(541 ) 



Sps- The chief aim of this communication is to indicate how we can 

 connect the transformation-views of the nets (Cig), {C^^) with that of 

 the net (Cg), which is by far the simplest. Because the three views 

 furnish at every moment a filling of the intersecting space, this 

 investigation can lead to new threedimensional space-fillings, even 

 though they be not entirely regular. 



To be able to design a transformation- view of the net (C, J we must 

 know for each of the component cells C^g the place of the centre 

 and the position about the centre; as the coordinates of the centres 

 of the cells are given above, we have only to occupy ourselves 

 further with the position about the centre. We designate that position 

 by means of the four diagonals of each C\e and we then notice that 

 these four lines for each of the two kinds of inscribed cells C\e are 

 also diagonals — groups of non-adjacent diagonals — of the circum- 

 scribed cells C\, whilst for the cells C\g of the third group they are 

 parallel to the axes of coordinates. 



If we suppose the centre of a cell C^.^y"^) of the third group to 

 be at the same time the centi-e of a cell CY>, the edges of which 

 are parallel to the axes of coordinates, the Cjg'^'^'^ is inscribed in this 

 new eighteen in such a sense, that the vertices of C^-^-) are the 



'-' lo 



centres of the eight bounding cubes of C^^\ For an obvious reason 

 we call this C{y-^ polarh/ inscribed in C^f^ — and now to distin- 

 guish, we call the cells of the two other groups bodily inscribed in 

 the cells 6'^_^). For, as was observed above, in each of the eight 

 bounding cubes of C*^^-* a bounding tetrahedron of C^-^-' is inscribed, 

 whilst each of the remaining eight bounding tetrahedra of C^^^^^ 

 has with respect to each of the four pairs of opposite bounding cubes 

 of C'f> three vertices of one and one vertex of the other cube as 

 vertices. 



In this way each of the cells C^^'^^) of the net (CjJ is packed 

 up in a C\ as small as possible, of which the edges are parallel to 

 the axes of coordinates; here the fourdimensional cao^'.^ of the "erect" 

 cells Cjg of the third group are cells C'^\ those of the "inclining" 

 cells Ci6 of the first and the second group are cells C(-). Whilst the 

 cases C^^-) of the inclining cells Cjg fill the space Sp^, the cases 

 C^^f' of the erect cells C^^ do so eight times, because C^^y-) is the 



8 16 



1 th 



— part of C^g*), — as is immediately evident when one divides the 



erect C^-^'^) and its case C^j^^ by spaces through the common centre 

 parallel to the pairs of bounding spaces of 6^-* into sixteen equal part^ 



