CH. VIII] BEES AND THEIR CELLS 167 



centre and its base a face of the cube. Place these pyramids 

 each on one face of the other cube. The resulting solid is a 

 dodecahedron with rhombic faces, 14 vertices, and 24 edges; 

 space can be filled with such solids. If such a dodecahedron is 

 bisected by a plane through the centre and perpendicular to a 

 diagonal of the cube, either half is of the form of the typical bee's 

 cell except that its depth is less than is usual in a honey comb. 

 Now the following argument* shows that rhombic dodeca- 

 hedrons are cell shapes naturally formed under certain physical 

 conditions. Consider a large space filled with as many equal 

 spheres as is possible t, which are crushed together symmetrically 

 till all the space between them is occupied; such an initial 

 arrangement is a natural one because it is of maximum density 

 and maximum stability. Initially each sphere touches twelve 

 adjacent spheres at the mid-points of the twelve edges of a cube, 

 its intermediate portions bulging out through the six faces of 

 the cube. When the spheres are crushed together, these twelve 

 points of contact move inwards along the radii, and the six 

 intermediate portions are squeezed into the over-arching spaces 

 which lie between the points of contact of the surrounding 

 spheres. Since there are four spheres round every face of the 

 cube, these portions will be squeezed into four-sided pyramids, 

 the faces of each being coterminous with those of the adjacent 

 pyramid, and both being the ultimate position of the original 



# 



* S. Bryant, London Mathematical Society Proceedings, 1884-85, vol. xvi, 

 pp. 311—315. 



t A large box can be filled with a number of small equal spheres arranged in 

 horizontal layers, one on top of the other, in various ways. It might be filled 

 bo that each sphere rests on the top of the sphere immediately below it in the 

 next layer, touches eaoh of 4 adjacent spheres in the same layer, and touches 

 one sphere in the layer above it, thus each sphere is in contact with 6 others: 

 such an arrangement gives the smallest number of spheres with which the 

 box can be filled. We might also fill the box with spheres arranged so that each 

 of them is in contact with 2 spheres in the next lower layer, with 4 in the same 

 layer, and with 2 in the next higher layer. Or we might fill the box with 

 spheres arranged so that each of them is in contact with 4 spheres in the next 

 lower layer, with 4 in the same layer, and with 4 in the next higher layer. Such 

 an arrangement occurs if the box be shaken steadily, and thus it is the most 

 stable arrangement ; it gives the greatest number of spheres with which the box 

 can be filled. In this arrangement each sphere is in contact with 12 others. 



