( 280 ) 



the sections of the cells C le and C eoe by an arbitrarily chosen space 

 are polyhedra with the property thai through each edge passes 

 respectively one diagonal plane or a couple of these. As four spaces 

 passing in Sp 4 through the same face arc cut by any space of Sf) t 

 in four planes through a line with the same cross-ratio, the sections 

 of C 600 by a space not containing an edge will be characterized by 

 the property that the couples of faces and diagonal planes through 

 an vd^Q possess a constant cross-ratio. Kor from the regularity of 

 Cjoo can he deduced that this cross-ratio is the 'same for all the faces, 

 as we have stated already. Now the section of 6 T li00 by a space 

 normal to an axis OE (through a vertex E ) is a regular icosahedron, 

 if only the intersecting space is quite close t<» E and this proves 

 that the constant cross-ratio of C B00 musl be equal to thai of the 

 icosahedron. 



5. Indeed, it is not difficult to show directly that the cross-ratio 

 of C BOfl is really J (3 - I 5). 



Let ABC be any face of G' fi00 ami 0, P, Q fig. 1) represent succes- 

 sively the centre of C eoa and the fourth vertices of the two limiting 

 tetrahedra ABCP, ABCQ passing through ABC. Then the plane 

 OPQ of the diagram will contain the centre of gravity G of the 

 face ABC and be perfectly normal to this face in this point. From 

 GP=GQ and 0P=0Q can be deduced that the quadrangle OPGQ 

 is a deltoid with OG as axis of symmetry. As furthermore the 

 normals GP' ami GQ' dropped from G on Of' and OQ are the 

 traces of the plane of the diagram with the two diagonal spaces, we 

 get for the cross-ratio I'tjL'S 



I'll QR fPR\ (tan a — tan 0V «V (a— 0) 



PS QS \PSJ \tana -f tan pj sin*(a-\~P) 



Now if the cnl'^r of C eoo is our unit and we represent for brevity's 



sake I 5 by e we have (sec my " Meltrdbnensionale Geometrie", 

 vol. II, p. 200 



OP = - {e -f 1), OP' -=. - {e \- 3), OG = - {e + 3) y3, PG = - [/G. 

 2 4 o 3 



From this ensues 



1 1 



£ = 60\ sin a — (e -p 1) |/6, cos a = -Vl — 'óe 

 8 4 



and therefore 



(PQRS) = (-J-) = ~ ( 3 — e ) = 0,381966 . . . ') 



') In the same way the cross-ratio of the lour planes through an rdgv of the 

 icosahedron can be found. 



