12 Lord Kelvin. [Jan. is. 



that A, B, C are points of the first assemblage nearest to P. Again Q 

 is a comer of a cube of which P is the centre ; and if Q A, QB, QC are 

 three conterminous edges of this cube, D, E, F are points of the second 

 assemblage nearest to Q. The rhombohedron of which PQ is body- 

 diagonal and PA, PB, PC the edges conterminous in P, and QD, QE, 

 QF the edges conterminous in Q, is our present rhombohedron. The 

 diagram of 9 (fig. 7), imagined to be altered to proper proportions 

 for the present case, may be looked to for illustration. Its three face- 

 diagonals through P, being PD, PE, PF, are perpendicular to one 

 another. So also are QA, QB, QC, its three face-diagonals through 

 Q. The body-diagonal of the cube PQ, being half the body-diagonal 

 of the cube whose edges are PD, PE, PF, is equal to PD x / : 

 and PA, PB, PC are also each of them equal to this, because A, B, C 

 are centres of other equal cubes, having P for a common corner. 

 January 30.] 



14. The tetrahedrons used in the model are those into which the 

 parallelepiped is cut by three planes through the axial diagonal, 

 which in this case cut one another at angles of 60. We wish to be 

 able to shift the tetrahedrons into positions corresponding to those of 

 the triangles in fig. 1, which we could not do if they were cut out of 

 the solid. I, therefore, make a mere skeleton of each tetrahedon, 

 consisting of a piece of wire bent at two points, one-third of its 

 length from its ends, at angles of 70, being sin" 1 v/3, in planes 

 inclined at 60 to one another. The six skeletons thus made are equal 

 and similar, three homochirals and the other three also homochirals, 

 their enantiomorphs. In their places in the primitive parallelepiped 

 they have their middle lines coincident in its axial diagonal PQ, and 

 their other 6x2 arms coincident in three pairs in its six edges 

 through P and Q. Looking at fig. 7 we see, for example, three of the 

 edges CP, PQ, QE, of one of the tetrahedrons thus constituted ; and 

 DQ, QP, PB, three edges of its enantiomorph. lo the model they are 

 put together with their middle lines at equal distances around the 

 axial diagonal and their arms symmetrically arranged round it. 

 Wherever two lines cross they are tied, not very tightly, together 

 by thin cord many times rpnnd, and thus we can slip them along 

 so as to bring the six middle lines either very close together, nearly 

 as they would be in the primitive parallelepiped, or farther and 

 farther out from one another so as to give, by the four corners of 

 the tetrahedrons, the twenty-four corners of all possible configurations 

 of the plane-faced space-filling tetrakaidekahedron. 



15. The six skeletons being symmetrically arranged around an 

 axial line we see that each arm is cut by lines of other skeletons in 

 three points. For an important configuration, let the skeletons be 

 separated out from the axial line just so far that each arm is divided 

 into four equal parts, by those three interaectional points. The 



