DERIVED FEOM THE REGULAR POLYTOPES. 91 



limits, the relation between the two chief constituents of an hnpd. 

 net can be thrown into the following form: 



Theorem LXV. "In the hmpd. nets the constituent of HM n origin 

 unequivocally determines that of Cr n origin and vice versa. If the 

 former is e k e k . . .e k e k HM n , the latter is represented by 



e k 1 -\ e k 2 -i- • - e k v _ x -\ e k-\ Cr n . 



We divide the proof of this theorem in two parts. In the first 

 part we suppose k p different from n, in the second we trace the 

 influence of the occurrence of e a HM n . 



Let the set of operations to be applied to the Cr ni in order to obtain 

 a polytope able to form an hnpd. net with e k e k . . . e k e k HM ni 



be represented by e k . e k > . . .e k . e k . . Then according to the 



results obtained in the preceding section the limiting S{n)^V%) of Cr n 

 is transformed into e,, e,, . . .e,.. e k , S(n)(~V%). whilst on the other 



h i k 2 A q _ i A q \ J > 



hand the S(iip*V%) of HM n is transformed into 



As the negative sign of the second symbol is accounted for by the 

 position of the two polytopes at different sides of the common limit 

 deduced from S(n) the coincidence requires that we have 



as the theorem states. 



We now suppose that the operation e n is added to the set of 

 e k expansions to be applied to the HM n , i. e. that we drive the 

 two groups of HM n apart by prisms. Then the enlargement of the 

 side H + H__ of the triangle CH + H_ (fig. 18), formed by the 

 centres C, U + , H_ of any triplet of constituents of different kind 

 in mutually (/) n _ t contact, caused by the intercalation of the prism 

 implies enlargement of the two other sides, as the triangle must 

 remain similar to itself. This enlargement of CH + and CH_ cannot 

 be effected by the application of the operation e n between the two 

 constituents of different form (see pag. 90); so it must be caused by 

 application of the operation e n _ i to the polytopes of Cr n origin. In other 

 words: the theorem to be proved also holds for the case that e n 

 occurs under the operations e k to be applied to the HM n groups. 



Moreover from theorem LXIV we deduce: 



Theorem LXVI. "The totality of the vertices of any hmpd. net 

 can always be represented by means of one net symbol, viz. that 

 corresponding to the constituent of Cr n origin." 



We still remark that the number of hmpd. nets in S n is 2 H_1 . 



