DEIUVED EROM THE REGULAR POLYTOPES. 



85 



itself becomes in general a fraction. Therefore it is impossible to 

 introduce here the notion of truncation integer. 



The following list contains the truncation fractions for the hmpd. 



in 8- 



two 



-}/, , AJc 



3 , KJ£, ^5 , 



truncations 



at 



S 3 l[3lf] 



$4 



£[3311] 

 £[3111] 

 £[5311] 



£[83311] 

 £[33111] 

 £[31111] 



£ 5 (£[55311] 

 £[53311] 

 £[53111] 



£[75311] 



here r , T ' 

 the vertices 



T 3 , T 4 represent successively the 

 and the truncations at the limits 



T 



^0 



T 3 



1 4 



2 





9 



3 





1 



1 





3 



2 





1 



2 





2 



3 





1 



3 





2 



5 





4 



2 





1 5 



5 





2 



8 





5 



1 5 





8 



2 



1 



1 5 



3 



3 



2 



1 2 





5 



25 





1 2 



1 4 



1 



25 



25 



5 



1 4 



1 6 



1 



2 5 



25 



5 



1 8 



4 



1 



35 



7 



7 



£[333311] 

 £[333111] 

 £[331111] 

 £[311111] 

 £[555311] 

 £[553311] 

 £[533311] 

 &( £[553111] 

 £[533111] 

 £[531111] 

 £[775311] 

 £[755311] 

 £[753311] 

 £[753111] 

 £[975311] 



T 



T 



Ta Ti 



2 



1 







9 



3 







1 



4 







3 



9 







4 



5 



2 





9 



9 



9 





5 

 9 



2 

 3 



4 

 9 



1 



3 



1 



2 







3 



5 







2 



5 



rr 

 i 



1 5 



2 



1 5 





7 



8 



4 



1 



1 5 



1 5 



1 5 



5 



7 



8 



2 





1 5 



1 5 



1 5 





8 

 15 



3 

 5 



4 

 1 5 



1 



5 



3 



2 



2 



1 



5 



3 



5 



5 



3 



7 



10 

 2 1 



2 

 2 1 





1 



1 1 



4 



1 



2 1 



2 1 



2 1 



7 



1 1 



4 



2 



JL 



2 1 



7 



T 



7 



4 



1 3 



2 



1 



7 



2 1 



7 



7" 



1 4 



5 



2 



1 



2 7 



9 



9 



9 



D. Expansion and contraction symbols. 



98. For £ = 2,3, 



# 



2 



n 



1 any limit J/" fc (2) of the 



M n C2) from which the HMJ$V*) has been deduced bears a limit of 



HM n <*V*\ this limit £ [11 . . 1] being a HM^V*) and therefore an 

 (J) k for £=3,4, . . ., n — 1, but an edge for h = 2. Now we 

 will define the expansion e k oîHM n (%V%) — for/- = 2, 3, ..., n — 2 — 



as the influence of the motion of the limits £[H. .1] contained 

 in the limits M^ of M n ^ caused by a translational motion of these 

 limits J/// 2 ' and what they contain, to equal distances away from 

 the centre O of M n ^ 2 \ each M,} 2) moving in the direction of the 

 line O M joining O to its centre M, these iI// 2) remaining equi- 

 pollent to their original position, the motion being extended over 

 such a distance that the two new positions of any vertex which 

 was common to two M k ^ shall be separated by the length 2\/2. 

 In order to justify this definition we have to show for what reasons 

 Ave deviate here from the custom followed until now-, to bring the 

 operation e k in relation with the limits (l) k of the polytope itself. 



