56 ANALYTICAL TREATMENT OF THE POL1TOPES REGULARLY 



Vertex gap poly tope. By extension of the vertex 2,0,0,0 of [2,0,0,0] 

 Ave get 12, 0, 0, as new origin 0' . By substituting #, — 0, {i= 1 , 2, 3,4), 

 n the first place and a x = 2 , a t = 0, (/ = 2, 3, 4), in the second (with 

 he movable digit G taken negatively) we put in evidence the two 

 »ets of vertices 6 [4, 2, 0] and 1 S [4, 2, 0], i.e. with respect to O' the 

 vertices [6] [4, 2, 0] contained in the net symbol. But this symbol 

 still contains other vertices lying at the same minimum distance 

 2\/14 from Ü ', i.e. all the vertices represented with respect to that 

 point by [0, 4, 2, 0] and no other. So we find e.g. the point 4, 6, 2, 0, 

 with the coordinates 16,0,2,0 with respect to the original axes, 

 by considering the vertices 1 2 ^ + 4, 1 2 a 2 — 6, 1 2 a s + 2, 12 # 4 

 and putting a ± = a 2 =l and a 3 = a^ = 0, etc. So the result is 

 that the constituent of vertex import is a [6, 4, 2,0] = e x e 2 C 16 and 

 therefore identical with the constituents of polytope import. 



Case ce l e 2 J¥(C u ). Net symbol 



4 



[10^-f 4, 10tf 2 + 4, IO03 + 2, 10fl 4 + 0],2:« 4 . even. 



1 



Here the constituent of polytope import is [4, 4, 2, 0] = ce i e 2 (7 16 . 

 As in the preceding case of e 1 e 2 N(C i o) the constituents of body 

 and of face import are lacking. Moreover by the contraction the 

 original edge and therefore also the constituent P c of edge import 

 is annihilated, i.e. P c is reduced to its base C. We verify this 

 analytically as follows. By extension of the midpoint 1,1,0,0 of 

 the edge (2,0) 0,0 of [2,0,0,0] we get 5,5,0,0 as new origin 

 O'. Now the vertices at minimum distance from O' contained in 

 the net symbol are found by putting a i = 0, (i= 1, 2, 3, 4), giving 

 4, 4 [2,0], and a 1 = a 2 =], a 3 = a Ii = (with the two digits 4 

 taken negatively) giving 6,0 [2,0], i.e. with respect to O' the two 

 squares 1,1 [2,0] and — 1, — 1[2,0] forming two opposite faces 

 of a cube with O' as centre. 



Finally we remark that the contraction c does not affect the con- 

 stituent of vertex import. This is easily verified by determining the 

 vertices at minimum distance from the point O' with the coordi- 

 nates 10,0,0,0 presenting itself here. 



82. Case e 2 e 3 jr(c 16 ). As the operation e 3 presents itself here we 

 have to find besides the constituent [2' 1' 1' 1] \/2 = e 2 e 3 C iQ of 

 polytope import those of face, of edge and of vertex import, and 

 in order to be able to gather all the vertices of these constituents 

 we have to use two of the three net symbols. But we prefer to 



