1 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



So, all in all we have got the limiting bodies 



8tC , 24 P 8 , 32 1\ , 16 CO; 



so their number is 80. 



As the numbers of faces of t C. P 8 , P 3 , CO are respectively 14, 

 10, 5, 14, the total number of faces is 



i (8 X 14 + 24 X 10 + 32 X 5 + 16 X 14) = 368. 



So the final result is (192, 480, 368, 80), in accordance with 

 the law of Euler. 



Remark. In the case of the measure polytope C s of 8^ repre- 

 sented by [1, 1, 1, 1] the spaces represented by 



a) .... x ± = 1 



b) .... œ i + x 2 = 2 



c) .... x ± + x 2 + w B = 3 



d) .... œ ± + x 2 + x z + #? 4 = 4 



contain respectively a limiting cube, a face, an edge, a vertex of 

 (7 8 . So we find here in the case of the chosen example 



8 tC of body import, 

 24 P 8 „ face 

 32 P 3 „ edge 



16 CO „ vertex 



J? 



52. Example [1 + 3 v '2, 1 + 2 i/2, 1 + 2 v/2, 1 + V%, 1]. 

 The number of vertices is 2 5 . 5! : 2! = 32. 120 : 2 = 1920. 

 The number of edges passing through each vertex is six, as can 

 be derived from the symbol 



1 + 3 V/2, 1 -f- 2 V/2, 1 +2 V/2, 1 + V2, 1 (— ) , 



containing live brackets and the negative sign after 1. So the 



1920 X G 

 number of edges is — - — — - — = 5760. 



In this case the limiting polytopes can only lie in spaces # 4 with 

 equations of the form 



a) ... ±x\ = 1 4- 3V2, 



6) . . . + œ i ± x 2 =2 + 5 V/2, 



c) . . : + œ ± ± Wz ± c2? 3 =3 + 7 V/2, 



c/) . . . + ^ + a? 2 + œ z + # 4 =4 + 8 V/2, 



*)...+ # 4 + # 2 + #3 + 5?4 + x h = 5 +8 V/2, 



corresponding respectively to 



