DERIVED FROM THE REGULAR POLYTOPES. 1 3 



5 tT of body import, 



10 P 6 „ face „ , 



10 P 3 „ edge „ , 



5 CO „ vertex „ 



8. We add an other example, this time of a poly tope in # 5 , and 

 choose (432110), showing all possible particularities. x ) 



The number of vertices is 6 ! divided by 2! i. e. 720 : 2 — 3G0. 

 The number of edges passing through each vertex is six, for in 



4"TTTT1) 



each of the six brackets indicates two coordinates w T ith difference 



360 y( 6 

 unity. So the number of edges is —^ — = 1080. 



Si 



Here we have to consider the equations 



X x X 2 X 3 00 [^ c2?5 ==z 11 O l 00 q == U , 



x x -j- w. 2 -f- x s -\- o? 4 = 10 or x 5 -f- a? 6 = 1 , 



œ x -\~ oo 2 -j- oo 3 = 9 or x k — | — ^? 5 — | — x G = 2 , 



x \ -\~ oo 2 =l or x d -j- x k -j- 00r o -|- # ö = 4 , 



a?! = 4 or # ? - - w* - - w k - - « .-, -f- 0C& - 7 . 



«) 



b) 

 c) 

 d) 



e) 



a). The equation x 6 == gives a? 4 , a? 2 , #? 3 , # 4 , w- = (43211), or — 

 if we diminish all the coordinates by one — we find in x 6 = — 1 

 the polytope x x , w 2 , w Sf w i9 x h = (32100), i. e. an e x e 2 8(b), occur- 

 ring six times 2 ). 



6). For x b -f- Wq — 1 we have the two possibilities w 5 = 1, w 6 = 

 and w 5 = 0, w 6 = 1, combined with x x , x 2 , x 3 , x k — (432 I), which 

 may be reduced to x x , x 2 , x 3 , x k = (3210) by subtracting unity from 

 all the coordinates. So we find a rectangular fourdimensional prism 

 P t0 with tO as base, occurring fifteen times. 



c. Here we have to combine the two systems w ± , w 2 , w B = (432), 

 and a? 4 , w hi Wq = (110). So we get a polytope with 6 X 3 = 18 

 vertices arranged in six equilateral triangles in planes parallel to 

 the plane x x = 0, x 2 = 0,^ = containing the face A^ A 5 A % of 



1 ) The fourth and the sixth columns of Table I contain the characteristic numbers 

 and the limiting elements of the highest number of dimensions of the new polytopes. 

 The meaning of the small subscripts in column four and of the fractions in column five 

 will be explained in part G of this section. 



2 ) The characteristic numbers of this form — compare Table I — can be deduced in 

 the manner indicated in art. 7; see farther under a). 



In the memoir quoted of M 1S . Stott the regular simplex of space S w is indicated by 

 the symbol C 5 ; we prefer to use here S (5), as this allows us to discriminate between the 

 regular simplex S (8) of space S 7 and the measure polytope C s of <S 4 , etc. 



