iJ S ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



In order to get a better insight into the constitution of the fourdi- 

 mensional hrnpd. nets we tabulate the contact between the different 

 constituents. To that end we introduce first a short notation with 

 respect to the nets themselves and to their constituents and the 

 tlneedimensional limits of these. We denote the hwpd. nets in 8^ 

 by the collective symbol NH k and distinguish them mutually from 

 each other by putting before that symbol the system of expansion 

 operations applied to the second constituent 6V 4 ; so the four nets 

 found above are e 3 NH h , e i <? 3 NH^, e 2 e 3 NH k , e i e 2 e 3 NH^. More- 

 over we indicate the four constituents of each net, i.e. the three 

 principal ones taken in the order of succession assumed in 

 theorem LXVII and the prism, by A , B, C, D and we represent 

 their different limits (/) 3 by means of subscripts in connexion with their 

 import; so A 3 , A t , A will represent the limits of body, truncation, 

 vertex import of A, whilst B { (i = 3 , 2, 1 , 0) and ^(^ = 3, 2,0) 

 will represent the limits of (/), import of B and of (/),, import of 

 C, and B 3 , B 2 , D will stand for the bases of D and the upright 

 limits (/) 3 of that prism which correspond to the faces of face import 

 and of vertex import of the bases. So we find the following small 

 table, where the numbers under the columns show how many 

 (/) 3 of each kind each polytope admits: 



Net 



1 A 



A t 



A | 



1 *3 I 



JB 2 



B i 1 



B* | 



P 1 



ü 3 1 



c, | 



P 1 



^3 | 



A | 



A, 



e,NH, 



T 



T 



— 



T 



p* 



P, 



C 







— 



C 



T 



P* 



— 



e x % // 



tT 



tT 







tT 



p* 



P* 



RCO 







P* 



RCO 



tT 



P« 



p» 



H e-z » 



T 



CO 



T 



CO 



■p» 



P* 



tc 



T 





tc 



T 



P* 





e r e, e s // 



tT 



to 



tT 



to 



P<s 



P* 



wo 



tT 



P* 



wo 



tT 



Pc, 



^3 





8 



8 



8 



16 



32 



24 



8 



16 



32 



8 



2 



4 



* 



This table shows that the contact between the four different 

 constituents is the same in the four nets, i.e. that we have in general 



A 3 = D s , A t = B z , A, = C z , B 2 = D 2 , B = C , C 2 = 2 , 



whilst B is in contact by its limits B^ of edge import with other 

 polytopes B, this transformed edge contact being preserved. So the 

 different threedimensional limits cover each other two by two. 



The contact between the different constituents can also be deduced 

 from the following small table in which we repeat the constituents 

 of the net in an other form: 



Net 



C 



B 



A 



B 



^1 ^2 f> \\ n 



[1111] 



v% 



[I'l'll] 



ii 



[l'l'l'l] 

 [2'2'i'r 



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[I'll 1]V2 



[2'1'1 1] „ 



[a'i'i'i] „ 



[3'2'1'1] „ 



„[3311 

 „[3111 

 „[5311] 



Km] 



"311] [1" 



[Hl][l] 



:bh][i] 



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ii 



