6 8 ANALYTICAL TREATMENT OF THE POLYTOPES REGUL AULY 



new body are the reciprocal polars of the lines bearing the edges 

 of the original one, that the vertices of the new body are the poles 

 of the planes bearing the faces of the original one, etc. 



This definition of reciprocal polyhedra can be extended imme- 

 diately to space S nf where we have to use a concentric spherical 

 space (with go" -1 points) as polarisator. If this concentric spherical 

 space is the circumscribed one, the limiting spaces S n _ ± of the new 

 poly tope pass through the corresponding vertices of the original one 

 and are normal in these points to the lines joining these points to 

 the centre. We use this most simple disposition in order to show 

 that the property of having one length of edge is transformed into 

 that of the equality of the dispatial angles. To that end we consider 

 (fig. 11) the plane determined by any edge AB and the centre O 

 of the original polytope and remark that the polar spaces S n _^ of 

 A and B project themselves onto that plane in the lines a and d, 

 in A and B normal to O A and OB respectively; so the space of 

 intersection S n _ 2 °f these two spaces S n _ i projects itself in the 

 point C common to a and b and the angle ACB is the dispatial 

 angle between the two spaces S n _ ± ; but this angle is the supple- 

 ment of the angle A OB which is constant, O A = OB and AB 

 being constant. 



By applying this inversion to any semiregular polytope of simplex 

 extraction the characteristic number symbol of it is inverted too. 

 So the symbol (15, 60, 80, 45, 12)of^ #(6) — see the table — 

 passes into (12, 45, 80, 60, 15). *) 



If, in inverting a definite polytope of simplex descent in S n , 

 we assume as polarisator the imaginary spherical space for which 

 the vertices of the simplex from which the polytope was derived 

 admit as polar spaces S n _ ± the opposite limiting spaces S n _ i of that 

 simplex, and (a i9 a 2 . . . . a n +±) is the coordinate symbol of the 



x ) It is a very good exercise to deduce the limiting bodies of the reciprocal polytopes 

 of S/ t by polarizing the properties of the edges passing through the vertices of the ori- 

 ginal simplex polytopes. So, if Le 1 e 2 e s stands for "the limiting bodies of the reci- 

 procal polytope of q e 2 e 3 S (5)", if T (1 3 , l 2+ i, %i _i_ \ ±. i) indicates a tetrahedron of 

 which one vertex bears three equal edges, one two equal and one unequal edges, two 

 three different edges, if P 1 deltoid means pyramid on a deltoid base, -P 2 2 -j-i double py- 

 ramid on an isosceles triangle as base, Rh rhombohedron, the results to be obtained 

 are represented by the equations 



Z, ei = 20 r(i 2 , 3 2+1 ), 



Le z = 20 Rh, 

 Le i c 2 = {y0 r(l 3 , 1 2+1 , 2 1+4+l ), 



Lc l e 3 = L (— e 2 e s) — G0 Pi deltoid) 

 L Gi e 2 e 3 = 120 T (2 2+1 , 2 2+1 ), 

 Lce^ = 10 P2 3i 

 Lce x e 2 = 30 T (4 2+1 ). 



