S ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



vertices of the polytope for which the expression 2x ± — x 2 becomes 

 either a maximum or a minimum we find the maximum 3 -\- 5\/2 for 

 t r 1 =l+2V / 2, c^ 2 = — (1 -f- V2) and the minimum — (3 + 5V2) 

 of the same absolute value for x\^= — (\-\-2\/2), x 2 =l-{-\/2. 

 So, for values of p between 3 -f- 5\/2 and — (3 -j- 5\/2) the 

 space 2x i — at 2 =p intersects the polytope, whilst it cannot contain 

 a limiting body but at most a limiting face only for the extreme 

 values + (3 -f- 5\/2) of p, as each of the two couples of equations 

 ^ = 1+2^2, x 2 = — (1+V X 2) and ^ = — (I -f 2\/2), 

 a> 2 = 1 -\- V2 determines a plane. Here too, as far as the vertices of 

 the polytope are concerned, any linear equation c i x\ -j- c 2 x 2 -f- . . . =p 

 represents k different equations if the non vanishing coefficients c t admit 

 /■ different absolute values. Here too the theorem is not reversible. 

 As to the theory of the determination of the number of faces (n = 4) 

 and the number of limiting bodies (n = 5) compare the end of art. 6. 

 Remark. In accordance with the central symmetry of the polytope 

 [a i9 a 2 ,. . ., «J any two parallel spaces S n _ if represented by the 

 equations œ i -\- x k -j- x L ~\- . . . = + p and lying therefore on different 

 sides at the same distance from the origin, bear either both or none 

 of them a limit (/) u _i of the polytope. So, in the determination 

 of the limits (/) n _i we can restrict ourselves here to the equations 

 x i -j-- x k -\- %i -j" • • • = maximum. 



51. We now treat at full length two examples, one in S± and 

 one in 8 5 . 



Example [1 + 2i/2, 1 + i/2, 1 + \/2, 1] ! ). 



The number of vertices is 2 4 . 4! divided by 2!, i. e. 

 16. 24:2 = 192. 



The number of the edges passing through each vertex is five. 

 For the pattern vertex 



1 4- 2\/2 , 1~+ V2 , 



1+ v% , 



1 



is adjacent to the five vertices 







1 + Y/2 , 1 + 2V/2 , 

 1 + y/2 , ' 1 + V2 , 

 1 -f iV2 , 1 

 1 + 2V/2 , 1 + }/2 , 



1 + 2V2 , 1 + V % > 



1 + V2 

 1 + 2V/2 , 

 1+ V2 , 



1 

 1+ V2 : 



1 



1 



1 + V2 

 1 -j- V2 

 , — 1 



*) In vol. XI of the „Wiskundige Opgaven" we have recently treated the polytope 

 [1 + 31/2, 1 + 2J/2, 1 + J/2, lj and its projections on its four kinds of axes (pro- 

 blem 78) and deduced the symbol of characteristic numbers of the polytope [1 + (n — 1) J/2, 

 1 + (n— 2) J/2,. . ., 1 + K'2, 1] of Sn (problem 80). For the latter point compare also 

 my paper „On the characteristic numbers of the polytopts e 1 <? 2 . . . e n _ 9 e n _ i S(n+l)and 

 e l e i ... e n _2 e n —\ M n of space S „" (Mathematical congress, Cambridge, August 1912). 



