( 64 ) 



so we tacitly represent that limiting body considered as lying in its 

 own tlireediniensional sj)ace. For, in the {)rqjection of a fourdinien- 

 sional polytope on a iilane tiie question of visihleness has no sense, 

 as fourdimensional space .stiiTunmls a plane situated in it in the same 

 way as threedimensional space surrounds a line situated in it. 



3. We now examine what we have to expect in general as to 

 the pentagonal projection of tin? semiregular poly topes deduced from 

 the »S'(5) by expansion and contraction. For shortness we introduce 

 for the group of the.se polytOj)es the symbol *S(5j ; moreover we make 

 use in future of the symbols 7', O, tT, (JO, tO, P^, I\ for the linuting 

 bodies of these poly topes. 



'/. The poly topes >S{5) partake with ,S(5j the property of present- 

 ing in pentagonal projection edges of five directions only. For it is 

 easy to prove that the three operations ^^i, t^^, t^.,, taken either separately 

 or in combination, can introduce oidy new etb^es pmri/lei tu the orifji- 

 nal ones. 



h. All the edges of <S(5) being of the same length we tind here 

 in projection once more two different lengths with the proportion 

 .s- : d and the two different complementary angles of inclination 

 obtained above. 



c. As the ten faces of -S'(5; split up in projection into two quin- 

 tuples of different form, the equivalent faces of SId) must do so 

 likewise. We shall even experience in the treatment of the different 

 particular cases that square faces always present a third form of 

 projection. 



d. In projection the limiting bodies of >S{p) behave differently 

 according to their im[)Ort. The general rule that equivalent limiting 

 bodies correspoJid in projection only holds for polyhedra of vertex 

 and of body import : while both the group of edges and the group 

 of faces of >S(5) admit two different projections, the limiting bodies 

 of edge and of face import must do so likewise. 



But what is of the greatest value with respect to the construction 

 of the projections desired is that all the limiting bodies of >S'(5) are 

 "arranged pentagonally" around the projection of the centre of the 

 original tivecell, i.e. that the four rotations indicated under f/. of the 

 preceding article bring any one of these limiting bodies successively 

 into coincidence with four others. If we assert moreover that the 

 effect of the operations of ex[)ansion and contriiction are extremely 

 easily obtained in ])entagonal projection, it must i)e cleai' that the 

 execution of what was planned with respect to the poly topes >b'(5) 

 is mere children's play. 



