58 



ANALYTICAL TREATMENT OF THE POLYTOPES EEGULARLY 



We now come back to the particularity of the nets 5 y/ ft , b VI b 

 hinted at above. We see now at a glance that these two nets are one 

 and the same, the polytope of the p th small column of the one 

 being equal but oppositely orientated to the polytope of the 7 — p th 

 small column of the other (p = 1 , 2, . . . , 6). So in each the six 

 constituents present themselves in only one of the two possible 

 orientations, which implies that none of them can be central sym- 

 metric, as in the range of the n -j- 1 different constituents of a 

 net of /S a in the order of succession obtained by a regular appli- 

 cation of the processes of art. 27 adjacent polytopes of a central 

 symmetric one differ in orientation only. Or otherwise : two opposite 

 limits [l) n _x of a central symmetric constituent are covered by two 

 congruent but oppositely orientated polytopes, i.e. if we project 

 on the line CO joining the centre C of any polytope of the net 

 to the centre O of any limit (/) n __! of this polytope all the poly- 

 topes of the net the centres of which lie on that line, the projection 

 of any central symmetric polytope with its centre on CO acts as 

 a "turn table" with respect to that projection. 



31. The simple rule of theorem XV enables us to extend the 

 list of nets to any value of n we like. So Ave would find for n = 6 

 and n=l respectively the 17 and the 29 cases represented as to 

 their roman rank index, their partition cycle and the character of 

 their constituents in the following small tables, where the three 

 subdivisions of each last column give successively the number of 

 central symmetric constituents, the number of the asymmetric con- 

 stituents .occurring in pairs and the number of asymmetric con- 

 stituents occurring in one orientation only. 



n 



6. 



I 



7 





6 





II 



16 



1 



6 





III 



25 



1 



6 





IV 



34 



1 



6 





V 



115 



J 



6 





VI 



124 







7 



VII 



VIII 



IX 



X 



XI 



XII 



133 



1 



6 





22 3 



1 



6 





1114 



1 



6 





1123 







7 



1213 



1 



6 





1222 



1 



6 





XIII 



XIV 



XV 



XVI 



XVII 



11113 



11122 



11212 



111112 



1111111 



