK2)2 '''+ i '^+ , '^+ i '^+ 1 ' 2 "' ^ 



giving by adequate substitutions 



[6 + 5V / 2,4+ V/2, 2+ /2, V/2], 



| [6 + 3 V/2, 4 + 3 V/2, 2 + 3/2 



■J- [6 - 4/2, 4 - • 2/2, 2 ■ ■ 2/2 

 . i 



— A [6- -4/2,4- -2/2,2- -2/2 



6 + 3 /2, 4 + 3 /2, 2 + 3 /2 



/2], 

 2/2], 



/2], 

 2/2 J, 



l. e. 



[6 + 5J/2, 4+J/2, 2+K2, J/2]- [6 + 3j/2, 4 + 3J/2, 2 + 3J/2, J/2]-, [6+4J/2, 4+2^2, 2 + 2j/2,2j/2], 

 representing together the 1152 vertices of the polytope e t e 2 e 3 C n . 



Verb. Kon. Akad. v. Wetensch. 1* Sectie Dl. XI No. 5. 



E 5 



