üö ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



Proof. In the original N{IlM n , Cr n ) any two Cr n in contact 

 are either in edge contact, or in vertex contact, in other words 

 the contact of the highest order between two Cr n of the net is edge 

 contact. Now this contact of the highest order can only be annihi- 

 lated by separation of the Cr n i. e. by applying the expansion e n to 

 them. As this operation is excluded (as leading to a net with two 

 kinds of vertices) the edge contact between the polytopes of Cr n origin 

 is maintained, though it is changed in character by the operations 

 e k , 1 < k << n, contact by edge being replaced by contact of an 

 (/)„_! limit of edge import. This proves in the first place that there 

 is only room for one new constituent different from a prism, viz. 

 a new polytope with respect to the Cr n of vertex import; vertex 

 contact being annihilated in any net deduced from N(HM U , Cr u ), 

 this third constituent alioays makes its appearance. Now we have 

 only to prove still that this third constituent is the contraction form 

 of the second; we prove this in two different ways, in the first 

 place by considering the contact with the second, in the second 

 place by considering the contact with the first constituent. 



According to theorem LXVI here also the third constituent 

 is determined by the limits (/)„_! of contact with the %n adjacent 

 polytopes of Cr n origin, the centres of which are the vertices of 

 a cross polytope with the centre of the vertex gap as centre. So, 

 here also, if the 2?i limits (l) n _ i are to determine a polytope with 

 vertices of one kind and edges of one length, there are three 

 possibilities : either the third constituent is equal to the second, or 

 it is the contraction form of the second, or it has the second for 

 contraction form. Here also the first and the last suppositions are 

 inadmissible for the reasons indicated in the case n = 3. So the 

 theorem is proved. 



We add the following second proof, which we consider even more 

 convincing, as a confirmation of the result obtained. In the notation 

 of the problem the limit of vertex import of e k e k . . . e k e k HM n 



is — compare the proof of theorem LXIII — represented by 



"->',> k p~ k P -i k 2~h k i 



(2/; -J-I, 2/;— 1, . . ., 33. .3, 1.1. .1), 

 i. e. 



n ~ k p k p- k p-i /c 2- /r l /r l 



( 2p , 2/>— 2, . . ., 22. .2, 00. .0), 

 or reversed 



h /c 2~ fc i k p-k p -i n ~ k v 



— ( 2p , 2p — 2, . . ., 22. .2, 00. .0), 

 i.e. -ce k ± e, x . .e k 4 e k _ x S(n){ 2 ^). So the limit (/)„_* of 



1 ^ j) — 1 p 



