DERIVED FKOM THE REGULAR POLYTOPES. 9 



which may be indicated by the brackets and the negative sign 

 after 1 in the symbol 



1 + 2V/2 , 1 -f- V/2 , 1 + V A 2 , 1(— ). 



So the number of edges is — = 480. 



In order to find spaces which may contain limiting bodies we 

 have to consider the equations 



a) ... + x v = 1 -f 2V/2, 



b) ... + œ ± + x 2 = 2 + 3V/2, 



c) . .7 ±^i±fl? 2 + a? 3 =3 -j- 4\/2, 

 rf) ... +^i + ^2 + ^3±^4 = 4(1 4- V2). 



ö). The equation ^ = 14- 2V/2 gives us for the other coordi- 

 nates the system represented by x 2 > a? 3 , x k = [1 -|- V/2, 1 -|- V/2, 1], 

 i.e. an ^ (7. This / C presents itself 2. 4 times, as in the equation 

 + cc l = 1 -\~ 2\/2 the sign may be either positive or negative 

 (factor 2), while the index i may be any of the four indices 1, 2, 

 3, 4 (factor 4). 



b). The condition x i -J- os 2 = 2 -\- 3V/2 gives a?, , x 2 = (1 -f- 2V/2, 

 1 -)~ V/2) and a? 3 , # 4 = [1 -|- V/2, 1], i.e. we have for the coor- 

 dinates in their natural order of succession 



œ u x 2 , p.,, x k = (1 + 2V/2, 1 -f \/2) [1 + V2, 1] 



representing an octagonal prism P 8 with end planes parallel to 

 0(X 3 X 4 ) and edges normal to these planes parallel to the lines 

 x i -\-x 2 = constant in 0(X 1 X 2 ); this P 8 occurs 2 2 . 6 times, as we 

 dispose in + w i + cr j = ^ ~h 3V/2 over two couples of signs (factor 

 2 2 ) and the pair of indices i, j stands for any of the combinations 

 of the four indices by two (factor 6). 



c) In the supposition x ± -\- x 2 ~\- x 3 = 3 -\~ 4 V / 2 we find in the 

 same way 



m u x 2 , x 3 , x, = (1 -f- 2\/2, 1 + V/2, 1 + V/2) [1], 



i.e. a triangular prism P 3 occurring 2 3 . 4 times. 



d) Finally for Ea?=4(l -[~ V/2) we get 



w u x 2 , a? 3 , a? 4 = (1 + 2V/2, 1 + V/2, 1 + V/2, 1), 



which — compare the last result of art. 46 — is a CO, occur- 

 ring 2 4 times. 



