DERIVED FROM THE REGULAR POLYTOPES. 1 9 



Vertices. There is only one kind of vertex (3) (2) (1) (1) (0) (0). 

 According to the rule given in art. 6 the number of vertices is ! 

 divided by 2 2 , i.e. 180. 



Edges. There are three groups of edges, represented in extended 

 and in un extended *) symbols by 



(32) (1) (1) (0) (0) = (32), (3) (21) (1) (0) (0) = (21), 

 (3) (2) (1) (10) (0) = (10). 



We indicate a new method of determining the numbers of these 

 edge groups. In the case of (10) the coordinates corresponding to 

 the two digits between the same brackets can be x Xi x k where i, k 

 is any combination of the subscripts 1, 2, 3, 4, 5, 6 by two, giving 



(6) 2 = ' == 15 possibilities; these two coordinates having been 



JL . tó 



chosen the four remaining ones can be assigned anyhow to the 

 four digits (3), (2), (1), (0), giving 4! = 24 possibilities. So the 

 number of edges (10) is (6) 2 .4! =360. In the case of (21) the 

 number 360 must be divided by 2 on account of the two equal 

 syllables (0),(0), in the case of (32) this number must be divided 

 by 2 2 on account of the two pairs of equal syllables (1), (1) and 

 (0),(0). So we have 



90 edges (32) -f 180 edges (21) + 360 edges (10) 



i. e. altogether 630 edges. 



Faces. There are six groups of faces, represented in extended and 

 in unextended symbols by 



(321)(1)(0)(0) = (321) = Pe , (32)(1)(10)(0) = (32) (10) = A , 

 (3)(211)(0)(0) = (211)=jö 3 , (3) (21) (10) (0) = (21) (10) = Pi , 

 (3)(2)(110)(0) = (110)=/, 3 , (3) (2) (1) (100)= (100) = p s . 



Taken in the order of succession of the rows the numbers of 

 these polygons are 



(6) 3 . 3!: 2= 60, (6) 2 (4),. 2! = 180, (6) 3 .3!:2 = 60, 

 (6) 2 (4) 2 . 2! = 180, (6) 3 . 3! =120, (6) 3 . 3! =120, 



i. e. we find 



300^3 -f 360^ 4 -f 60 2h = ? 20 faces - 



Limiting bodies. There are seven groups of limiting bodies, viz.: 



l ) As we have seen in the preceding article the unextended symbols are deduced 

 from the extended ones by omitting the syllables of one digit. 



2* 



