DERIVED FROM THE REGULAR POLYTOPES. 39 



Case of the polytope (P) IU deduced from the measure poly tope. 

 The n 2 spaces Sy n _ x found above are spaces of symmetry for 

 (P) m ; so here again the only question is if (P) m can admit a space 

 of symmetry Sy n _ i which is no Sy n _ 1 for the M n from which (P) m 

 has been derived. We suppose that there is such an Sy n _ ii repre- 

 sent by M' n the mirror image with respect to that Sy n _^ of the 

 M n from which (P) )n has been derived by a set of e k and c opera- 

 tions, and remark now that — as Sy n _ x is space of symmetry for 

 the figure consisting of (P) m and the two measure polytopes M n , 

 M' n — it must be possible to derive {P) tn from M' n by the same 

 set of operations. This particularity presents itself in the case of 

 the octagon e { (;; 4 ) only, as the p^ itself may be represented either 

 as [1, 1] or as [1, 0] V / 2. So we find: 



n 



Theorem XLII. "The measure polytope [1 1 . . 8 1] of S n and 

 the polytopes deduced from it by expansion and contraction admit 

 ?i 2 spaces Sy n _ x of symmetry, the n spaces x i = and the n [n — 1) 

 spaces œ L + x k . = 0- Only in the case of the plane we have to add 

 for e x (pz) the four new axes of symmetry passing through pairs of 

 opposite vertices of the octagon". 



70. Moreover we find: 1 ) 



Theorem XLTII. "The order of the group of anallagmatic displa- 

 cements of the measure polytope M H of S n and the polytopes 

 deduced from it by expansion and contraction is 2 n ~ i . nV' 



"The order of the extended group of anallagmatic displacements 

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

 metry included , is 2 n . n ! In this extended group the first group 

 of order 2 n ~ 1 . n\ forms a perfect subgroup". 



For n = 2 these general results have to be completed in the 

 known way for the octagon." 



For the simple proof we refer to the article quoted. 



71. Finally we have to apply to the polytopes and nets of measure 

 descent the scale of regularity due to M r . Elte. As to the theory 

 we can only repeat here wdiat has been remarked in the art s . 42 

 and 43, with omission of all that refers to the central symmetry 

 of some of the polytopes of simplex extraction. So theorem XXV 

 must take here the simpler form : 



Theorem XLIV. "Any two limiting elements {l) d belong to the 

 same subgroup or to different subgroups, in the sense of the scale 



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



