L04 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



So we find in the case B= [5'4 43 3 2 2'2'1 l]\/2 of S l0 : 



C = [4'4'4'3'3'2'2'2Tl]Y/2 

 A ==i[97 5 5 533111 r 



A (i) = ,",[7555331 11] [l]\/2 

 A (2) = „[5 5 533111] [11] „ 

 ^< 3) = „[5533111] [211] „ 

 A^ = „[533111] [2'2Tl] „ 

 A^ = „ [aa 1 1 1] [2'2'2Tl] „ 

 ^ 6) = „[311 1] [3'2'2'2T1] „ 

 A™ = „[111] [3'3'2'2'2Tl] „ 



Bv applying this theorem we Unci immediately the sixteen nets 

 of jS 5 , as they have been registered in the eighth part of Table X 

 with the heading "constituents in an other notation". Moreover 

 Table XI gives the corresponding results for the 32 nets of S 6 . 



F. Polarity. 



108. By polarizing an ^-dimensional hmpd. with respect to a 

 concentric spherical space (with oc n_1 points) as polarisator we get 

 a new polytope admitting one kind of limit (/) n _i and equal 

 dispacial angles, to which corresponds the in versed symbol of 

 characteristic numbers of the original polytope. Moreover, if 

 WfL\, a 2 , . • ., <z n _i, # n ] * s ^ ne coordinate symbol of the original 

 h i, rpd., this symbol also represents the limiting spaces S n ._ i of the 

 new polytope in space coordinates. 



The fact that there is no hmpd. proper in S 3 and # 4 implies the 

 corresponding fact with respect to the new forms. So, if by the 

 subscript s is indicated that space coordinates are meant, we have: 

 \[ I Ll] s = (4, 6,4) = 2 7 , i[311]. s = (8,18,12) = r with py- 

 ramids on the faces, 

 L[llll] s =(16,32,24,b)=if 4 , i[3311] s =(24,96,120,48)=Jf 4 



with pyramids on the cubes, etc. 



109. Theorem LXÏX. "Any hmpd. in S n has the property 

 that the vertices V i adjacent to any arbitrary vertex V lie in the 

 same space 8 n _ i normal to the line joining this vertex V to the 

 centre O of the polytope. The system of the spaces 8 tl _. { corres- 

 ponding in this way to the different vertices of the hmpd. include 

 an other polytope, the reciprocal polar of the original polytope with 

 respect to a certain concentric spherical space, unless the chosen 



