84 ANALYTICAL TREATMENT OP THE POLYTOPES REGULARLY 



(775)(53)(331—1),— (7553831), -(75533)(311),(75533X31-1),— 



(7533) (3311), (7553) (331—1), (755) (33311), (755) (3331 — 1), 



(75)(53331 1), (751(53331— 1),— (5533311), — (553331 — 1), 



(977553)1 [11], — (9775) (533) £ [11], — (977) (5533) £ [11],— 

 (97) (75533) £ [11], (97) (75) (5333) i [11], — (775533)|[1 1] 

 (775) (5333) \ [11], — (755333) \ [11], (9775)1' 



(97) (755) \ [311 

 (775) (53) \ [311 



311 



, (97) (75) (53) i [311],— (7755)i [311], — 



(7553)1 [311], — (75) (533)1 [311], 



(977)i[331lJ,-(97X75)i[3311],-(775)i[3311],-(755)i[3311] — 



(75)(5*'3)i[3311],-(553)i[33ll], (97)i[3331 l],-(75)i[33311], 



1 [533311]. 



C. Extension number and truncation fractions. 



97. Theorem LV1II. "The polytopes ■J{a 1 fl 2 . . . «„_!«„] of S nt 

 all with edges 2\/2, can be found by means of a regular exten- 

 sion of the measure polytope Mjj® followed by a regular trun- 

 cation at the two groups of non adjacent vertices of M n , either 

 with or without truncation at the limiting (/) 3 , or at the limiting 

 (/) 3 and (/) 4 , or at the limiting (/) 3 , (/) 4 and (/) 5 , etc. or at the 

 limiting (/) 3 , (/) 4 , (/) 5 , etc. and (/) n _ 2 ." 



This theorem is an immediate consequence of the character of 

 the equations of the spaces S n _ x bearing the limits (/) nr _ 1 of the hmpd. 



The extension number is once more the largest digit of the sym- 

 bol of coordinates, i.e. a 1 ; so here it is always odd. 



On account of the lopsidedness of the hmpd. we measure the 

 amount of truncation on the corresponding half diameter limited 

 at the centre O of the polytope. So in the case ^[775533311] the 



5 



truncation corresponding to the space JS> 8 with the equation 2^ = 27, 



i. e. the truncation at the limits M k of M 9 extended, is -^^, if P 



PQ 

 PO' 



is the centre of the M k and Q the point of intersection of OP 



5 



and the indicated space jS s . As E ^ is 35 for M Q extended we find 



l 



PQ PO--QO 35--27 8 e , . , . . 8 



bo trie truncation fraction is 



PO PO 35 " 35' «—- 85 



in this case. 



This case shows clearly that in general the fraction number admits 

 as denominator the product of the extension number by the number 

 of coordinates figuring in the equation of the truncating space. So 

 reducing this denominator to the extension number the numerator 



