Part 2

Considering now the three fours of parallel edges of the parallelepiped, let the straight lines of one set of four be altered to equal and similar parallel curves in the manner which I have described; and proceed by the same rule for the other two sets of four edges. We thus have three fours of parallel curved edges instead of the three fours of parallel straight edges of our primitive parallelepiped with corners (each a point of intersection of three edges) unchanged. Take now the quadrilateral of four curves substituted for the four straight edges of one face of the parallelepiped. We may call this quadrilateral a curvilineal parallelogram, because it is a circuit composed of two pairs of equal parallel curves. Draw now a curved surface (an infinitely thin sheet of perfectly extensible india-rubber if you please to think of it so) bordered by the four edges of our curvilineal parallelogram, and so shaped as not to cut any of the substance of any molecule of the assemblage. Do the same thing with an exactly similar and parallel sheet relatively to the opposite face of the parallelepiped; and again the same for each of the two other pairs of parallel faces. We thus have a curved-faced parallelepiped enclosing the whole of one molecule and no part of any other; and by similar procedure we find a similar boundary for every other molecule of the assemblage. Each wall of each of these cells is common to two neighbouring molecules, and there is no vacant space anywhere between them or at corners. Fig. 7 illustrates this kind of partitioning by showing a plane section parallel to one pair of plane faces of the primitive parallelepiped, for an ideal case. The plane diagram is in fact a realization of the two-dimensional problem of partitioning the pine pattern of a Persian carpet by parallelograms about as nearly rectilinear as we can make them. In the diagram faint straight lines are drawn to show the primitive parallelogrammatic partitioning. It will be seen that of all the crossings (marked with dots in the diagram) every one is similarly situated to every other in respect to the homogeneously repeated pattern figures: A, B, C, D are four of them at the corners of one cell.

§ 16. Confining our attention for a short time to the homogeneous division of a plane, remark that the division into parallelograms by two sets of crossing parallels is singular in this respect--each cell is contiguous with three neighbours at every corner. Any shifting, large or small, of the parallelograms by relative sliding in one direction or another violates this condition, brings us to a configuration like that of the faces of regularly hewn stones in ordinary bonded masonry, and gives a partitioning which fulfils the condition that at each corner each cell has only two neighbours. Each cell is now virtually a hexagon, as will be seen by the letters A, B, C, D, E, F in the diagram Fig. 8. A and D are to be reckoned as corners, each with an interior angle of 180°. In this diagram the continuous heavy lines and the continuous faint lines crossing them show a primitive parallelogrammatic partition by two sets of continuous parallel intersecting lines. The interrupted crossing lines (heavy) show, for the same homogeneous distribution of single points or molecules, the virtually hexagonal partitioning which we get by shifting the boundary from each portion of one of the light lines to the heavy line next it between the same continuous parallels.

Fig. 8 bis represents a further modification of the boundary by which the 180° angles A, D, become angles of less than 180°. The continuous parallel lines (light) and the short light portions of the crossing lines show the configuration according to Fig. 8, from which this diagram is derived.

§ 17. In these diagrams (Figs. 8 and 8 bis) the object enclosed is small enough to be enclosable by a primitive parallelogrammatic partitioning of two sets of continuous crossing parallel straight lines, and by the partitioning of ‘bonded’ parallelograms both represented in Fig. 8, and by the derived hexagonal partitioning represented in Fig. 8 bis, with faint lines showing the primitive and the secondary parallelograms. In Fig. 7 the objects enclosed were too large to be enclosable by any rectilinear parallelogrammatic or hexagonal partitioning. The two sets of parallel faint lines in Fig. 7 show a primitive parallelogrammatic partitioning and the corresponding pairs of parallel curves intersecting at the corners of these parallelograms, of which A,B,C,D is a specimen, show a corresponding partitioning by curvilineal parallelograms. Fig. 9 shows for the same homogeneous distribution of objects a better conditioned partitioning, by hexagons in each of which one pair of parallel edges is curved. The sets of intersecting parallel straight lines in Fig. 9 show the same primitive parallelogrammatic partitioning as in Fig. 7, and the same slightly shifted to suit points chosen for well-conditionedness of hexagonal partitioning.

§ 18. For the division of continuous three-dimensional space into equal, similar, and similarly oriented cells, quite a corresponding transformation from partitioning by three sets of continuous mutually intersecting parallel planes to any possible mode of homogeneous partitioning, may be investigated by working out the three-dimensional analogue of §§ 16-17. Thus we find that the most general possible homogeneous partitioning of space with plane interfaces between the cells gives us fourteen walls to each cell, of which six are three pairs of equal and parallel parallelograms, and the other eight are four pairs of equal and parallel hexagons, each hexagon being bounded by three pairs of equal and parallel straight lines. This figure, being bounded by fourteen plane faces, is called a tetrakaidekahedron. It has thirty-six edges of intersection between faces; and twenty-four corners, in each of which three faces intersect. A particular case of it, which I call an orthic tetrakaidekahedron, being that in which the six parallelograms are equal squares, the eight hexagonal faces are equal equilateral and equiangular hexagons, and the lines joining corresponding points in the seven pairs of parallel faces are perpendicular to the planes of the faces, is represented by a stereoscopic picture in Fig. 10. The thirty-six edges and the twenty-four corners, which are easily counted in this diagram, occur in the same relative order in the most general possible partitioning, whether by plane-faced tetrakaidekahedrons or by the generalized tetrakaidekahedron described in § 19.

§ 19. The most general homogeneous division of space is not limited to plane-faced cells; but it still consists essentially of tetrakaidekahedronal cells, each bounded by three pairs of equal and parallel quadrilateral faces, and four pairs of equal and parallel hexagonal faces, neither the quadrilaterals nor the hexagons being necessarily plane. Each of the thirty-six edges may be straight or crooked or curved; the pairs of opposite edges, whether of the quadrilaterals or hexagons, need not be equal and parallel; neither the four corners of each quadrilateral nor the six corners of each hexagon need be in one plane. But every pair of corresponding edges of every pair of parallel corresponding faces, whether quadrilateral or hexagonal, must be equal and parallel. I have described an interesting case of partitioning by tetrakaidekahedrons of curved faces with curved edges in a paper published about seven years ago. In this case each of the quadrilateral faces is plane. Each hexagonal face is a slightly curved surface having three rectilineal diagonals through its centre in one plane.

The six sectors of the face between these diagonals lie alternately on opposite sides of their plane, and are bordered by six arcs of plane curves lying on three pairs of parallel planes. This tetrakaidekahedronal partitioning fulfils the condition that the angles between three planes meeting in an edge are everywhere each 120°; a condition that cannot be fulfilled in any plane-faced tetrakaidekahedron. Each hexagonal wall is an anticlastic surface of equal opposite curvatures at every point, being the surfaces of minimum area bordered by six curved edges. It is shown easily and beautifully, and with a fair approach to accuracy, by choosing six little circular arcs of wire, and soldering them together by their ends in proper planes for the six edges of the hexagon; and dipping it in soap solution and taking it out.

§ 20. Returning now to the tactics of a homogeneous assemblage, remark that the qualities of the assemblage as a whole depend both upon the character and orientation of each molecule, and on the character of the homogeneous assemblage formed by corresponding points of the molecules. After learning the simple mathematics of crystallography, with its indicial system for defining the faces and edges of a crystal according to the Bravais rows and nets and tetrahedrons of molecules in which we think only of a homogeneous assemblage of points, we are apt to forget that the true crystalline molecule, whatever its nature may be, has sides, and that generally two opposite sides of each molecule may be expected to be very different in quality, and we are almost surprised when mineralogists tell us that two parallel faces on two sides of a crystal have very different qualities in many natural crystals. We might almost as well be surprised to find that an army in battle array, which is a kind of large-grained crystal, presents very different appearance to any one looking at it from outside, according as every man in the ranks with his rifle and bayonet faces to the front or to the rear or to one flank or to the other.

§ 21. Consider, for example, the ideal case of a crystal consisting of hard equal and similar tetrahedronal solids all same-ways oriented. A thin plate of crystal cut parallel to any one set of the faces of the constituent tetrahedrons would have very different properties on its two sides; as the constituent molecules would all present points outwards on one side and flat surfaces on the other. We might expect that the two sides of such a plate of crystal would become oppositely electrified when rubbed by one and the same rubber; and, remembering that a piece of glass with part of its surface finely ground but not polished and other parts polished becomes, when rubbed with white silk, positively electrified over the polished parts and negatively electrified over the non-polished parts, we might almost expect that the side of our supposed crystalline plate towards which flat faces of the constituent molecules are turned would become positively electrified, and the opposite side, showing free molecular corners, would become negatively electrified, when both are rubbed by a rubber of intermediate electric quality. We might also from elementary knowledge of the fact of piezo-electricity, that is to say, the development of opposite electricities on the two sides of a crystal by pressure, expect that our supposed crystalline plate, if pressed perpendicularly on its two sides, would become positively electrified on one of them and negatively on the other.

§ 22. Intimately connected with the subject of enclosing cells for molecules of given shape, assembled homogeneously, is the homogeneous packing together of equal and similar molecules of any given shape. In every possible case of any infinitely great number of similar bodies the solution is a homogeneous assemblage. But it may be a homogeneous assemblage of single solids all oriented the same way, or it may be a homogeneous assemblage of clusters of two or more of them placed together in different orientations. For example, let the given bodies be halves (oblique or not oblique) of any parallelepiped on the two sides of a dividing plane through a pair of parallel edges. The two halves are homochirally similar; and, being equal, we may make a homogeneous assemblage of them by orienting them all the same way and placing them properly in rows. But the closest packing of this assemblage would necessarily leave vacant spaces between the bodies: and we get in reality the closest possible packing of the given bodies by taking them in pairs oppositely oriented and placed together to form parallelepipeds. These clusters may be packed together so as to leave no unoccupied space.

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.

§ 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.

§ 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}.

This will be understood by looking at Fig. 12, showing S{1} and neighbours on each side of it in the rows of S{1}S{2}, S{1}S{3}, and in a row parallel to that of S{2}S{3}. 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 S{4} may be brought into proper position to touch S{1}, S{2}, and S{3}_ 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 S{1}, S{2}, S{3}, 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 S{1} 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 S{4} from above into position to touch S{1}, S{2}, and S{3}, 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, S{4} 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 S{1}, the point of contact on the other will be the point on it corresponding to the point or one of the points of S{1}, where its tangent plane is parallel to the tangent plane at p. If S{1} 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 S{1}_; 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.

§ 28. In the homogeneous assemblage defined in § 24, each solid, S{1}, is touched at twelve points, being the three points of contact with S{2}, S{3}, S{4}, and the three 3’s of points on S{1} corresponding to the points on S{2}, S{3}, S{4}, at which these bodies are touched by the others of the quartet. This statement is somewhat difficult to follow, and we see more clearly the twelve points of contact by not confining our attention to the quartet S{1}, S{2}, S{3}, S{4} (convenient as this is for some purposes), but completing the assemblage and considering six neighbours around S{1} in one plane layer of the solids as shown in Fig. 12, with their six points prq″p′r′q″′ of contact with S{1}; and the three neighbours of the two adjacent parallel layers which touch it above and below. This cluster of thirteen, S{1} and twelve neighbours, is shown for the case of spherical bodies in the stereoscopic photograph of § 4 above. We might of course, if we pleased, have begun with the plane layer of which S{1}, S{2}, S{4} are members, or with that of which S{1}, S{3}, S{4} are members, or with the plane layer parallel to the fourth side S{2} S{3} S{4} of the tetrahedron: and thus we have four different ways of grouping the twelve points of contact on S{1}_ into one set of six and two sets of three.

§ 29. In this assemblage we have what I call ‘close order’ or ‘close packing.’ For closest of close packings the volume of the tetrahedron (§ 24) of corresponding points of S{1}, S{2}, S{3}, and S{4} must be a minimum, and the least of minimums if, as generally will be the case, there are two more different configurations for each of which the volume is a minimum. There will in general also be configurations of minimax volume and of maximum volume, subject to the condition that each body is touched by twelve similarly oriented neighbours.

§ 30. Pause for a moment to consider the interesting kinematical and dynamical problems presented by a close homogeneous assemblage of smooth solid bodies of given convex shape, whether perfectly frictionless or exerting resistance against mutual sliding according to the ordinarily stated law of friction between dry hard solid bodies. First imagine that they are all similarly oriented and each in contact with twelve neighbours, except outlying individuals (which there must be at the boundary if the assemblage is finite, and each of which is touched by some number of neighbours less than twelve). The coherent assemblage thus defined constitutes a kinematic frame or skeleton for an elastic solid of very peculiar properties. Instead of the six freedoms, or disposables, of strain presented by a natural solid it has only three. Change of shape of the whole can only take place in virtue of rotation of the constituent parts relatively to any one chosen row of them, and the plane through it and another chosen row.

§ 31. Suppose first the solids to be not only perfectly smooth but perfectly frictionless. Let the assemblage be subjected to equal positive or negative pressure inwards all around its boundary. Every position of minimum, minimax, or maximum volume will be a position of equilibrium. If the pressure is positive the equilibrium will be stable if, and unstable unless, the volume is a minimum. If the pressure is negative the equilibrium will be stable if, and unstable unless, the volume is a maximum. Configurations of minimax volume will be essentially unstable.

§ 32. Consider now the assemblage of § 31 in a position of stable equilibrium under the influence of a given constant uniform pressure inwards all round its boundary. It will have rigidity in simple proportion to the amount of this pressure. If now by the superposition of non-uniform pressure at the boundary, for example equal and opposite pressures on two sides of the assemblage, a finite change of shape is produced: the whole assemblage essentially swells in bulk. This is the ‘dilatancy’ which Osborne Reynolds has described in an exceedingly interesting manner with reference to a sack of wheat or sand, or an india-rubber bag tightly filled with sand or even small shot. Consider, for example, a sack of wheat filled quite full and standing up open. It is limp and flexible. Now shake it down well, fill it quite full, shake again, so as to get as much into it as possible, and tie the mouth very tightly close. The sack becomes almost as stiff as a log of wood of the same shape. Open the mouth partially, and it becomes again limp, especially in the upper parts of the bag. In Reynolds’ observations on india-rubber bags of small shot his ‘dilatancy’ depends, essentially and wholly, on breaches of some of the contacts which exist between the molecules in their configuration of minimum volume: and it is possible that in all his cases the dilatations which he observed are chiefly, if not wholly, due to such breaches of contact.

But it is possible, it almost seems probable, that in bags or boxes of sand or powder, of some kinds of smooth rounded bodies of any shape, not spherical or ellipsoidal, subjected persistently to unequal pressures in different directions, and well shaken, stable positions of equilibrium are found with almost all the particles each touched by twelve others.