( 388 ) 

 net s C 8 ) in the second communication, and it can be of great service 



here, as the cells Cs bodily circumscribed to the inclined cells C\ 6 

 form a net (C 8 ). However, it often proves to be more practical to 

 start from any section presenting itself, to hunt for other sections, 

 possible in the position of the intersecting space under consideration, 

 admitting a face agreeing in shape and in size with one of the faces 

 of the chosen section, and to investigate if it is possible to arrive 

 in this manner at a space-filling either by these two polyhedra only 

 or by means of still more forms equally possible. 



Space- fillings normal to OR g . Let us imagine in threedimensional 



space a net of cubes with edge two, built up by cubes alternately 



white and black so as to form a threedimensional chessboard, with 



an infinite number of cubes, and let us describe in all white cubes 



a righthanded, in all black cubes a lefthanded tetrahedron. Then the 



interstitial spaces between these tetrahedra can be filled up by regular 



octahedra, forming with the tetrahedra the mixed net of tetrahedra 



and octahedra with common length of edge 2|/2. If we describe in 



all white cubes the tetrahedra truncated at vertices and edges of the 



second, in all black cubes the tetrahedra truncated at vertices and 



edges of the fourth of the live figures of the second column of 



plate II, the interstitial spaces can be filled up by regular octahedra 



1 3 



of two different sizes, i.e. with edges -|/2 and -V"!. If we describe 



1 2 



in all cubes the combination of cube and octahedron in equilibrium 



represented by the third of the live figures, the remaining interstitial 



spaces can be once more filled by regular octahedra of the same 



size, this time with the edge l/2. These generally known results 



are obtained immediately by means of the method of juxtaposition, 



if we only bear in mind that two bodily inscribed sixteencells, the 



(2) 



boxes G's of which have a limiting cube in common, are cut by 

 any space normal to the space of that cube in polyhedra being one 

 anothers mirror-image with respect to the plane of intersection as 

 mirror, from which it ensues immediately that of the five figures 

 of the second column the first and the fifth correspond to one 

 another, also the second and the fourth, whilst the third stands for 

 itself. By the juxtaposition, which comes here to the filling up of 

 the interstitial spaces, we then find that the two extreme figures of 

 the second column are to be combined with the two extreme figures 

 of the first column, that the middle figure of the second column 

 demands the middle figure of the first column, whilst the two 



