(495 ) 



diagonal AA' in the tirst point of division A^, at a distance — \/n 



n 



from A, the section is a simplex with edge |/2. So the two sim- 



plexes, generated when an arbitrary point P of AA' is substituted 



for point A^, have edges of a length of AP^27i and A'P[/2n, 



wherefore we indicate them, also with reference to the number of 



vertices, by Sn{AP\/27i) and S'niA'Pl'^^n). So the theorem holds: 



"If we shove an Af„, of which the diagonal AA' is normal to a 

 given space aS/;„— i, in the direction of that diagonal through that 

 space /S)^_i, so that the spaces Spn—\ of the bounding poly topes 

 Mn—\ move parallel to themselves, the section of Spn—i with the 

 moving polytope 3In is at every moment the part of that space 

 Spn-i that is enclosed within two concentric, yet oppositely orientated, 

 regular simplexes SnipV^n) and >S'„ (7/ K2?2) where p and p' are 

 connected in such a way that the sum p -\- p' is equal to |/?i. 

 During that movement of <¥« the common centre of gravity of the two 

 simplexes remains in its place and the spaces Spn—2 of the bounding sim- 

 plexes Sn-\ and S'n—\ move parallel to themselves ; whilst simplex 

 Sn expands itself from this common centre of gravity to a simplex 

 Sn {n V2), simplex S'n inversely contracts from a simplex aS"„ {n l/2) 

 to this point". 



At the moment when this process has got halfway and the two 

 simplexes are of the same size we find : 



"The section Dn—\ is the part of the intersecting space Spn-\ 

 enclosed by two definite equal concentric yet oppositely orientated 



regular simplexes Sn ( — n\/2 j and S'n ( — n 1/2 j ." 



Thus for ?z = 3 the regular hexagon with sides — j/2 is the figure 



Li 



3 

 enclosed by two triangles with sides — \/2 — think of the well- 



di 



known trademark — , thus for n ^ 4 the regular octahedron with 

 edges yi is the figure enclosed by two tetrahedra with edges 2 y'2 — 

 think of the two tetrahedra described in a cube and the octahedron 

 common to both. So in general the problem in the space of n 

 dimensions is reduced to another problem in space of n — 1 

 dimensions and moreover the connection of the result with regular 

 simplexes is explained. 



If we think the simplex Sn to be white and the simplex S'n to 

 be black, the n bounding spaces Spn—2 of Dn-\ originating from Sn 

 will be white, those originating from S'n will be black. From this 

 ensues that it must be possible to colour the 2n bounding spaces 



