58 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



furnishing (4 + 2V% 2 -f- 2V2, 2 -f- 2V 7 2), for the third po- 

 sition of the first triangle *). Indeed the part of the second net 

 symbol corresponding to [4 -f- 2V\2, 2,2,0], i. e. 



[4 + 2^2,2,2,0], (4 + 2^)2^ + 1, 2ff 3 + l, 2« 3 +i, 2^+1, 2a, odd, 



4 



gives for âj = a 2 = a â = and fl 4 = — 1 the set of vertices 

 represented by 



(4 + 2Y/2— 0,4 + 2^2— 2, 4+.2V2 — 2) 4 + 2V-2 — 4— 2V2, 



i. e. (4+2V/2, 2 + 2V/2, 2 + 2V<2) 0. 



Edge gap prism. By extension the centre 1 , 1 , , of the edge 

 (2,0)0,-0 of [2, 0, 0, 0J gives 2 (2-f- V2), 2 (2 -f-V / 2>, 0,0 

 for the coordinates of O' . By reducing the first net symbol to this 

 point as new origin Ave get 



[4--J/2, 2 + 1/2, 2 + K2, K2], (4 + 2^2)2^ — 1, 2r/o — 1, 2a s , 2a±, £a, even. 



l 



By putting a t = 0, (i = 1, 2, 3, 4), and taking the permutable 

 digits in the indicated order and with the positive sign we find the 

 vertex — \/2, — (2-f-V / 2) J 2-j-\/2, \/2 lying at minimum distance 

 2\/(4-J- 2\/2) from O'. As this distance is smaller than 4 -j- V2 

 we are obliged, in order to find all the vertices contained in that 

 symbol lying at that distance from O' , to put a s = a k = and to 

 take either a x = a. 2 = or a ± = a 2 = 1. So we find the 32 ver- 

 tices | [2 + V2, V2] [2 -f \/2, V2], where the | refers to the 

 first syllable corresponding to the coordinates ct\, œ 2 only. Now we 

 have furthermore to examine the other two net symbols. For O' as 

 origin the second net symbol is 



[4 + 2K2, 2 , 2 , ]) 



[4- |/2, 2-K2, »--!/». ^4' (4 + 2K2) ** '* 2a3 + 1 ' 2 ^ + 1 ' f ° dd ' 



*) Until now we have only used implicitly the condition that the planes of the 

 generating polygons are perfectly normal to each other, in the equation p] 2 + P2 2 — P 2 - 

 As the plane x± + #2 + ^3 — 0» ^4 = is parallel to those of the first triangle, the 

 plane x^ = x 2 = x ?) perfectly normal to it must he parallel to these of the second. We 

 verify this by the following table of the nine vertices of the prismotope 



4+ y 2, 2+ V 2, 2+ K2, K2 2+ */2, 4+ J/2, 2+ 1/2, Vï 

 4+ J/2, 2+ 1/2,2+ J/2, — J/2 2+ J/2, 4+ J/2, 2+ J/2, — K2 



2+ J/2, 2+ 1/2, 4+ J/2, K2 

 2+ 1/2,2+ J/2, 4+ K2, — V2 

 2+21/2, 2+2J/2, 4+2K2, 



4+2J/2, 2+2J/2, 2+2J/2, I 2+21/2, 4+2^2, 2+2J/2, 



the three rows forming the positions of the first triangle and the three columns (of sets 

 of coordinates) those of the second. So for the triangle of the first column we have 

 x 1 — x 2 = 2, x 2 = x z , etc. 



By continuing this research it can be verified, that each of the three net symbols con- 

 tains the six vertices of a P3 with two positions of the first triangle, i.e. two rows of 

 the table of the nine vertices, as end planes. 



