DERIVED FROM THE REGULAR POLYTOPES. 1 1 



a). .2. 5polytopes(l + 3V / 2)[l+2V / 2, 1 + 2V2, 1+V2, 1], 

 3)..2 2 .10 „ (1 + 3V2, 1 + 2\/2) [1 + 2V2, 1+^2, I], 

 c)..2 8 .10 „ (1 + 3^2,1 + 2^2, l-f2V2)[l-f V2, 1], 

 d)..2\ 5 „ (1 + 3^2,1+2^2, 1+2V2, 1 + V2)[1], 

 ?)..2 5 . „ (1+3^2,1 + 2^2, 1+2^2, 1+V2, 1), 



Of these groups of polytopes the first, of polytope import, can 

 be studied by itself; it proves to be a form with the characteristic 

 numbers (192, 384, 248, 56), an e x e 2 C%. The second group 

 consists of prisms on [1 + 2 \/2, 1 + \/2, 1] = tCO as base, 

 the third group of prismotopes (3 ; 8), the fourth group of prisms 

 on (1 + 3 Y/2, 1 + 2 V% 1 + 2 V%, 1 + V%) = CO as base. 

 According to art. 46 the iifth group, of vertex import, contains 

 forms e ± e^S(6) m So we find 



10 e ± e 2 C 8 + 40i> fco + 80 (8; 3) + 80 P co + 32 e i e 3 £(5) = 



= 242 polytopes, 



and, as e i e 2 C 8 , P tC0f (8; 3), P co , ^i £3 $(5) admit respectively 

 56, 28, 11, 16, 30 limiting bodies 



£ (1 X 56 + 40 X 28 + 80 X I* + 80 X 16 + 32 X 30) = 



= 2400 polyhedra. 



So, according to the law of Euler, the number of faces is 6000, 

 and the final result a (1920, 5760, 6000, 2400, 242). *) 



53. We pass now to the more direct method going straight on 

 from vertices to limits with the highest number of dimensions, and 

 apply it to the second example 



[1 + .3 i/2, 1 + 2 i/2, l+2i'2, 1 + i/2, 1] 

 of the preceding article. But in order to make the symbols less 

 clumsy and thereby the method more manageable we represent 

 once more 1 -\-pV2 by p. 



The number of vertices was and remains 1920. 



According to the symbols the edges split up into four groups, 

 viz. (3' 2'), (2''1'), (l'l), [1]. Here (3' 20 means that any deter- 

 minate pair of coordinates each affected by a given sign take the 

 interchangeable values 3' and 2', the other coordinates retaining 

 the same values; whilst [I] means that any determinate coordinate 

 takes successively the values + 1 and — 1 , the other coordinates 

 remaining unaltered. 



J ) The fourth and the sixth column of Table IV contain the characteristic numbers 

 and the limiting elements of the highest number of dimensions. The meaning of the 

 second column, of the small subscripts in column four and of the fraction in column five, 

 will be explained later on. 



