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octahedron adjacent to lliis vertex, in the ease of the C t00 a regular 

 icosahedron adjacent to this vertex, situated in a diagonal space. 

 Through any face ABC of the G' floo pass two diagonal spaces, as ABC 

 lies in two spaces ABCP, ABCQ and therefore also in the diagonal 

 spaces corresponding • to P, Q. Through any face ABC of the C\ n 

 passes only one diagonal space, as the fourth vertices P, Q of the 

 limiting spaces ABCP, ABCQ through .1/>V are opposite vertices 

 and these points lead here to the same diagonal space. 

 The diagonal spaces of the C soa include a regular C li0 . 



3. By ''diagonal space >^j>,.—\ " of an «-dimensional polytope we under- 

 stand anv space Sp.,—\ having with the boundary of this polytope 

 only limiting spaces Sp n 2 di common. 



Of the three regular polytopes, the simplex < s '„fi with n -f- 1 

 vertices and u -\- I limiting spaces Sp„—\, the measure polytope M„ 

 with 2" vertices and '2/i limiting spaces Sp„—\ , and the cross polytope 

 Cr n with reversely '2n vertices and 2' 1 limiting spaces Sp„—\, only 

 the last one possesses diagonal spaces Sp,,—] ■ Through any space 

 Sp»—2 hearing a limiting simplex S(„—\) passes one diagonal space 

 Sp„—\ , containing the centre and bisecting the angle between the 

 two limiting spaces Sp,—\ passing through this Sp„ •>• 



The fact that of the three regular polytopes only the cross polytope 

 possesses diagonal spaces Sp H — 1 is once more closely connected with 

 this that through each of the vertices pass 2"~ l and therefore 

 more than n limiting spaces Sp n 1 . If we take away from the 



limiting simplexes S n) passing through any vertex the spaces Sp H —2 

 passing through this vertex, so as to retain the 2"^ 2 spaces Sp»—* 

 opposite to this vertex, we find the cross polytope Cr n —\ adjacent to 

 this vertex, situated in a diagonal space Sp n —i ■ Here too through 

 any space Sp n —2 containing a limiting simplex S( n — 1) pass two limit- 

 ing spaces Sj) n —i • tkU» as the new vertices P and Q of the simplexes 

 S( n ) situated in these limiting spaces arc opposite vertices of Cr„ 

 leading to the same GV„_i , through each limiting simplex S( n — 1) 

 passes only one diagonal space Sp n —\- 



4. By intersecting a fourdimensional polytope, each face of. which 

 is situated in <l diagonal spaces, by a space not containing an al^e 

 of the polytope, we gel as section a polyhedron with the property 

 that each of its edges is contained in d diagonal planes. For, if the 

 intersecting space meets a face of the polytope, it meets also the d 

 diagonal spaces passing through that face, and this always furnishes 

 an va\^ of the section and (/ diagonal planes passing through it. So 



