38 ANALYTICAL TREATMENT OF THE POLITOPES REGULARLY 



in the case of N centres of limits /7/ 3 , 



„ c\ N 9 c e x iV, c e { e 2 N „ „ „ M ó , M , 



„ e 2 N, e i <>, N „ „ „ M s , M { , M , 



whilst — as we remarked above — in the cases where e- s occurs 

 all the groups of centres contribute to the system of vertices. 

 In the case of the groups /)/,, M a space filling double pyramid 

 on a square base may be considered as the constituent of the reci- 

 procal net, in the case of the three groups M 3 , M if M we are 

 obliged to consider as constituent a polyhedron (5, 9, 6) which may 

 be got by dividing the double pyramid mentioned into four equal 

 parts by bisecting the pairs of parallel sides of the square base x ). 



G. Symmetry, considerations of the theory of groups , regularity. 



69. We determine the spaces of symmetry Sy ll _ i and consider 

 successively the case of the measure poly tope M n of S n and that of 

 any polytope [P) m deduced from M n by the operations of expansion 

 and contraction. 



Case of tli e measure polytope. Let us suppose Sy n _^ is a definite 

 space of symmetry of M n and let A i be a vertex of M n not con- 

 tained in Sy n _ x . Then the mirror image of A { with respect to 

 tyn—i i s an other vertex A 2 of M ni which implies that A ± A 2 is 

 either an edge or a central diagonal of a certain limit M k of M n 

 where Tc may be — the case of the edge included — one of the 

 numbers 1, 2,. . } n — 1. Let S k be the space containing that M k . 

 Then any edge A v A' through A x of M n not belonging to M k is 

 normal in A x to S k and therefore to A i A 2 ; so these n — k edges 

 A ± A' are parallel to Sy n _ i and M n can be generated by prismati- 

 zing M k in these directions, i.e. Sy n _ i is a space of symmetry of 

 M u , if and only if its section /S > / ,_ 1 with S k is a space of symmetry 

 of M ki which condition is fulfilled in the cases h = 1, k = 2 

 only. Lor in all the remaining cases £=8,4,..., n — -1 (and 

 also for k = n) the two simplexes S{k) the vertices of which are 

 the groups of vertices of M k adjacent to A { and to A 2 are equal 

 but of opposite orientation, which proves that the space /S^ of S k 

 normally bisecting A L A 2 is no space of symmetry of M k . 



Lor k = 1 the line A ± A 2 is an edge, for k = 2 it is a diago- 

 nal of a face. So the two groups of spaces Sy ll _ 1 are the n spaces 

 #. = and the n (n — 1) spaces x i + x k = 0; so the number of 

 spaces Sy n _ i is n 1 . 



*) We defer further developments about reciprocal nets to an other paper also des- 

 tined to complement art. 39; compare "Nieuw Archief voor Wiskunde", vol. X, p.273 # 



