DERIVED FROM THE REGULAR POLYTOPES. 



75 



n 



\ [11111] HM S \ 



-- e. 2 HmJ 

 -- e 3 UMI ' 

 -- e 4 HMJ 



= 5 

 K55811] 



| [33 3 1 1] 

 \ [3 3 1 1 1 J 

 i[31111] 



e 2 e s HM h 

 [5331 1] e,_ e k HM- a 



e 3 c k II M b 

 <? 2 e 3 e k HM h 



\ [5 31 11} = 



\ [7 5 3 11] 



We introduce for these forms and for the corresponding ones 

 in spaces of a higher number of dimensions the collective "half 

 measure polytope descendent", which we abreviate to hmpd. 



B. The characteristic numbers. 



92. Tt is not difficult to determine the characteristic numbers 

 of TIM n for a general n. For, if a p and a p denote the numbers 

 of limits (l) p of M n and HM n respectively, we have the relations 



a 2 

 4 a» 



a 



"2 a 





=z 



a 2 



— 



a 9 



= ^3 



i 

 a 4 



= a k 



+ i(»)4 a 



a 



a 



p 



+m 



p+l a 



where at the right the numbers are arranged in two columns of 

 which the first contains old, the second new limits. Indeed the 

 process transforming M n into HM n — which may be called an 

 alternate truncation — destroys half the number of vertices, all 

 the edges, all the faces, and maintains all the other limits (/) 3 , 

 (/)4,.. . ,(/) n _ \ of M n but in an altered shape, bringing new sets 

 of edges, faces, limiting bodies, etc. into existence. Now each face 

 of M n produces an edge of HM n , each limiting body of M n 

 — becoming a T — produces four triangular faces of HM n and 

 finally in general any set of p 4- 1 vertices of M n adjacent to a 

 vertex destroyed produces a regular simplex S (p-\-l) forming a limit 

 (/) p of IlM n , for p = 4, 5, . . ., n — 1. This accounts for all the 

 relations given above. Now, as the characteristic numbers of M n 

 are given by the equation 



a p = (n) p 2 n p , (j? =0,1, 2, ...,?* 

 we find for HM„ : 



i) 



