52 



MINERALOGY 



VI. Rhombic Dodecahedron; a:a:ooa; (no), Fig. 81. 



If the pole is now moved to the didigonal axes, Fig. 73 a, four faces 



a, a', b, b', will fall in one plane, producing a form with 12 rhombic 



faces, Fig. 81. The faces are grouped four around the ditetragonal 



axes, three around the ditrigonal, and the didigonal axes bisect 



the edges. There is but one rhom- 

 bic dodecahedron with the angles 

 fixed at 120. It is also a fixed 

 form. 



VII. Octahedron; a: a: a; (in), 

 Fig. 82. 



The seventh and last possible 

 form in this type is where the pole 

 is moved to the ditrigonal axes, 

 when the six faces a, b, c, e, etc., 

 of the general form grouped around 

 this axis will fall in one plane, pro- 

 ducing a form bounded by eight 



equilateral triangular faces, Fig. 82. Four faces are grouped 

 around the ditetragonal axes, the ditrigonal axes terminate in the 

 center of the face, and the didigonal axes bisect the edges. All 

 dihedral angles of the regular octahedron are fixed at 70 31' 42" ; 

 it is therefore a fixed form. 



FIG. 82. The Octahedron, (111). 



Relation of the Seven Forms 



When any one of the 48 triangular segments into which the planes 

 of symmetry divide space is considered, Fig. 83, it has been shown 

 that the pole na : a : ma, the hexoc- 

 tahedron, may be located anywhere 

 within the area, and when it ap- 

 proaches the sides or angles, either 

 one or both of the variables m and n 

 approach their limiting values 1 and 

 co . If the pole approaches one of 

 the sides, only one of the variables 

 approached its limit, or the two oa:a.-o>a 

 variables are of the same value. The 

 hexoctahedron, tetrahexahedron, tetragonal trisoctahedron, and 

 the trigonal trisoctahedron, are known as the variable forms, since 

 their parameters contain a variable. The position of the pole of 



FIG. 83. 



