MODELS FOR USE IN TEACHING ELEMENTARY CRYSTALLOGRAPHY. 73 



form, forty-eight for the cubic, twenty-four for the hexa- 

 gonal, sixteen for the tetragonal, and eight for the rhombic. 

 For the rhombic model the card may take the form of any 

 acute angled triangle. For the tetragonal model the acute 

 angled triangle must have one angle less than 45 degrees, 

 for the hexagonal less than 30 degrees. 



As it is rather difficult to cut a card which will fit into 

 the cubic model, I have calculated the shapes required for 

 some of the principal forms. It will be noticed that in 

 figure 1 the mirrors are lettered H, V and S, respectively. 

 The model should be placed with H horizontal and V ver- 

 tical. The following are suitable triangles, the edges 

 which are to come into contact with the mirrors being 

 indicated by means of similar letters : 

 Octahedron (111) sides in proportion of H= ^3 V = 1 S = 2 

 Cube (100) „ „ H=1V = 1S=V2 



Dodecahedron (110) „ „ H=v2V=lS- vS 



Hexoctahedron (123) „ „ H = *745 V = *570 S = '915 



Cards of the first three shapes show very instructively 

 the fact that the simpler forms of the system may be 

 regarded as limiting cases of the general form. 



The models are instructive in other ways. For instance, 

 in the case of the cubic model, if a rod be held in the 

 position of a centronormal to any particular crystal face, 

 the multiple reflections show the positions of all the centro- 

 normals to all the faces of the form. It will be found that 

 there are seven distinct ways in which the rod may be 

 held corresponding with the seven types of form possible 

 in the group. 



Thus, if the rod be laid in the dihedral angle of H and S, 

 the positions of the six centronormals of the cube, coincid- 

 ing with the three quaternary axes of symmetry, appear. 

 If the rod is laid somewhere on the face H, the twenty- 

 four centronormals of a tetrahexahedron appear. If the 



