02 ANALYTICAL TREATMENT OF THE POLYTOPIES EEGTJLArXLY 



By making the a t to disappear the first and the third l ) 



. /5-1/2 1 — j/2 — 3— JX2 —3 — j/2\ 



symbol give the sets of vertices ( — — ~> — - — > - — -» ^— — J, 



•n-j-i/2 1 + ^/2 — 3-J-K2 — 3-j-|/2\ i r i • i i . 



( !— — • ^r 1 -' -^- -' ^ J each ot which corresponds to 



a (2100), i. e. to a tT. So the result is a P lT , all the vertices of 

 the second symbol lying at larger distance from 0' than the circum- 

 radins V i 3 of this P tT . 



Face gap prismotope. Here the three net symbols are 



[4,2,0,0], (3 + V '2) 2a t — f , 2« 2 — f, 2a 3 — f , 2tf 4 , Z«, even, 



i 



£[3,8,1,1]. (8 + V2)2^+£, 2tf. 2 +l 2fl,+ |, 2a t +l, S«j odd, 



1 



£[3,3,1,1], (3 + V2) 2^+1, 2^+|, 2fl3+£, 2« 4 -R> S^eveo. 



By taking in the first symbol #, = 0, (i === 1,2, 3, 4)*, in the 

 second #, = , (7 = 1, 2, 3), a k = — 1 , in the third a, = 0, 

 (J = 1,2,3,4), we get the three hexagons 



(2 — f\/â, — |V2, — 2 — 2\/2)' 

 (2 + 1^2, i\X2, _2 + |V2) — V2 j. 

 (2 + -1V/2, t y/2, — 2+|V/2) V/2 



So the result is a [6 ; 3]. 



Edge gap prism. Now the three net symbols become 



[4,2,0,0], (3 -f V2) 2^—1, 2fl 2 — 1, 2# 3 , 2« 4 , Z^ even, 



£[3,3,1,1], (3- -V2)2a i , 2a 2 , 2a 3 -\- 1 , 2« 4 + 1 , 2^ odd, 



1 



-£[3,3,1,1], (3 4-1/2)2^ , 2« 2 , 2^3-J- 1, 2« 4 + 1, S^ even. 



i 



By taking in the first symbol a 3 = a k = and either a i = a 2 = 

 or fltj = a. 2 = 1 , in the second ^ 1 = a 2 = and # 3 , a, v = ( — 1, 0), 

 in the third a x = a 2 = and either a 3 = a k = or # 3 = # 4 = — 1 

 and by combining with the not disappearing immovable digits the 

 greater permutable ones, generally affected by the sign tending to 

 decrease the absolute value of the coordinate but — on account of 

 the sign before £ [3, 3, 1, 1] of the second and the third symbol — 



') That one of the three symbols must remain inactive in the generation of the body 

 gap prism is an immediate consequence of the lemma of art. 80. 



