7 8 ANALYT KAL TllEATMENT OF THE POLYTOPES RËGULA11LY 



c). Example (111000), This central symmetrie form has one kind 

 of edge (10), one kind of face (110)-= — (100), but two sub- 

 groups (1110) = — (1000) = ^and (1 100) = O of limiting bodies 

 and once more one kind of limiting polytopes (11100) = — (11000). 



So we find (g5g). • 



Bemark. The degree of regularity of the polytopes of S n found 



1+4 3 



here is at least ' 2 = — - and therefore for n = 3 at least i. 



n 2n * 



So the Archimedian polyhedra of the stereometry are sewiregular 



in the right sense of the word, if we take semiregular to mean 



that the degree of regularity is -^ at least but less than unity. 



44. As the scale used for the determination of the regularity is 

 independent from the number of vertices, edges, faces, etc. of the 

 poly tope, the same method may be applied to nets of polytopes, 

 by considering a net in S n as as polytope limited by an infinite 

 number of limits (l) n in S n+i . This new application depends only 

 on the problem how to determine the different kinds of vertices, 

 edges, faces, etc. of the net. 



All the nets considered here have vertices of the same kind and 



edges of the same length. So for a net in S n the fraction of régu- 

 la i ± 3 



larity is at least — -~, i. e. ; — — . So in the most frequent 



J n-\-l 2(« + l) 4 



number of cases in which a constituent of the nets admits two 



or more differently shaped faces we have only the choice between 



2 3 



, ■■■ and — — : — — of which the first value corresponds to the 

 n '-}- 1 2 (« -f- 1) r 



case of only one kind of edge, the second to that of two or more 



différents kinds of edges. 



In order to make the determination of the fraction of regularity 

 of the nets in # 4 and S 5 as easy as possible we enumerate in 

 Table III the different limits (/) 4 , (/) 3 , (/) 2 of the nets in # 4 and 

 the different limits (/) 5 , (/) 4 , (/) 3 , (I), of the nets in S 5 . In the part 

 corresponding to n = 4s we find under the seven headings I, II, . . , VII 

 the subdivisions 4, 3, 2 standing for (/) 4 , (l) d , (/) 2 , in the part 

 corresponding to n = 5 likewise under I, II, . . ,XII the subdivisions 

 5, 4, 3, 2 standing for (/) 5 , (/) 4 , (l) 3 , (/) 2 . These limits are indicated 

 in abridged notation: under 5 the symbols 1, ce it ce 2 , etc. denote 

 >S Y (6), ce^Q), ce 2 S(6), etc.; under 4 the symbols 1 , ce i9 etc. signify 

 S(b), ce ± S(b), etc. 



The results of Table III are inscribed in Table II in the sixth column 



