DERIVED FROM THE REGULAR POLYTOPES. 49 



constituent only, we get a frame *) dissimilar to that of other nets, 

 with any two adjacent repetitions of the unique constituent in 

 contact by a limit (/) n _i, i.e. a face here; these "exceptions" 

 disappear, if we adhere to the analytical idea, according to which 

 N{tO) admits four groups of constituents. 2 ) 



28. If we indicate by p p the number of the digits of the zero 

 symbol of the central polytope (P)° leaving the remainder p when 

 divided by r, i. e. if p p represents in general the number of the digits 

 p of the zero symbol , but in the particular case of p the sum of the 

 numbers of the digits zero and the digits r (the latter being absent 

 in the case q x = r — 1 of the original zero symbol), we have the: 



Theorem XIII "The two operations stated above which may lead 

 to new constituents of the same net do not affect the circular order 

 of succession of the terms of the series /0 r -i> /V_ 2 , • • • > Pi> Po- This 

 series with the sum n -\- 1 will be called "partition cycle of n ~\- 1 , 

 mod. r" of the net and be represented by ,.( / o r _ 1 , p r - 2 > • • •»/ ? i> A))h"- 



This theorem is self evident. For the first of the two processes 

 does not affect the series at all, whilst the second transforms it into 



Po, Pr-ii P r -2f • ■ > Pi- 



We apply the two processes to an example in order to show the 

 circular permutation of the partition and suppose to that end that 

 in space /% there is a net with the period 4 admitting the con- 

 stituent (3222100). Then application of the two processes gives 

 successively 



Partition cycle 



(3222100) 



(4322210) 



(4432221) = (3321110) 



(4332111) == (3221000) 



(4322100) 



(4432210) 



(4443221) = (3332110) 



(4333211) = (3222100) 



1312 

 1312 

 2131 

 1213 

 1213 

 1213 

 3121 

 1312 



Here every new symbol in the first column is derived by the first 

 process from the one in the line immediately above it which con- 



*) In art. 39 the system of the centres of all the tO of N(tO) will prove to form the 

 vertices of a net of rhombic dodecahedra , which latter net is not of simplex extraction. 



2 ) In order to avoid misunderstanding we stipulate expressly that it is not our inten- 

 tion to replace the notion of kind of constituent by that of group, but that we wish 

 to stick to the notion of kind of constituent, complemented by that of group as soon 

 as the partition cycle (see the next article) is a power cycle (see page 57). 



Verhand. Kon. Akad. v. Wetensch. (1<*« Sectie) Dl. XI. C 4 



