( 278 ) 



of iht' icosahedron a regular pentagon adjaceni to this vertex, situated 

 in a diagonal plane. Through any edge All of* the icosahedron puss 

 two diagonal planes, as All lies in two (aces AIM* and ABQ and 

 therefore also in the diagonal planes corresponding to P, Q. Through 

 any edge All of the octahedron passes only one diagonal plane, 

 as the third vertices P, Q of t he faces A III', ABQ through AB are 

 opposite vertices and those points lead here to the same diagonal plane. 

 The diagonal planes of the icosahedron include a regular dode- 

 cahedron. 



2. By "diagonal space" of a fourdiniensional polytope we under- 

 stand any space having only laces in common with the boundary 

 of that polytope. 



There are two regular cells admitting diagonal spaces, the C\ e 

 and the (',„„. Through any face of the C lt passes one diagonal space, 

 containing the centre and bisecting the dispatial angle of the two 

 limiting bodies passing through the face. Through any face of the 

 Cgoo pass two diagonal spaces; the angle formed by these spaces 

 and that formed by the two limiting spaces through the face have 

 the bisecting spaces in common, and the cross-ratio between the 

 couple of diagonal spaces and the couple of limiting spaces has 

 again - as we will prove afterwards - i (3 — 1/5) for one of its 

 six mutually connected values. 



The fact that only the two mentioned regular cells possess diagonal 

 spaces is again closely connected with this that through each of the 

 vertices pass more than four limiting spaces and we are obliged 



to add here that these limiting spaces are letrahedra '). If we 



lake away from the limiting tetrahedra meeting in a vertex the 

 faces passing through that vertex, so as to retain of each the face 

 opposite to this vertex, we find in the case of the G' la a regular 



') This addition is necessary here. For the spatial sections of the regular C 24 

 do not admit diagonal planes, though any vertex of this cell is situated in six of 

 its limiting octahedra. As Mrs. A. Boole-sjtott pointed out to me these spatial 

 sections admit what we may call "would-be diagonal planes." If we consider — 

 sec lig. (54 of vol 11 of my "Mehrdimensionale Geometrie" — of the six octahedra 

 meeting in A the squares adjacent to A, we get the six faces of a cube, the 

 vertices and the edges of which are vertices and edges of C. 2i , whilst the faces 

 of it are not faces of t. 2i . If C 24 is cut by a space intersecting this cube, the 

 vertices of the section which are points of intersection with edges of the cube 

 will lie in a plane without all the sides of the polygon of intersection with these 

 points as vertices being edges of the section. In the fourth part of my comrr.u- 

 nicalioii "On fourdimensional nets and their sections by spaces'* 1 hope to be able 

 to come back to this point. 



