1910] on the Chemical Significance of Crystal Structure, 827 



The cubes of the partitioning, having served their purpose, may now 

 be removed, leaving one of the 230 types of homogeneous point- 

 systems (Fig. 2). Imagine next that each point of the system expands 

 uniformly in all directions until it touches its neighbours ; a system 

 of spheres packed together in contact is thus obtained (Fig. ;>), and, 

 on examination, it is found that no way exists of packing these equal 

 spheres moie closely together than the one thus derived. The 

 system is therefore termed the cubic closest-packed assemblage of 

 equal spheres and, being derived in the manner described, still 

 retains the high symmetry of the cube : the fragment shown, in fact, 

 outlines a cube. Three directions at right angles in it, those which 

 are parallel to the three cube edges, are seen to be identical in kind ; 

 this identity in kind in the three rectangular directions, a, b and c, 

 is conveniently expressed by the ratio, ^:&:^ = 1:1:1. 



On removing spheres from one corner of the cubic closest-packed 

 assemblage of equal spheres a close triangularly arranged layer is 

 disclosed, and, by similarly treating each corner of the fragment of 

 assemblage, the cube outline gives place to one of octahedral form. 

 The assemblage is now seen to be built up by the superposition of 

 the disclosed triangularly arranged layers, the hollows in one layer 

 serving to accommodate the projecting parts of the spheres in adjacent 

 layers. AYhen this operation is performed it is perceived, however, 

 that two ways of stacking the layers homogeneously are possible. 

 The first of these, in which the fourth layer lies immediately over 

 the first, the fifth over the second, and so on, yields the cubic closest- 

 packed assemblage. The alternative mode of stacking, in which the 

 third layer lies immediately over the first, the fourth over the second, 

 and so on, exhibits the same closeness of packing as the first, but 

 possesses the symmetry of the hexagonal crystal system ; it is accord- 

 ingly termed the hexagonal closest-packed assemblage of equal spheres 

 (Fig. 4). Examination of the hexagonal assemblage shows that the 

 horizontal directions, in the planes of the layers, are not identical in 

 kind with vertical directions, perpendicular to the planes of the layers. 

 Corresponding dimensions in these two directions, a and c, are in the 

 ratio of 



a :c = 1 : Vd) = 1 : <>-81G."). 



The final step in the treatment of the closest-packed assemblages 

 of equal spheres consists in converting them into the corresponding- 

 assemblages of cells fitting together without interstices which have 

 been already mentioned ; it may be carried out in these, and in all other 

 cases, by causing the component spheres to expand uniformly in all 

 directions until expansion is checked by contact with the expanding 

 parts of neighbouring spheres. The cubic closest-packed assemblage 

 then becomes a stack of twelve-sided polyhedra, rhombic dodecahedra, 

 which are so fitted together as to fill space without interstices. It is 



