80 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



wé find that the edge (21) 100 is common to (P) l5 (P) 2 , (P) 3 , (P) 4 , (P) 6 

 and the edge 21(10)0 to (2% (P) 3 , (P) 6 , (P) 7 , (P) 8 . So both edges 

 are common to 3 e 2 , ce 1 , e 3 . 



But this fact is not yet decisive, as the possibility exists that 

 the grouping of the sets of five polytopes around the edges (21) 

 and (10) is different. In order to decide this point we draw up 

 the following table of threedimensional contact, where 1, 2,. . ., 10 

 stand for (P) ls (P) 2 , . . . , (P)i an d contact by a prism is indicated 

 by a small asterisk. 



1 



2 



3 



4 



5&0 



7 



8 



9 



10 



2 



1 



1 



2 



1* 



1 



3 



2 



4 



3 



4 



4 



3 



2* 







5* 



5* 



5* 



5* 



5* 



5* 



5 



3* 



6 



6* 



6* 



6* 



6* 



6* 



6* 



6 



4 



8 



7 



7 



8 



7 



9 



8 



10 



7 



8* 



9* 



1 0* 



9 



10 



10 



9 



This table shows that if we arrange each of the two sets of five 

 polytopes as follows in three groups 



each polytope is in bodily contact with the polytopes of the other 

 groups of its horizontal row, whilst two polytopes in the same column 

 are equal. So there is no difference whatever in the threedimen- 

 sional contact, i. e. there is only one kind of edge 



Case 5 V . Here the point 321000 is common to the 17 polytopes 



e i e 2 



çe ± e 2 e s . (3, 

 ■ <3, 

 .(3, 



.(3, 



— e± e.y 



— e A . 



4 ' 



e.e 2 



2 

 2 

 2 

 2 



1 

 1 

 1 

 1 







) 1 



3) 



, , 



, — 3 + 3, , 

 , — 3 + 3, — 3 + 3, 

 , — 8 + 8, — 8 + 8, — 8- 



3 + 4 — 3 + 3,-3+3,-3 + 3) 

 (3, — 3 + 5, — 3 + 4, — 3 + 3, — 3 + 3, — 3 + 3)11 

 (3, — 3 + 5, — 3 + 4, — 3 + 3, — 3 + 3, — 6 + 6) 3 



ce, e 2 e 3 . (3, — 3 + 5, — 3 + 4, — 3 + 3, — 6 + 6, — 6 + 6) 8 



^,...(3, — 3 + 5, — 3+4, — 6 + 6, — 6 + 6, — 6 + 6)1 1 



) 3 

 ) 3 



1 

 1 



