( 644 ) 



rated. The space ,r, = 1 contains of this tetrahedi'on the equilateral 



triangle I\P,P, with the side 12; the other faces I\PJ\J\1\P„ 



P^P,P^ l.ying in the spaces ,/-,:= J ,,/■„=: i , .r, = 1 are isosceles 



1 

 triangles with basis \2 and sides -~\^H). So tiiis section is not a 



regular tetrahedron Init a regular ti'luiKjuhd' pi/niinid, of which the 

 perpendicular let down out of the vertex I\ on to the groundplane 

 P,P^P^ is an axis with liic period three; because the foot of 

 this perpendicular lies on the diagonal AB of the right cube at a 

 distance from .4 forming a sixth part of AH and as J /^j is likewise 

 a sixth of AB' this axis is parallel to the diagonal /^/^' of the eight- 

 cell. It is now easy to deduce the changes of the section following 

 from the displacement of the intersecting space by investigating 

 either the parallel displacement of the edges of the section over the 

 faces of the eightcell or the parallel displacement of the faces of the 

 section through the bounding cubes of the eightcell. If the intersecting 

 space has removed itself as far as double the distance from A, then 

 — as is evident from both considerations — the tetrahedron of 

 intersection has simply been multiplied by two from A. Passing on 



2 

 trom this section — it seems preterable to watch more closely the 

 12 * 



2 



edges. If the edges PJ\ and /^/^j of the section have arrived 



A A 



in the positions I\Pj and P/ P^ ot the section ^ when the inter- 

 secting sj)ace has come at the threefold distance from the starting- 

 point A, it is sutticiently evident that the connection of the points P^P^' 

 must furnish a new edge. So we see gradually how the entire 



rhombohedron forming the section ^ develops itself. We \q[ point 



to the tact that the section in each j)Osition of the intersecting 

 space during its {tarallel motioji has an axis with period three, 

 parallel to the diagonal HP' and at last passing into this line. 

 Indeed, the diagonal AB of the bounding cube lying in space ,i\ = J 

 being an axis of revolution with the period three for that cube, so the 

 plane through AH and AH' is a "jilane of revolutio]i" with the 

 period three for the eightcell. As now the moving intersectiiig space 

 is and remains norujal to the line OC lying in this plane — see 

 the lirst of the 20 diagrams — the line of intersection of this plane 

 with the intersecting space, which line is of course normal to i^C, 

 must be an axis with the period three for the section. As was found 



