1 ANALYTICAL TREATMENT OF THE POLYTOPES EEGULARLY 



an edge passes through a vertex; so the total number of edges is 

 half this product. Now the number of edges passing through a 

 vertex is equal to the number of vertices lying at distance unity 

 from the chosen vertex, and this number is easily determined, as 

 will be shown by examples for n = 4 and n = 5 separately. 



In order to be able to find the number of the limiting bodies 

 (71 = 4) and that of the limiting polytopes (n = 5) we prove in 

 general the following theorem : 



Theorem II. „The non vanishing coefficients c { of the coordinates 

 X; in the equation c^ x x -j- c 2 x 2 -f- ...=/? of a limiting space S n _ i 

 of the polytope deduced from the regular simplex S(n-{-l) of S n 

 must all be equal to each other". 



Proof. The linear equation c x x x -\- c 2 x 2 —J— . . . =jo represents, 

 as far as the vertices of the polytope are concerned, more than one 

 equation, if the coefficients c i are different. We show this by a 

 simple example. If in the case (32110) of # 4 we start from the 

 equation 2 x\ -j- x 2 = p and we try to determine the vertices of the 

 polytope for which the expression 2 a? 4 -f- x 2 becomes either a maxi- 

 mum or a minimum we find, the maximum 8 for x x = 3, x 2 = 2 

 and the minimum 1 for x x = 0, x 2 = 1. So for values oîp situated 

 between 8 and 1 the space 2 x x -j- x 2 = p intersects the polytope, 

 while it contains a limiting face only — and not a limiting body — 

 for the extreme values 8 and 1 of p, as each of the couples of 

 equations œ ± = 3, x 2 = 2 and œ ± = 0, x 2 = 1 determines a plane; of 

 these planes the first contains the triangle x\ = 3, x 2 = 2 and x 3 , x k , x- = 

 (110), the second the hexagon x x = 0, x 2 = 1 and x 3 , x k , x b = (321). 



Erom this example can be deduced generally that the equation 

 c i x i -j- c 2 x 2 -\- . . . .= p represents k different equations, as far as 

 the vertices of the polytope are concerned, if the non vanishing 

 coefficients c { admit together k different values. 



The theorem is not reversible, i. e. not every linear equation with 

 equal coefficients c t represents for the maximum or the minimum 

 value of p a limiting space jS n _ i of the polytope in S n . So in the 

 case of the simplex (10000) of /S > 4 only the five spaces x t = bear 

 limiting tetrahedra of #(5), while the ten spaces x t -\- x lc -— 

 bear faces (100), the ten spaces x L -f- x h -(- x t = bear edges 

 (10), etc. 



In order to find the number of the faces (n = 4) and that of 

 the limiting bodies (n = 5) we determine the form of the limiting 

 bodies (n = 4) and that of the limiting polytopes (n = 5). For the 

 number of faces {n = 4) is half the sum of the numbers of the 

 faces of all the limiting bodies, and the number of limiting bodies 



