(550 ) 
4. From the two ways mentioned in the preceding paper leading 
to the knowledge of the threedimensional space-fillings generated by 
the intersection of the net (C,,) we choose here the more theoretical 
one, in which is deduced from the section of the intersecting space 
with a definite cell C,, how this space must affect the other cells’C,,. 
To this end we indicate in diagram 6 how the boxes C,@) filling 
twice the space Sp, project themselves on the chosen axes OF,,, 
OK,,, OF ,,, OR, Of the projections on the axes OF, (2, 1,1, 1) C,, 
OK,, OR,*) coinciding with the named ones the third case OK, 
distinguishes itself from the other by this that the centre of the 
eightcell (2, 6,6, 2) does not project itself in the projection of a ver- 
tex: in connection with this each space-filling corresponding to OK, 
deviates from the general rule according to which the difference of 
the fractions corresponding to sections partaking in a selfsame space- 
filling is itself a fraction with numerator unity, the denominator of 
which is equal to the number of equal parts of the projection of an 
eightcell. So according to this rule this difference is sean the first 
. le 
and the fourth case, in the third case of the fig. 6, and it would 
una : : 1 
have been = in the case of OK, but 7s now only = 
Now we have indicated in general which sections of C,, must 
generate together a space-filling we can proceed to the treatment of 
the individual cases; we shall then see that in any of these combi- 
nations of sections each face occurs twice in the same position, 
as the juxtaposition of the pieces requires. 
Space-fillings normal to OW, Here we have to deal successively 
ma EE a ie ae 
with the combinations of fractions (5) (5e) Ga so we find 
three space-fillings, that of granatohedra, that of granatohedro truncated 
at the six octahedral vertices and of small cubes, that of cubes with 
edges twice as long. 
Space-fillings normal to OK. Expressed in the fractions belonging 
eles Oa Ve | pe), See 
to C, here the combinations : ) 
0, Ek ee ra ava Nr ’ 
Loe 16> 671 16 16 AC iG 
2 6 10°14 ; 
se | present themselves; as no threedimensional section 
LEN eb 16 
1) In order to make the diagrams fit better on the plate, the order of succession 
is changed there by interchanging (2,1, 1,0) Os and OK. 
