46 ANALYTICAL TllEATMENT OF THE POLYTOPES REGULARLY 



the distance of the two limits (/) n _ 2 or of the two vertices, which 

 coincided for r = q, has now become k and this distance may not 

 surpass unity. So we have also to discard all the cases r > q x -|- 1 . 

 So the result is that we can only use the values r = q± and r = q i -\-l, 

 or inversely : the only values of the largest digit q i of the zero 

 symbol of the central polytopes are r and r — 1 . Now as any 

 polytope of the net can be promoted to central polytope we have 

 in general : 



Theorem XII. "Any possible net with period r contains only 

 constituents with zero symbols having for largest digit q x either 

 r — 1 or r. Two adjacent repetitions of a constituent for which 

 q^ = r — 1 are free from each other, whilst two adjacent repeti- 

 tions of a constituent for which q± = r are in contact, in general 



n — 1 



by a limit (/) n _ 2 > but in the particular case (211 ... 10) by a point." 



27. But now unexpectedly a difficulty presents itself. In the case 

 q x = r any two adjacent repetitions of a definitely orientated consti- 

 tuent are in {l) n _ 2 -contact or in point contact , in the case q ± == r — 1 

 these two repetitions are free from each other. In both cases we 

 need other constituents to fill up gaps, in other words all the nets 

 are mixed. But this result is at variance with the existence of the 

 net N(po) in the plane, of the net N(tO) in space. So we have 

 to look out for a way out of this difficulty. This way will present 

 itself immediately, if we examine how to find the other constituents 

 of a net, the central polytope and the period of which are given. 



Let the zero symbol of the central polytope (P)° of a net with 

 period r be represented once more by (qi,q 2 , . . . , q n , 0), where 

 Ave have either q 1 = r or q ± = r — 1. Then we can ask by what 

 processes we can deduce from the symbol 



{rh + (Ji, rb, -f q 2i . . . , rb n + q n) rb n + i -f 0), 



representing the net decomposed into the repetitions of the central 

 polytope, other constituents. There are two of these processes com- 

 pleting each other in this sense that the first can be used in the 

 case q n = 0, the second in the case q n = 1. 



1°. In the case of the zero symbol (q if q 2 , . . ., q n _ x , 0, 0) con- 

 taining more than one zero we can write the decomposing symbol 



(rb, + fj u rb, -f- g 2 , . . . , rb.^, + q n _ if rb n + 0, rb n+i -f- 0), 2 b, = 



in the form 





