RK- 



'") 



///. Explanation in details of the connection of Dn~^\ 

 loith 7'egular arid regularly truncated simplexes. 



7. We consider in the space ^S'/;» a rectangular system of coordinates 

 with an arbitrarj^ point as origin and OX^, OX^,... OX,, as axes, 

 and we now call the 2"^'' part of that space which is the locus of 

 the point with only positive coordinates the "?2-edge (Jlj JY^- •• ^")"- 



If A, A' are two opposite vertices of a measure-poljtope Mn of 

 Sp„ and if AA^, AA^, . . . AAn ai'e the edges passing through ^ and 

 A'A\,A'A\,...A'A'n the edges parallel to these but directed 

 oppositely, then Mn can be regarded as the part of the space Spn 

 common to the two n-edges A {A^ A^ . . . An) and A' {A\ A\ . . . A'„). 



If we intersect this figure of the two oppositely orientated ?z-edges 

 and the raeasure-polytope J/„ common to both by an arbitrary space 

 Sjhi—i , the two ?z-edges are intersected along two oppositely orieiitated 

 simplexes and the section of Mn with that space S^^n-i appears as 

 the part of that space that is enclosed at the same time by both 

 simplexes situated in that space. If the selected space is normal to 

 the diagonal AA' , connecting the vertices of the >i-edges, the simplexes 

 are regulai' and they have the point of intersection P of the intersecting 

 space /S/>„— 1 with A A' as common centre of gravity. So the general 

 theorem holds : 



"The section of Mn with a space Spn—\ normal to a diagonal 

 can always be regarded as a part of that space SjJn—i enclosed by 

 two definite concentric, oppositely orientated, regular simplexes of 

 that space". 



If we wish to make use of this theorem we must determine in a 

 more detailed way the length of the edges of those oppositely orien- 

 tated regular simplexes with common centre of gravity. 



8. If we think the intersecting space Spn—i to be normal to the 



'). Tliis theorem shows distinctly why the sections of an octahedton parallel to 

 two faces must be identical to those of a cube by planes normal to a diagonal in 

 points of the middle tkird part of that line. The same in other words: If we 

 truncate a cube with the unity of edge at two opposite vertices by planes normal 

 to the connecting line in the points dividing this diagonal into three equal parts 

 and if we compress an octahedron with edges |/!2 in the direction of the normal 

 on two parallel faces as far as half the thickness, then we cause the same solid 

 to be generated in two different ways. 



