1 



DERIVED PROM THE REGULAR POLYTOPES. 07 



(f — Vi, | — V2, -t- | — V%) [V2] —, • 



representing together the vertices of a prismotope [6; 3]. 

 Vertex gap polytope. Here we find finally 



[4-)- j/2,4 + >/% 2 + ^2, j/2 ], (10-|-6j/2) a x , « 3 , a 8 , a 4 , 5>; odd, 



l 



:g t XX 3 VU i +U . +U} ( 3 + 3K2) 2ai+1, 2aa+1, 2as+1> H+1> ?* even> 



Ïs + 'Ss+Vm + ^M + ^H 5 + 3^2)^+1, 2^+1, 2,3+1, K+l,^ odd, 



giving — as it ought to do — quite the same result as in the 

 preceding example. 



84. Probably after all the indications contained in the treatment 

 of several special cases Table VII would be quite clear by itself 

 but for the first column of the part corresponding to the second 

 extreme polytope and the last column but one; so we have to 

 add a few words about these two columns. 1 ) 



In the two special cases treated in art. 81 the vertex polytopes 

 proved to be polytopes all the vertices of which can be represented 

 by one symbol, i. e. polytopes of measure polytope- extraction, viz. 

 ce i e 2 e s C 8 = e ± e 2 C 16 . But in the five cases studied in the art 8 . 82, 

 83 we had to deal with vertex polytopes the vertices of which 

 cannot be represented by one symbol only, i. e. with forms which 

 do not belong to the measure polytope family. These forms were 

 said to be derivable from the cell C 2k by applying respectively the 

 sets of operations ce ± e 2 , e 2 , e B , e x e x e s . Now in part F of this section 

 will be shown, not only that «//the forms appearing here as vertex 

 polytopes — whether their vertices are represented by one, two 

 or three symbols — can be deduced from cell G. lk by applying 

 the operations e k and c, but also by which set of operations any 

 required result is to be obtained. This set of operations is indicated 

 for all possible cases in the first column of the part of Table VII 

 corresponding to g . So in the second case of art. 82 we found 

 e 2 C 2k \ but as the general theory (compare Theorem LV) demands 

 ce x e s C 24 which is equal to e 2 C 2!i , we have inscribed ce x e 3 C 2k . 2 ) 



The remark of the second foot note of art. 78 — that several 

 nets deduced from iV((7 16 ) are equal to nets deduced from JY(C. 2i ) — 



J ) The very last column will be explained later on. 



*) The deduction of the symbols contained in the Table by applying the operations 

 e jt . and c to the cell C04, i.e. to [1, 1, 0, 0] J/2, will be given in the last section of 

 this memoir. 



5* 



