( 393 ) 



sider two bodily adjacent eightcells of the net, the line joining the 

 centros of which is equipollent to PQ, in order to see that these 

 eightcells are intersected in two congruent prisms with common 

 base, whilst the sections with the inscribed inclined sixteencells are 

 symmetric figures with respect to that base, [f we consider difference 

 in form only we have to deal therefore with two or three, if we 

 also take into account difference in orientation we have to deal with 

 four or six polyhedra. 



If we deprive the problem of the intersection of the system ol 



cells ('s } partially penetrating one another, polariv circumscribed to 

 the erect sixteencells, of the one superfluous dimension, we find a 

 system of cubes with edge four, the centres of which are the vertices 

 of a net of cubes with edge two, whilst the edges are parallel to 

 those of the cubes of the net. This system is then to be intersected 

 by a plane normal to a diagonal. By means of a simple diagram 



1 2 



we then find that to the three sections a, a 4- ,a-\- of the net 



1 3 ' 3 



-(2 



C& correspond the six sections 



a 1 



2 + 6 ' 



of the 



a o 



• ' ¥ + <i 



system Cs . But of these six different sections of C& only two give 

 rise to sections of erect sixteencells, viz. those the fractions of which 

 2 4 



lie between and 



6 6 



So we get as the two most regular of the 



space-fillings presenting themselves here the two indicated by figures 



in heavy type in the following scheme: 



1 2 3 4 5 T 



.0, -. — . . . En Cs 



',(4) 



In ( 



(21 



in cr 



6 



0. 



6 <; 

 * 



6 <) 



3 3 



■/2j 



In Ck ' 



1 3 5 9 9 11 



12' 12' Ï8' IS'] 2' 12 



l 3 5 



Of these space-fillings the first consists of regular hexagonal double- 

 pyramids (last figure of column five of plate II) and as to shape only 

 one other form, an irregular octahedron (the octahedron of the sixth 

 column with OPQ as base), whilst the second one is built up by 

 three different bodies in that supposition. The diagrams 10 and II 

 represent a projection of both on the base of the prisms forming 

 the sections of lie including eightcells. 



From lig. 12, added [tartly to till the page, which shows the 

 sections of a cube by planes normal to an interior diagonal, can be 

 deduced finally that the segments of lines PQ, RS, TU of the three 

 figures of column live and of the last three figures of column 

 six — of plate 11 have the same length. 



26* 



