( 537 ) 



yet fill the whole space Sp^. For to make of a C^ the inscribed 

 Cjj we must truncate from this measure polytope at each of 

 the eight vanishing vertices a fivecell rectangular at this point, of 

 which the four edges passing through this point have a length 2. 

 Because a vertex which vanishes for one of the sixteen cells Cg, 

 to which it belongs, does this for all, there will remain round 

 this point sixteen alternate white and black rectangular five- 

 cells and these will form together a Cf^^^ of which this point is 

 the centre. Thus a space-filling for Spt is formed by three equally 

 numerous groups of cells (7^,21/2) ^^jth the property that all cells 

 Cia of the same group can be made to cover one another by translation. 

 To show how striking the regularity of the net of the C^, is we 

 must suppose three cells C\~^^^K of which no two belong to the same 

 group, to be removed parallel to themselves to a common centre, 

 the origin of coordinates. We then see immediately that the vertices 

 of the three Cy/-^ together form the vertices of a C^^). B'or 



lb '-' 24 



the two inscribed cells Cf^^^ together again furnish the vertices 

 (=b 1, dz 1, ± 1, ± 1) of the original eightcell C^^^and the coordinates 

 of the vertices of the third cell C^^y^^ are 



16 



(+ 2, 0, 0, 0), (0, ± 2, 0, 0), (0, 0, + 2, 0), (0, 0, 0, ± 2), 

 from which is evident what was assumed (compare " Mehrdimensio- 

 nale Geometrie" , vol II, p. 205). 



We shall presently use this observation to trace the connection 

 between the four groups of axes of the three systems of cells Cjg 

 with the groups of axes of Cg . 



2. To transform the net of the cells C^ into a net of cells C,, 

 we must again suppose the cells of the former alternately coloured 

 white and black in order to break up each of the black cells into 

 eight congruent pyramids with the centre of the eightcell as common 

 vertex and the eight bounding cubes as bases. )^y adding to each 

 white eightcell the eight black pyramids having a bounding cube 

 in common with it, the net of the cells C^^) is generated ; in reality 

 to the sixteen vertices of the eightcell supposed to be white with the 

 origin of coordinates as centre, viz. to the points (db 1, d= 1, ± 1^ dr 1) 

 the eight vertices mentioned above 



(± 2, 0, 0, 0), (0, ± 2, 0, 0), (0, 0, ± 2, 0), (0, 0, 0, ± 2) 

 are added. 



The transformation of the net of the Cf^ into that of C„ can also 

 take place in the following simple way. Divide each of the cells C^2) 



37* 



