The rhombic dodecahedron has six tetrahedral angles, and eight trihedral angles; and it is obvious, on consideration, that at each of the former six dodecahedra meet in a point, and that, where the four tetrahedral facets of each coalesce with their neighbours, we have twelve plane films, or interfaces, meeting in a point. In a precisely similar fashion, we may imagine twelve plane films, drawn inwards from the twelve edges of a cube, to meet at a point in the centre of the cube. But, as Plateau discovered[379], when we dip a cubical wire skeleton into soap-solution and take it out again, the twelve films which are thus generated do _not_ meet in a point, but are grouped around a small central, plane, quadrilateral film (Fig. 134). In other words, twelve plane films, meeting in a point, are _essentially unstable_. If we blow upon our artificial film-system, the little quadrilateral alters its place, setting itself parallel now to one and now to another of the paired faces of the cube; but we never get rid of it. Moreover, the size and shape of the quadrilateral, as of all the other films in the system, are perfectly definite. Of the twelve films (which we had {338} expected to find all plane and all similar) four are plane isosceles triangles, and eight are slightly curved quadrilateral figures. The former have two curved sides, meeting at an angle of 109° 28′, and their apices coincide with the corners of the central quadrilateral, whose sides are also curved, and also meet at this identical angle;—which (as we observe) is likewise an angle which we have been dealing with in the simpler case of the bee’s cell, and indeed in all the regular solids of which we have yet treated. By completing the assemblage of polyhedra of which Plateau’s skeleton-cube gives a part, Lord Kelvin shewed that we should obtain a set of equal and similar fourteen-sided figures, or “tetrakaidecahedra”; and that by means of an assemblage of these figures space is homogeneously partitioned—that is to say, into equal, similar and similarly situated cells—with an economy of surface in relation to area even greater than in an assemblage of rhombic dodecahedra. In the most generalised case, the tetrakaidecahedron is 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. In a certain particular case, the quadrilaterals are plane surfaces, but the hexagons slightly curved “anticlastic” surfaces; and these latter have at every point equal and opposite curvatures, and are surfaces of minimal curvature for a boundary of six curved edges. The figure has the remarkable property that, like the plane rhombic dodecahedron, it so partitions space that three faces meeting in an edge do so everywhere at equal angles of 120° [380].