64 ANALYTICAL TREATMENT ÖE THE POLYTOPES REGULARLY 



[6 f |/2,4 + J/2,H^.^] . (12 + 6^2) a x 

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



, « 3 , a 3 



, « 4 ,2 a i even, 

 l 



i 



— j[6+ j/2, 4 + i/2,2+i/2,V2]\' ( 6 + 3 ^ 2 ) 2ö5 i + X ' 2 ^2 + *. 2 ^3 + 1,2« 4 + 1,2X odd, 



[6 + 2^2, 4 ,2 ,0]) - 4 



l[6+ ^2, 4 + ^2,2 + 1/2,1/2]}' ( 6 + 8 ^ 2 ) 2 **i + l.^» + l,*a 8 + M«4 + !.<*«» even, 



which are to be reduced to the new origins, indicated in the pre- 

 ceding example £i£ 4 iV(C 16 ). But in the case of the body gap we will 

 mention only the first net symbol and the lower part of the third, 

 which lead to the desired result. 

 Bod// gap prism. We find 



f6 + i/2,4 + K2, 2+^2,K2], (6 + 31/2)2^ — i,2« 3 — |, 2« 3 — ^, 2« 4 — ^,vai even, 



l 



\ [6 4 j/2, 4 + ]/%, 2 + 1/2, V%\ (6 + 3 ^2) 2^ + 1 , 2« 3 + i 2« 3 -f- 1, 2^ + J, 1«; even, 



l 



giving by means of the suppositions of the preceding example the 

 prism P t0 , the two bases of which are 



(8 — .JV2, i_ \V2 9 — 1— \Vl,~ 8 — \V2\ 

 ($+iV2,I+iV2,—l+l t V2,-d+±V2). 



Face gap prismotope. Here we have 



[6 + |/2,4-fK2, 2-f j/2,j/2] , (6 + 3*/2) 2« x — f, 2« 3 — f, 2« 3 — f, 2« 4 , Sa x even, 



[6 + 2^2, 4 ,2 ,0]) 4 



i [6 + ^2,4+^2,2 + ]/2, j/2] ' ( 6 + 3 ^ 2 ) 2 «i + *• 2 «2 + *» 2 % + h ^H + 1. -*«* odd > 



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



[6-|- ]/2 t ^ + y2 y 2 + yi t \/2]Y ( ö + 3 ^ 2 ) 2 ^ + *' 2ff 2 + ^ 2ö 3+3 J 2« 4 + l,^even, 



giving by means of the suitable substitutions easily found succes- 

 sive! \ 



(2 + V2, 



V2. 



, — 2-\-V2y- 



- \/2, 



( 2 







2 )- 



-2V2, 



(2 + V2, 



V2 



, — 2 + V/2) 



V», 



( 2 







2 ) 



2\/2, 



which can be combined to 



(2 — V-2, — K2, — 2 — K2)[K2]—, (2 + K2,K2, — 2 + K2)[l/2]— ,(2,0, — 2)[2K2], 



representing together a prismotope [6 ; 6]. 



