52 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



I. . . . 

 II.... 1 



III... .--4 



"2^ + 2,2« 2 + 0,2a 3 + 0,2tf 4 +- 0' 



2a, + 1 + 1 ,2« 2 +l + l,2fl 3 +l + l,2« 4 +l + l 

 2^ + 1+1,2^ + 1 + 1,2^+1 +l,2<z 4 +l+l 



,2# t even, 

 , „ odd, 

 , „ even. 



Of these three sets I represents the erect group, while II and III 

 form the two inclined groups. 



If we wish to represent analytically the fourdiinensional nets derived 

 from N (C i6 ) we have to start from the three symbols I, II, III, 

 and to study the influence of the operations e k , c. As to the repre- 

 sentation of all the vertices of these new nets by coordinate symbols 

 these influences can be split up into two inadequate parts; of these 

 the first deals with the variation in form of any C iQ of each of the 

 three groups, whilst the second is concerned w 7 ith the variation of 

 the distance of any two C i& . We treat each of these two parts for itself. 



a) Variation in shape. We know the influence of the operations 

 e,,, c on the coordinate symbol [2000] of the central (7 J6 (2j/2) f 

 the erect group and from this we can deduce the corresponding 

 influences on the (? 16 (2j/2) of each of the inclined groups by means 

 of the transformations of coordinates by which [2 0] passes into 

 \ [11 11] and —£[U11]. 



The formulae corresponding to the first transformation are 



fyi 



W\ ~\ <^2 ~\ ^3 |~ ^4 



2^2 



— — X \ «?o ^3 ^4 



2y 3 



■ o?| t2?o " i 6 ^3 ~~ ~~ ' ^4 



2y 4 



c6| c6.2 ft-3 ^^ <^4 



by changing the sign of y x we get formulae corresponding to the 

 second transformation. In the following small table we put on record 

 the result of the first transformation: 



l [2 o o 0] \ [1 1 1 1] 



^[4200] 1 [3 3 1 1] 



<? 2 [4 2 2 0] [4 2 2 0] 



ér s [l'l 1 l]i/2 .' \\%'\ 1 1] and —\\_\'\'\\ \/% — 11 



e 1 c 3 [6 4 2 0] [G 4 2 0] 



e l e. i [2'V\ Y\y2 \ [3 + 2y2, 3 11] and — J[3 + j/2, 3 + ^2, 1', J/2 — 1] 



e 2 e. 6 [2' 1' L' 1] y/2 [4 + 2*/2, 2 2 0] and — \ [4 + J/2, 2 + */2, 2 + y2, j/2] 



e l e 2 e i id'2'l'\]y2 [6 + 2^2, 4 2 0] and — ^[6 + ^2, 4 -f- J/2, 2 -f V% 1/2] 



ce l [2 2 0]= d;^ 2) ... [2200] 



«? 3 [2 2 2 0] — 1 [3 1 1 1] 



ee 3 [1 1 1 1] y 2 = (j^ 2) - [2 0]j/2 and — £ [1 1 1 1] y I 



cé? x <? 2 [4 4 2 0] — i[5 3 1 1] 



ce l e i [i'l'Y Y\yi [2 -f 2y2, 2 0] and — Jfl'1'1 1] \/2 



«f a e 8 [l'l'l'l]V2 — i [3 + 21/2, 1 1 1J and — J [3 + j/2, I'll'] 



ce 1 « 9 ^[2'2'l'l]l/2 — it 5 + 2K2 5 3 1 1] and — \ [5 + J/2, 3 + J/2, 1 + J/2, 1 + J/2] 



