504 Sir W. Thomson on the Division of Space 



be filled with (or divided into) equal and similar rhombic 

 dodecahedrons. Hence it might seem that the rhombic 

 dodecahedron is the solution of our problem for the case 

 of all the cells equal in volume, and every part of the boundary 

 of the group either infinitely distant from the place considered, 

 or so adjusted as not to interfere with the homogeneousness 

 of the interior distribution of cells. Certainly the rhombic 

 dodecahedron is a solution of the minimax, or equilibrium- 

 problem ; and certain it is that no other plane-sided poly- 

 hedron can be a solution. 



4. But it has seemed to me, on purely theoretical con- 

 sideration, that the tetrahedral angles of the rhombic dodeca- 

 hedron*, giving, when space is divided into such figures, 

 twelve plane films meeting in a point (as twelve planes from 

 the twelve edges of a cube meeting in the centre of the cube) 

 are essentially unstable. That it is so is proved experi- 

 mentally by Plateau (vol. i. § 182, fig. 71) in his well-known 

 beautiful experiment with his cubic skeleton frame dipped in 

 soap-solution and taken out. His fig. 71 is reproduced here in 

 fig. 1. Instead of twelve plane films stretched inwards from 

 the twelve edges and meeting in the centre of the cube, it 

 shows twelve films, of which eight are slightly curved and four 

 are plane f, stretched from the twelve edges to a small central 

 plane quadrilateral film with equal curved edges and four 

 angles each of 109° 28'. Each of the plane films is an 

 isosceles triangle with two equal curved sides meeting at 

 a corner of the central curvilinear square in a plane perpen- 

 dicular to its plane. It is in the plane through an edge and 

 the centre of the cube. The angles of this plane curvilinear 

 triangle are respectively 109° 28', at the point of meeting of 

 the two curvilinear sides : and each of the two others half of 

 this, or 54° 44'. 



5. I find that by blowing gently upon the Plateau cube 

 into any one of the square apertures through which the little 

 central quadrilateral film is seen as a line, this film is caused 



* The rhombic dodecahedron has six tetrahedral angles and eight tri- 

 hedral angles. At each tetrahedral angle the plane faces cut one another 

 successively at 120°, while each is perpendicular to the one remote from 

 it; and the angle between successive edges is cos -1 ^, or 70° 32'. The 

 obtuse angles (109° 28') of the rhombs meet in the trihedral angles of the 

 solid figure. The whole figure may be regarded as composed of six square 

 pyramids, each with its alternate slant faces perpendicular to one another, 

 placed on six squares forming the sides of a cube. The long diagonal 

 of each rhombic face thus made up of two sides of pyramids conterminous 

 in the short diagonal, is V2 times the short diagonal. 



t I see it inadvertently stated by Plateau that all the twelve films are 

 " legerement courbees." 



