§ 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.
Fig. 9.
§ 18. For the division of continuous three-dimensional space[5] 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[6] 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.
Fig. 10.
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.