MOLECULAR ARCHITECTURE 367 



closely than the one of which we have traced the derivation. 

 Being derived from a cubic partitioning of space and as it retains 

 the high symmetry of the cube, it is termed the cubic closest- 

 packed arrangement of equal spheres. The fragment shown in 

 the figure outlines a cube ; it can be seen that the three direc- 

 tions at right angles, those of the cube edges, are identical in 

 kind ; this identity in the three rectangular directions <?, b and c 

 is conveniently expressed by the ratio a : b : c = i : i : i. 



But we have said that there is another closest-packed assem- 

 blage of equal spheres which possesses hexagonal symmetry. 

 This second assemblage is closely related to the cubic form 

 already described, as we will demonstrate. When spheres are 

 removed from one corner of such a cubic fragment as fig. 2, 

 a close triangularly arranged layer is disclosed and it can be 

 seen that the assemblage is built up of such layers superimposed 

 upon one another in such a manner that the fourth layer is 

 directly over the first, the fifth over the second and so on. 

 But there is an alternative mode of stacking these triangularly 

 arranged layers, in which the third layer lies directly over 

 the first, the fourth over the second and so on, the structure 

 thus obtained being just as close-packed as the other but 

 exhibiting hexagonal instead of cubic symmetry. Fig. 3 shows 

 a fragment of such a structure. In both the cubic and the 

 hexagonal types of assemblage, we can regard the structure 

 as built up of triangularly arranged layers and select as a 

 horizontal dimensional unit {a) the diameter of a sphere drawn 

 through two contacts and as a vertical dimensional unit {c) the 

 distance separating planes drawn through the layer centres. 

 The ratio o{ a : c then becomes i : \/ (f ) = i : o"8i65. 



Before we can translate our spheres into atoms and our 

 close-packed assemblages into crystals, it is necessary to take 

 one more step. Suppose the component spheres of each assem- 

 blage described above to expand uniformly in all directions 

 until further expansion is checked by contact with neighbouring 

 spheres, all insterstitial space having by that time been eliminated, 

 the spheres are all transformed into twelve-faced polyhedra, 

 those in the cubic assemblage having the form shown in fig. 4 

 and those in the hexagonal assemblage the form of fig. 5. The 

 regular uniform expansion of each particle of the original point 

 system to a regular dodecahedron is symbolical of the even 

 radiations of forces from the atomic centre. Each close-packed 



