DERIVED FROM THE REGULAR POLYTOPES. 75 



two cases, the case u = 2 found above and the case #==3 of the 

 tetrahedron J i A 2 J 3 Ai t with the planes through O parallel to a pair 

 of opposite edges. So we have proved the general theorem: 



n 



Theorem XXIV. "The simplex (1 00 . . 0) of 8 n and the poly- 

 topes deduced from it by expansion and contraction admit ^n^i-\-l) 

 spaces Sj/ I( _ i of symmetry, the spaces x- t = x k . Moreover in the 

 plane the e ± (p d ) admits the three new axes of symmetry x x = g 

 of the hexagon, whilst in space the ce i 1 1 = 0, e 2 T= CO, e i e. 2 T = tO 

 admit the new planes of symmetry œ i -\~ œ t = x k -\- cc L of the octa- 

 hedron". 



41. We now prove the following theorem x ): 



Theorem XXV. "The order of the group of anallagmatic dis- 

 placements of the simplex S(n -f- 1) of S tl and of the polytopes 

 deduced from it by expansion and contraction is jy(n -\- 1)!" 



"The order of the extended group of anallagmatic displacements 

 of these polytopes, reflexions with respect to spaces Sy n _ i of sym- 

 metry included, is (#-j-l)l In this extended group the first group 

 of order -i-(#-j-l)! forms a perfect subgroup". 



"For n = 2 and n = 3 these general results have to be com- 

 pleted in the generally known way". 



The simplest proof of this theorem is connected with the remark 

 that reflexion of the polytopes with respect to any space Sy n _± 

 corresponds to the interchanging of any pair of vertices of the sim- 

 plex. So the order of the group of reflexions (and anallagmatic 

 displacements) is equal to the number of permutations of the n -\- 1 

 vertices of S(n-\-l), i.e. (n-\-l)\, and the group of the anallag- 

 matic displacements is of an order half as large, i. e. of order ±(n-\-l)\ 



For the cases n = 2 and n = 3 we refer to F. Klein's "Vor- 

 lesungen über das Ikosaeder" (Leipsic, Teubner, 1884). 



42. The manner in which the polytopes considered here have 

 been derived from the simplex is a guarantee that all the vertices 

 are of the same kind and all the edges have the same length. But 

 this is all that can be asserted ; so e.g. the polyhedron tT has two 

 kinds of edges, edges common to two hexagons lying in planes 

 including a definite acute angle and edges common to a hexagon 

 and a triangle lying in planes including the obtuse supplementary angle. 

 So in judging of the regularity we have to look at the edges from two 

 different points of view ; we must not only take into account the 

 length but also consider angles on or faces through the edges, etc. 



*) Compare Report of the British Association, 1894, p. 563. 



