72 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



of the polytopes of the given net as vertices (see art. 51 of An- 

 dreini's memoir, quoted in art. 22). 



Under what circumstances polarization of a simplex net leads 

 to an other simplex net? The answer to this question is: "this 

 only happens in the plane with the nets N(p 3 ), JV(p 6 ), N(p 3 ,p Q ) 

 and the for two reasons discarded net N{p 3 ,p^\ For, if other- 

 wise the net contains two or more different constituents the reci- 

 procal net will contain two or more différents kinds of vertices, and if 

 the net is formed by one constituent only and this self space filler 

 is partially regular the vertices of the new net will be partially 

 regular. So the only possible case of tioo reciprocal simplex nets is 

 that of the pair N(p 3 ) and N(p 6 ) in the plane, the centres of the 

 two sets of triangles of N(p 6 ) being the vertices of an JV(p 6 ) , the 

 centres of the hexagons of N(p 6 ) being the vertices of an JV(p s ). 



In the treatise "Sulle reti, ecc.' 5 quoted once more above M r . 

 Andreini has indicated how to draw up a complete list of all the reci- 

 procal nets of threedimensional space ; in this research he comes to the 

 remarkable result (art. 59) that the rhombic dodecahedron and some 

 other less regular polyhedra into which this semiregular polyhedron 

 of the second kind can be decomposed form the constituents of 

 the different reciprocal nets. If we restrict ourselves to the cases 

 concerned with nets of simplex extraction this result is that the 

 constituent of the reciprocal net of 



N{T, 0) is the rhombic dodecahedron BD, 



N(T, tT) „ the rhombohedron (\ BD), 



N(0, CO) „ a double pyramid on a square (^ BD), 



N(l2, tO, CO) „ a pyramid on a lozenge (y 1 ^ RD), 

 iV (tO) „ a tetrahedron limited by four equal isosceles 



triangles (gL- BD). 



What corresponds to this remarkable result in space JSf n ? It goes 

 without saying that this question deserves an answer. But that ans- 

 wer can only be fragmentary, unless we surpass the limits between 

 which we wish to confine ourselves in this paper. So all we can 

 do now is to express the hope that we may be able to give a 

 complete answer to that question in a new paper of its own. Only 

 we cannot retain the remark that the constituent of the reciprocal 

 net of the net corresponding to the undivided partition n -\~ \ of 

 n-\-\ and that of the reciprocal net of the net corresponding to 

 the partition of n ~\- 1 consisting of units only are very interesting 

 polytopes, worthy of study for their own sake. 



