48 



MINERALOGY 



pairs of parallel fa'ces. Diagram Fig. 72 represents the symmetry ; 

 13 axes, 9 planes, and a center. The value and position of the axes 

 and planes may be understood best by the consideration of their 



relation to the edges of the cube or 

 hexahedron. 



There are three ditetragonal axes 

 ending in the center of the cube 

 faces, these are the crystallographi- 

 cal axes; four ditrigonal axes end- 

 ing in the corners; six didigonal 

 axes ending in the middle of the 

 edges; three planes of symmetry 

 bisect the edges of the cube, and 

 contain the crystallographical axes ; 

 they intersect in the center of sym- 

 metry and divide space around it into eight equal portions (oc- 

 tants). The remaining six planes contain the edges, each plane 

 passing through the center and opposite edges. The nine planes 

 of symmetry divide space around the center into 48 equal tri- 

 angular solid angles. 



FIG. 72. 



Forms 



I. Hexoctahedron ; na:a:ma; (hkl), Fig. 73. 



Here the values of the coefficients, m and n, are independent of 

 each other and not at their limiting values, i or oo . When n = 2 

 and m = 3, yielding the parameters 

 3 a : a : 2 a, they will locate a face in 

 each one of the 48 triangular segments 

 into which the planes of symmetry di- 

 vide space, or 48 scalene triangles. 

 This is the largest number of faces 

 possible on any crystal form. Eight 

 faces are symmetrically grouped around 

 the extremities of the ditetragonal 

 axes (crystallographical axes) ; six 

 around the ditrigonal axes (center of 

 the octants) ; four around the di- 

 digonal axes. All faces are similar scalene triangles, each of 

 which intersects one axis at unity, the second at a greater dis- 

 tance, the third at a still greater distance. Faces, edges, and 



FIG. 73. The Hexahedron, 

 3 a : a : 2 a. 



