12 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



It is easy to calculate the numbers of edges of each group. 

 Through the pattern point with the coordinates 3', 2' 2' V, 1 pass ■ — 

 on account of the two digits 2' — two edges (3' 2) and (2' 1'), 

 and one edge (1/1) and [1]. So there are in toto 

 1920 edges (3' 2'), 1920 edges (2' 1'), 960 edges (V 1), 960 edges [1], 

 i. e. 5760 edges. 



Remark. We may notice that [lj with one digit only is equi- 

 valent, as to the representation of edges, to (3' 2'), (2' 1'), (1' 1) 

 with two digits. This difference is explained by the different cha- 

 racter of the symbols: the digits between square brackets have given 

 absolute values, whilst the digits between round brackets satisfy a 

 linear equation, the sum of the digits being constant. This diffe- 

 rence will repeat itself throughout the whole section; so [1' 1] is 

 a face, an octagon, and (3' 2' 2') is a face, a triangle, etc. 



By applying the notions of "unextended" and "extended" symbols, 

 of the "syllables" of these symbols, etc., given for the offspring 

 of the simplex in art. 9, to the group of polytopes deduced from 

 the measure polytope we easily extend this direct method to faces. 

 According to the symbols the faces split up into eight groups, viz: 

 the triangles (3' 2' 2') and (2' 2' 1'), the squares (3' 2') (2' 1), (3' 2') (1' 1 ), 

 (3' 2')[1],(2'1')[1], the hexagon (2' 1' 1) and the octagon [1' 1]. 

 In the pattern vertex P concur one of each of the two groups of 

 triangles, one octagon and — on account of the two digits 2' — 

 two of each of the four groups of squares, two hexagons. So we find 



-.™^ (% triangles . 8 squares 2 hexagons . 1 octagon 



1920 3 \ ^ ^_ 



V 3 r 4 ^ 6 ~ 8 



= 1280 triangles -)- 3840 squares -\- 640 hexagons -f- 240 octagons, 

 i. e. 6000 faces. 



According to the symbols the limiting bodies split up into nine 

 groups : 



(3' 2' 2' 1'), (3' 2' 2')(1' 1), (8' 2' 2') [1], (3' 2') (2' 1' 1), (3' 2') (2' 1') [1], 

 (3' 2') [l'l], (2 ; 2' 1' 1), (2' 2' 1') [1], [2' 1' 1], 



i. e. taken in the same order of succession, of 



CO , P s , P 3 , 7 G , C , 



P 8 , tT , P 3 , tCO. 



So we find through P 



CO + 3 P d -f- 2 P 6 + 2 C -f- 2 P 8 -f tT -f 2 WO 



and therefore in toto 



