( 539 ) 



the triitli of the wellknown theorem, that the centres of the faces 

 of a 6'^-) — and therefore also the centres of the edges of each of 

 the two inscribed cells C{-y-^ — are the vertices of a Cy^\ 



lu 24 



3. Before examining more closely the nets of the cells Cg, C^^, C\^ — 

 or, as we shall express ourselves, the nets (6'J, (CiJ, (C^J — in 

 their mutual connection we put to ourselves the question whether 

 it is possible to fill S/>^ entirely with difereiit regular cells. Here 

 the table given above points to two possibilities. We can either com- 

 plete the sum of the angles 75° 31' 21 ' and 164° 28' 39" with 120° 

 to 360° or by combination of one of the two cells C^^, C^^ with 

 twice the other arrive at 360°. The latter is however already 

 excluded by the fact that C^^ and C^^ differ in bounding bodies, 

 which obstacle does not occur when one tries to arrange the three 

 cells C5, C16, Csoo with the same length of edges around a face. 

 Yet, though this is possible, neither in this way does one arrive at 

 the object in view. If the indicated space-filling had taken place then 

 two bounding tetrahedra of C^, having always a face in common, 

 would have to differ from each other in this, that one would at the 

 same time have to belong to a C^^ and the other to a Cgoo and this 

 is impossible. For one cannot colour the bounding tetrahedra of a 

 C5 alternately white and black for the mere reason, that the number 

 five of those tetrahedra is odd. So there is no space-filling of Sp^ 

 where different regular cells appear. 



4. We shall now consider more closely the systems of points 

 formed by the centres of the regular cells of the nets (C'g), (CiJ, 

 (Cj»4^ which we shall indicate by the symbols (Pj, (P^), (P^J. 



Of the systems of points (Pj, (P^J, (P,J„ which we might call 

 fourdimensional "assemblages of Bravais", (PJ is the simplest. If the 

 axes of coordinates are assumed through the centre of a definite cell 

 Cfj parallel to the edges of this cell, then (P^) is the system of the 

 points (2^1, 2a,, %-t^, 2a J with only even integer coordinates which 

 we indicate by means of abbreviated symbols by the equation 

 (P3) = (2a,-). 



Of the two other systems of points, (P^J can be most simply 

 expressed in (P^). Out of the second mode of transformation of the cells 

 C^) into the cells C^^^^ it was clear to us that (P^J is found by 

 joining the system (Pj to the system of the vertices of the cells 

 6^^- Now this system of the vertices can be deduced out of (Pg) by 

 a translation indicated in direction and magnitude by the line-segment 

 connecting the centre of the eightcell, which served to determine the 



