DERIVED FROM THE REGULAR POLYTOPES. 



53 



b) Variation in distance. We account for the variation of the 

 distance of any two sixteencells due to the extension of these cells 

 by multiplying the immovable parts of the digits of the three sym- 

 bols of coordinates given above for the three groups of sixteencells 

 by a certain constant. This constant is the extension number itself 

 when the operation e k is lacking, i.e. in the two general cases (e, c) 

 and (c,c) of nets deduced from N(C iG ); in the remaining general 

 cases (e,e) and (c, e) we have to add V 2 to that multiplier in order 

 to create room for the intermediate prisms with 2V/2 as height. 



As we start from [2 0] the extension number is half the sum 

 of the digits. So we find for the multiplier the values given in the 

 following table 





(e, e) 



(e 



,e) 





(c,c) 



fo 



e) 







e k . . 



.1+ V2 







CG^ . . 



.V/2 



e ± . 



.3 



g^ 6? 4 . . 



.8+ V2 



CG i . 



. .2 



CG^G^ . . 



.2+ V2 



e 2 . . 



.4 



G 2 G^ . . 



.4+ V2 



CG 2 . 



. .3 



CG 2 G/^ . . 



.8 -j- V/2 



%■ 



.1 + 2V 7 2 



C 3 G„ . . 



. l + 3V / 2 



CG 3 . 



. .2V/2 



CG 3 G k . . 



.3V/2 



e ± e^. 



.0 



Ga G .) Gr, . . 



.0+ vz 



CCa 6 o • 



. .5 



ce ± e 2 e^. . 



.5+ V/2 



e ± e s < . 



.3+2V2 



G 1 C 3 G k . 



.3 + 3\/2 



CG,G- S . 



. .2-f2V2 



C&i G 3 Gi t . . 



.2 + 3V/2 



e 2 e 3 . . 



.4+2V2 



6 1 G'S G/ L • • 



.4-f.3V / 2 



ce 2 e 3 . 



. .3-J-2N/2 



CG 2 G s G k . . 



.3-f-3 V/2 



G \ e 2 e <i- ' 



.6+2V2 



1 j> 3 4 * ' 



.6-J-3V2 



ce L e 2 e s . 



. .5-J-2V2 



CG^G 2 G Ó G^. . 



.5-j-3\/2 



SO. By means of the preceding developments we can find the 

 three net symbols for all the different nets deduced from N(C i6 ). 

 Bat this work can be reduced by the remark that it will do to 

 use only the net symbol of the erect group in the cases of the 

 seven nets 1 , e i9 e 2) g 1 g 2 , cg x , cg 2 , cg 1 g 1 , while we want these of two 

 groups only for the eight nets e 3 , e ± e B , e 2 e Zi e i e 2 e Bi ce 3 , ce ± e s> ce 2 e s , 

 cg ± g 2 g 3 , and all the three symbols in the remaining cases where e 4 

 occurs. The proof of this assertion is based on the following theorem, 

 where we distinguish the three sets of cases just indicated as the 

 set without g 3 and e k , the set with g 3 and without e 4 , and the set 

 with G k : 



Theorem LII. "Any of the three net symbols represents all the 

 vertices of the net in the set without e 3 and e 4 , two thirds of all 

 the vertices in the set with g 3 and without e 4 , one third of all the 

 vertices in the set with e 4 ". 



This theorem is an immediate consequence of the following lemma: 

 "Any limiting tetrahedron of the net JV((7 16 ) is common to two 

 C iQ belonging to different groups, any limiting triangle is common 

 to three C 16 no two of which belong to the same group". 



