328 THEFOKMS OF TISSUES [ch. 



set down fits into its nest between three others, and the four 

 form a regular tetrahedral arrangement. Just as it was obvious, 

 then, that by mutual pressure from the six laterally adjacent cells, 

 any one cell would be squeezed into a hexagonal prism, so is it also 

 obvious that, by mutual pressure against the three terminal 

 neighbours, the end of any one cell will be compressed into a solid 

 trihedral angle whose edges will meet, as in the analogous case 

 already described of a system of soap-bubbles, at a plane angle 

 of 109° and so many minutes and seconds. What we have to 

 comprehend, then, is how the six sides of the cell are to be combined 

 with its three terminal facets. This is done by bevelling off three 

 alternate angles of the prism, in a uniform manner, until we have 

 tapered the prism to a point; and by so doing, we evidently 

 produce three rhombic surfaces, each of which is double of the 

 triangle formed by joining the apex to the three untouched angles 

 of the prism. If we experiment, not with cylinders, but with 

 spheres, if for instance we pile together a mass of bread-pills (or 

 pills of plasticine), and then submit the whole to a uniform pressure, 

 it is obvious that each ball (like the seeds in a pomegranate, as 

 Kepler said), will be in contact with twelve others, — six in its own 

 plane, three below and three above, and in compression it will 

 therefore develop twelve plane surfaces. It will in short repeat, 

 above and below, the conditions to which the bee's cell is subject 

 at one end only; and, since the sphere is symmetrically situated 

 towards its neighbours on all sides, it follows that the twelve plane 

 sides to which its surface has been reduced will be all similar, 

 equal and similarly situated. Moreover, since we have produced 

 this result by squeezing our original spheres close together, it is 

 evident that the bodies so formed completely fill space. The 

 regular solid which fulfils all these conditions is the rhombic 

 dodecahedron. The bee's cell, then, is this figure incompletely 

 formed : it is a hexagonal prism with one open or unfinished end, 

 and one trihedral apex of a rhombic dodecahedron. 



The geometrical form of the bee's cell must have attracted the 

 attention and excited the admiration of mathematicians from time 

 immemorial. Pappus the Alexandrine has left us (in the intro- 

 duction to the Fifth Book of his Collections) an account of its 

 hexagonal plan, and he drew from its mathematical symmetry the 



