96 



CRYSTALLINE FOBMS. 



FIG. 48. 



they intersect each other, and the faces of the octohedron 

 disappear, a perfect cube is produced (fig. 46) . 



If the twelve sides of the octohedron (fig. 23) are replaced 

 by tangent planes, and these are extended till they mutually 

 intersect, the rhombic dodecahedron will be formed (fig. 47). 

 The cube may, by analogous changes, be converted into 

 the octohedron, tetrahedron, and rhombic dodecahedron. 

 For, as previously stated, if the eight solid 

 angles of the cube be replaced by equila- 

 teral triangles, and these are enlarged till 

 the planes of the original cube are de- 

 stroyed, the octohedron is the result. 



The tetrahedron may be formed by re- 

 placing the four alternate solid angles of 

 the cube by tangent planes, so that all its 

 original faces disappear (see fig. 32). 



By replacing the twelve edges of the cube (fig. 16) with 

 tangent planes till the new faces intersect each other, the 

 rhombic dodecahedron will be produced (fig. 24) . 



The octohedron with a square base is allied to the right 

 square prism, for if (fig. 35) two tangent planes are substi- 

 tuted for the solid angles, a, , and the 

 edges of the base are replaced by faces 

 perpendicular to the former, new forms 

 will result. If the faces of the octo- 

 hedron disappear, the right square prism 

 is formed (fig. 49) ; but if traces of them 

 remain, secondary forms, intermediate be- 

 tween the two primary forms, will be produced. 



The rectangular and rhombic bctohedrons, and the right 

 rectangular and rhombic prisms, are associated with each 

 other. Thus on replacing the solid angles a, #, and the four 

 edges of the base of the rectangular octohedron (fig. 36), 

 by tangent planes, and extending them till the planes of 

 the octohedron disappear, the right rectangular prism is 

 formed (fig. 50) ; and the rhombic octohedron (fig. 37), 

 by a similar change, is converted into the right rhombic 

 prism (fig. 51). 



By applying tangent planes to all the edges of the rhombic 

 octohedron, fig. 37, except those of the base, the rectangular 

 octohedron, fig. 36, may be produced ; and by reversing the 



FIG. 49. 



