of Homogeneous, Partitioning 's of Space. 89 



As regards the deformations under (2), (o), and (4) it is 

 immediately obvious that, with the assumptions we have 

 made, it is possible to express the volume and surface of each 

 polyhedron as a function of the length of the edge and at 

 any one angle in any of the rhombs, preferably the angle of 

 the pole a. For the cubo-octahedron it is most convenient 

 to refer the calculation to the whole edge of the pyramid and 

 the angle at the apex of the latter (fig. 4, a). As the volume 

 is to remain constant, we have to find the volume of the 

 minimum polyhedron, for which the value of a is known, 

 taking the edge as being of unit length, and have then to 

 calculate the edge E for various values of a, so that the 

 volume is the same as that of the minimum polyhedron with 

 the edge of unit length. The other quantities of interest to 

 us can then be expressed in terms of E and a, ; they are 

 the total surface and the length of the axis (for the cubo- 

 octahedron the distance between the square faces). The 

 expressions found are as follows : — 



I. Cube stretched along space diagonal and transformed 

 into rhombohedron. (Fig. 1, a & b.) 

 1 



E = 



^2sin^3-4sin^ L = 3E \A 



4 . 9 cc 



3 sm Y 



S = GE 2 sin a. 



II. Rhombo-dodeeabedron stretched along crystallographic 



axis. (Fig. 2,ak b.) 



B- — 1— ...- , 



L = 4E v cos a. 



\J 21 cos a. sin 4 1 



S = 8E 2 / sin a. -f sin ~ s/ 2 cos a). 



III. Rhombo-dodecahedron stretched along axis passing 

 through two opposite 3-faced corners. ~ (Fig. 3, a & b.) 



E= I 



L=E(l + 3' V /l-|sin^). 



S = 2E 2 (3sin a+v ^r2 S i n |} 



