( «45 ) 



already aboxc (lie line OB i.s really normal to (>6' and so the obtained 

 axis is parallel to OH. Because the plane through ^4/^ and Zj' contains 

 the perpendiculars (>(' and OR ont of O on to the intersecting 

 space and the space ,i\ == 1 of the righthand cube, each line of it and 

 therefore also OB must be normal to the plane determined by tiie 

 intersecting space in the space a\-=\\ so if we move the intersecting 



space m an op{)osite sense and return ti-oni by — , etc. to ^ — the 



rhombohedron forming the central section, and then moving in the 

 direction of the edge B'A thi-ough the eightcell, is truncated normal 

 to the axis l)y the plane determined in the space of that right cube. 

 In fact, in the above mentioned paper ( VerhandeUiuien, vol. IX, n°. 7) 

 has been found that the section is always a rhombohedron or a truncated 

 rhombohedron when the intersecting space is normal to a plane 

 through two opposite edges, which is here the case, as the plane 

 through AH and H' contains the edge AH' and the opposite edge. 

 We now indicate the bod}- corresponding to the fractional symbol 



u 



— bv /),, , where n can take one of the values I, 2, . .., 11, 12 and 

 12 ' 



l),t and /)\2 n represent the two oppositely orientated positions of a 



selfsame body, with a view to then investigating which of those parts 



make their appearance when the net (6'^) is cut by the central space 



'S.i\ -\- .v^ -\- .1', + ii\ = 0. From the distances of the points with the 



coordinates (2 tii), forming the system of centres of the net, follows 



immediately that the parts D^, I)^, D,, />,, />,„ appear together and 



that thus the corresponding threedimensional space-tilling consists 



of three — and if we notice the orientation even oftive — -different 



groundforms. Now, as we know, the form D^ alone already is able 



to till the space and so this is also the case with the forms D, and 



/>, and the forms IJ^ and JJ,, together. What is more, from the 



condition that in the obtained space-tilling with the three or five 



different groundforms the face of one of those forms must continue 



itself in faces of the surrounding forms, follows immediately that 



beside each /)., must lie a completing D^, beside each D^ a completing 



jDjo and that recomposition of those parts completing each other to a 



D^ must lead to a net of rlionibohedra D^. We really cause this net 



of rhombohedra to be generated in a simpler way if, before cutting 



the net (Cs) by the assumed space, we suppose the series of the 



spaces ,i\ = 2a^ -\- 1 to have disappeared, a thing to which the use 



of the plane of projection through the two edges, here AB' and the 



opposite one, has led us involuntarily in the paper quoted last. By 



thi^ the net (Cg) transforms itself into a threefold infinite net of an 



