DERIVED EROM THE REGULAR POLYTOPES. 73 



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



40. We begin by determining the spaces S n _ i of symmetry which 

 may be indicated by Sy n _ i and we consider to that end successively 

 the case of the simplex S(?i ~f- 1) of S n and that of any polytope 

 (P) s deduced from that simplex S(n -f- 1) by the operations of 

 expansion and contraction. 



Case of the simplex. The vertices of S(n -f- 1) lying outside a 

 space of symmetry Sy n _ i of this S(n -)- 1) occur in couples. Now 

 there must be at least one of these couples, as Sy n _ i cannot contain 

 all the vertices of S(n -\- 1), and on the other hand there cannot 

 be more than one of these couples, as S(n -\- 1) does not admit 

 parallel edges. So any space Sy n _ ± must bisect orthogonally one 

 edge of S (n -{-!), i.e. the number of spaces Sy n _ i is ^n(n~\- 1). 



It is not at all difficult to indicate the equations of the in(n -f- 1) 



n 



spaces Sy n _ i of (1 00. .0). For the space S n _ t bisecting normally 

 the edge A k A t joining the points A k and A t with the coordinates 

 [œ k = 1, œ notk = 0) and (% r = 1, x notl = 0) is represented by the 

 equation x k = x t . 



Case of the polytope (P) s deduced from the simplex. It goes 

 without saying that the ^n(n-\-\) spaces of symmetry x k = x t of 

 S(n-\-\) are at the same time spaces Sy n _ i for any polytope (P) s 

 derived from that S(n-\-l) by the operations e and c, and that 

 any two limits of that (P) s which are each others mirror images 

 with respect to any of these Sy n _ l are of the same import. So the 

 only question is, if the polytope (P) s can possess a space of sym- 

 metry which is no Sy n _ 1 for the S(?i-\-l) from which the (P) s 

 has been derived. To answer this question we suppose there is such 

 a space jSy n _ i and we examine the consequences to which this 

 supposition leads. According to this supposition (P) s is its own mirror 

 image with respect to that definite Sy n _ ± , which may be represented 

 by the symbol Sy n _ i , whilst the mirror image of the simplex S(n -\- 1) 

 from which (P) s has been deduced is an other simplex S' (n -f- 1) 

 concentric to S{n-\-Y). But then the figure consisting of (P) s on 

 one hand and the two simplexes 8{n-\-Y), S'{n-\-Y) on the other 

 is symmetric with respect to Sy n _ i ; so it must be possible to de- 

 duce (P) s by the same set of expansion operations from the new 

 simplex S'(n-\~l). Prom this we can draw two conclusions, one 

 with respect to the two simplexes, an other with respect to (P) s . If 

 we can deduce the same polytope (P) s from two different simplexes, 

 these simplexes must be concentric and oppositely orientated; if we 



