(2) Bring S3 to touch S1 and S2, at r and q respectively.
(3) Bring S4 (not shown in the diagram) to touch S1, S2, and S3.
(4) Place, any number of the bodies together in three rows continuing the lines of S1S2, S1S3, S1S4, 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 S2S3, S3S4, S4S2. 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 S1, S2, S3, S4. 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.
Fig. 12
[§ 25.] It is to be remarked that operations (1) and (2) leave for (3) no liberty of choice for the place of S4, except between two determinate positions on opposite sides of the group S1, S2, S3. The volume of the tetrahedron will generally be different for these two positions of S4, 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 S4.
This will be understood by looking at Fig. 12, showing S1 and neighbours on each side of it in the rows of S1S2, S1S3, and in a row parallel to that of S2S3. The plane of the diagram is parallel to the planes of corresponding points of these seven bodies, and the diagram is a projection of these bodies by lines parallel to the intersections of the tangent planes through p and r. If the three tangent planes through p, q, and r, intersected in parallel lines, q would be seen like p and r as a point of contact between the outlines of two of the bodies; but this is only a particular case, and in general q must, as indicated in the diagram, be concealed by one or other of the two bodies of which it is the point of contact. Now imagining, to fix our ideas and facilitate brevity of expression, that the planes of corresponding points of the seven bodies are horizontal, we see clearly that S4 may be brought into proper position to touch S1, S2, and S3 either from above or from below; and that there is one determinate place for it if we bring it into position from above, and another determinate place for it if we bring it from below.
§ 26. If we look from above at the solids of which Fig. 12 shows the outline, we see essentially a hollow leading down to a perforation between S1, S2, S3, and if we look from below we see a hollow leading upwards to the same perforation: this for brevity we shall call the perforation pqr. The diagram shows around S1 six hollows leading down to perforations, of which two are similar to pqr, and the other three, of which p′q′r′ indicates one, are similar one to another but are dissimilar to pqr. If we bring S4 from above into position to touch S1, S2, and S3, its place thus found is in the hollow pqr, and the places of all the solids in the layer above that of the diagram are necessarily in the hollows similar to pqr. In this case the solids in the layer below that of the diagram must lie in the hollows below the perforations dissimilar to pqr, in order to make a single homogeneous assemblage. In the other case, S4 brought up from below finds its place on the under side of the hollow pqr, and all solids of the lower layer find similar places: while solids in the layer above that of the diagram find their places in the hollows similar to p′q′r′. In the first case there are no bodies of the upper layer in the hollows above the perforations similar to p′q′r′, and no bodies of the lower layer in the hollows below the perforations similar to pqr. In the second case there are no bodies of the upper layer in the hollows above the perforations similar to pqr, and none of the under layer in the hollows below the perforations similar to p′q′r′.
§ 27. Going back now to operation (1) of [§ 23], remark that when the point of contact p is arbitrarily chosen on one of the two bodies S1, the point of contact on the other will be the point on it corresponding to the point or one of the points of S1, where its tangent plane is parallel to the tangent plane at p. If S1 is wholly convex it has only two points at which the tangent planes are parallel to a given plane, and therefore the operation (1) is determinate and unambiguous. But if there is any concavity there will be four or some greater even number of tangent planes parallel to any one of some planes, while there will be other planes to each of which only one pair of tangent planes is parallel. Hence, operation (1), though still determinate, will have a multiplicity of solutions, or only a single solution, according to the choice made of the position of p.
Henceforth however, to avoid needless complications of ideas, we shall suppose our solids to be wholly convex; and of some such unsymmetrical shape as those indicated in [Fig. 12] of § 25, and shown by stereoscopic photograph in [Fig. 13] of § 36. With or without this convenient limitation, operation (1) has two freedoms, as p may be chosen freely on the surface of S1; and operation (2) has clearly just one freedom after operation (1) has been performed. Thus, for a solid of any given shape, we have three disposables, or, as commonly called in mathematics, three ‘independent variables,’ all free for making a homogeneous assemblage according to the rule of [§ 22.]