338 THE FORMS OF TISSUES [ch. 



expected to find all plane and all similar) four afe 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 quadri- 

 lateral, 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 " tetra- 

 kaidecahedra " ; 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 quadri- 

 laterals nor the hexagons being necessarily plane. In a certain 

 particular case, the quadrilaterals are plane surfaces, but the 

 hexagons shghtly 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°*. 



We may take it as certain that, in a system of perfectly fluid 

 films, like the interior of a mass of soap-bubbles, where the films 

 are perfectly free to glide or to rotate over one another, the mass 

 is actually divided into cells of this remarkable conformation. 



* Von Fedorow had already described (in Russian) the same figure, under the 

 name of cubo-octahedron, or hejita-parallelohedron, Umited however to the case 

 where all the faces are plane. This figure, together with the cube, the hexagonal 

 prism, the rhombic dodecahedron and the "elongated dodecahedron," constituted 

 the five plane-faced, parallel-sided figures by which space is capable of being 

 completely filled and symmetrically partitioned ; the series so forming the founda- 

 tion of Von Fedorow's theory of crystaUine structure. The elongated dodecahedron 

 is, essentiaUj^ the figure of the bee's cell. 



