34 ANALYTICAL TREATMENT OV THE POLYTOPES REGULARLY 



[11] . [10JV2, [11] . [11], [10]V / 2 . [10] V2 



corresponding (fig. 1 5) to the projections 



ABCD\ ABCB\ EFGH\ 



EFGHY ABCDY E F G H) 



on the planes O X i X 2 and OX 3 X 4 , if in these symbols the suc- 

 cessive digits refer to x ± , œ 2 , œ 3 , x k . Of these the second, equal to 

 [11 11] occurs in one position only, whilst the two others admit 

 respectively six and three positions in accordance with the splitting up of 



F. Polarity. 



65. If we polarize an expansion or a contraction form derived 

 from the measure polytope MJ 2) of S n with respect to a concentric 

 spherical space (with co n _1 points) as polarisator we get a new poly- 

 tope admitting one kind of limit (l) n _ 4 and equal dispacial angles 1 ), 

 to which corresponds the inverted symbol of characteristic numbers 

 of the original polytope. Moreover, if \a l} a 2 , . . . , a n _ i3 a n ~] is the 

 coordinate symbol of the original polytope, this symbol represents 

 also the limiting spaces /S n _ i of the new polytope in space coor- 

 dinates. 



For the manner in which the process of truncation is transformed 

 by inversion compare page 69 of Section I. 



66. We now pass to: 



Theorem XL. "Any polytope (P) n of measure polytope descent 

 in 8 n has the property that the vertices V i adjacent to any arbi- 

 trary vertex V lie in the same space S n _ x normal to the line joining 



*) Compare for this inversion page 68 of Section I. 



By inversion of the measure polytope we find the cross polytope. Moreover we find 

 in 5 4 , in the notation of the foot note of page 63, if L e t e 2 e 3 stands now for the 

 "limiting bodies of the reciprocal polytope of e y e 2 e 3 C 8 ", 



Z. ei = 64T(l 3 ,3 2+1 ), 



Le 2 = Mpi 2+u 



7>3 = 64X, 

 Lei*i = 192 r(l 2 +i,l 2 +i,2i+i+i), 

 Le ies = ld2 symm. P l deltoid , 

 Le 2 e 3 = 192 symm. P 1 ^^, 

 Le 1 e 2 e 3 = SS^ F, 



Lce x = 32 P 3 2, 

 Lce 2 = LC24 = 24 0, 

 Lce 3 = LC 16 = 8 C, 



J Lcv 1 e 2 = 96r(2 2+1 ,2 2+1 ), 



Lce 1 e s =: 96 P 3 2 , 



Lcr 2 6'3 = 48P 4 1 (square) , 

 Lce 1 e 2 <? 3 = 192 P(l 3 ,3 2 +i), 



X representing a polyhedron limited by six faces, two groups of three equal deltoids 

 connected in such a way as to give rise to an axis of period 3, and Y a tetrahedron 

 limited by four unequal scalene triangles. For the shape of the tetrahedra Y compare 

 problem 79 of vol. XI of the "Wiskundige Opgaven", where the projections of these 

 tetrahedra on the four sets of axes of the polytope are given into the bargain. 



