ISOMETRIC SYSTEM. 



25 



planes and the same number of planes ; and this is the general 

 law for all the forms. This is a consequence of the equality 

 of the axes and their rectangular intersections. 



But in some crystalline forms there are only half the num- 

 ber of planes which full symmetry requires. In %. 39 a cube 

 is represented with an octahedral plane on half, that is, four, of 



the solid angles. A solid angle having such a plane is diag- 

 onally opposite to one without it. The same form is represented 

 in fig. 40, only the cubic faces are the smallest ; and in fig. 41 

 the simple form is shown which is made up of the four octahe- 

 dral planes. It is a tetrahedron or regular three-sided pyra- 

 mid. If the octahedral faces of fig. 39 had been on the other 

 four of the solid angles of the cube, the tetrahedron made of 

 those planes would have been that of fig. 42 instead of fig. 41. 



Other hemihedral forms are represented in figs. 43 to 45 ; fig. 

 43 is a hemihedral form of the trapezohedron, fig. 4, p. 7 ; 



6g. 44, hemihedral of the hexoctahedron, fig. 7, or a hemi-nex- 

 octahedron. Fig. 45 is a combination of the tetrahedron (piano 

 1) and hemi-hexoctahedron. 



In these forms figs. 41—44, no face has another parallel to it ; 

 and consequently they are called inclined hemihedrons. 



Fig. 46 represents a cube with the planes of a tetrahexahe- 

 dron, as already explained. In fig. 47, the cube has only one 

 of the plaives i-2 on each edge, and therefore only twelve in all ; 



