40 ANALYTICAL TREATMENT OF THE POLYTOPES .REGULARLY 



of regularity, according as their symbols of coordinates are equal 

 or different." 



As the application of Elte's scale 1 ) to polytopes and nets of 

 measure descent is rather easy it may suffice to give some examples, 

 both of polytopes and nets. 



a). Example [3' 3' 2' I' l]. Here we find four different groups of edges 

 (3', 2'), (2', 1'), (T, 1) (I, — 1). So the contributions to the numerator 

 are 1 from the vertices and A from the edges and the fraction is 



i+i 



, the minimum value in 8 5 , 



5 • 10 



b). Example [3321 0]V2 . Here three groups of edges appear, viz. 



3 



(3, 2) V2, (2, 1) V%(1 3 0)V/2. So we find once more — . 



c). Example [1 10 00]\/2. Only one kind of edge, viz. (1, 0)Y/2. 



So we have to examine the faces. As it is clear that we find 



triangles 0/2, \/2, 0) and squares [V2, V2] 0, the degree 



■;... 4 2 

 of regularity is 



10 



5 



d). Example *? 4 J^jr 5 ( 2 )). The groundform [1' 1 1 1 1] admits two 

 kinds of edges (l'l)l 1 1 and 1' 1 1 1[1] of a different character. 



So we find 



6 



e). Example 



4' 



cc^ **2 c 3 



<*,, JV(Jü r 5 < 2 >). Here we have to deal with four 



groups of constituents represented wich their frames in the table 



ft • [4321 0] (2ft , 2// 2 , 2p, , 2p, , 2 lh ) 4 



ff3 . . .[210][10] . . . (2ft ,2ft ,2ft ,2^+1,2^+1)4 

 Si . . . .[10] [2 10] . . (2ft , 2ft , 2ft+ 1, 2ft+ l,2ft+ 1)4 

 g« [43210].(2ft + 1, 2ft + 1, 2 ft + 1, 2 A +1, 2ft + 1)4 



So through the vertex 4, 3, 2, 1, pass 



[ 4, 3, 2, 1, 



[8 + 4, 3, 2, 1, 



[4 + 0, 4 + 1] [2, 1, 



[4+0,4 + 1,4+2] [1, 

 [4+0,4 + 1,4 + 2,4 + 3, 4 



[4 + 0, 4 + 1, 4 + 2, 4 + 3, — 4 + 4] ~ 2 



i. e. six polytopes and more in detail four cells [43210] and two 

 prismotopes [2 1 0] [1 0]. Now the edges (4 3) 2 1 and 4 3 2(10) 

 belong to both the prismotopes, whilst each of the edges 4(3 2)1 



0] 

 0]. 



0] 

 0]. 



4J 



A 2 

 B 



C 



A 



D 



r ) We stick here to the original scale (compare Proceedings of Amsterdam, vol. XV, 

 p. 200). 



