Whatever the number of pieces in a cluster in the closest possible packing of solids may be for any particular shape, we may consider each cluster as itself a given single body, and thus reduce the problem to the packing closely together of assemblages of individuals all sameways oriented; and to this problem therefore it is convenient that we should now confine our attention. § 23. To avoid complexities such as those which we find in the familiar problem of homogeneous packing of forks or spoons or tea-cups or bowls, of any ordinary shape, we shall suppose the given body to be of such shape that no two of them similarly oriented can touch one another in more than one point. Wholly convex bodies essentially fulfil this condition; but it may also be fulfilled by bodies not wholly convex, as is illustrated in Fig. 11. [Illustration: Fig. 11.] § 24. To find close and closest packing of any number of our solids _S_{1}_, _S_{2}_, _S_{3}_ ... of shape fulfilling the condition of § 23 proceed thus:-- (1) Bring _S_{2}_ to touch _S_{1}_ at any chosen point _p_ of its surface (Fig. 12). (2) Bring _S_{3}_ to touch _S_{1}_ and _S_{2}_, at _r_ and _q_ respectively. (3) Bring _S_{4}_ (not shown in the diagram) to touch _S_{1}_, _S_{2}_, and _S_{3}_. (4) Place, any number of the bodies together in three rows continuing the lines of _S_{1}S_{2}_, _S_{1}S_{3}_, _S_{1}S_{4}_, and in three sets of equi-distant rows parallel to these. This makes a homogeneous assemblage. In the assemblage so formed the molecules are necessarily found to be in three sets of rows parallel respectively to the three pairs _S_{2}S_{3}_, _S_{3}S_{4}_, _S_{4}S_{2}_. The whole space occupied by an assemblage of _n_ of our solids thus arranged has clearly _6n_ times the volume of a tetrahedron of corresponding points of _S_{1}_, _S_{2}_, _S_{3}_, _S_{4}_. Hence the closest of the close packings obtained by the operations (1) ... (4) is found if we perform the operations (1), (2), and (3) as to make the volume of this tetrahedron least possible. [Illustration: FIG. 12] § 25. It is to be remarked that operations (1) and (2) leave for (3) no liberty of choice for the place of _S_{4}_, except between two determinate positions on opposite sides of the group _S_{1}_, _S_{2}_, _S_{3}_. The volume of the tetrahedron will generally be different for these two positions of _S_{4}_, and, even if the volume chance to be equal in any case, we have differently shaped assemblages according as we choose one or other of the two places for _S_{4}_.