



82 THE POPULAR SCIENCE MONTHLY 



g on the diagram. As the space has two dimensions this has two varia- 

 bles, \:n:m. The first group at the corners of the triangle had faces 

 with like sides, the second at the sides of the triangle had faces with two 

 like sides, this third and last group has faces with unlike sides; they are 

 scaline triangles. They have forty-eight sides, a number that can factor 

 into 24, 12, 8, 6, 4 and 3. As they are arranged over the surface of 

 the triangle, these nearest the octahedron are six-faced octahedrons in 

 appearance, those nearest the cube are eight-faced cubes, those nearest 

 the trapezohedrons are two-faced trapezohedrons, and so on. 



For many years I have shown this symmetrical passage of these 

 seven forms into each other by using three colors, red for the octahedral 

 lines, blue for the dodecahedral, and green for the cubical, a device my 

 old pupil, Geo. H. Williams, used in his " Elements of Crystallography.'' 

 The upper corner is all octahedral, the middle horizontal band is half so r 

 the base not at all octahedral, and so of the other corners symmetrically. 



The law of symmetry permits any symmetrical half of these faces to 

 appear independently on the crystal, and the crystal fulfils the law of 

 symmetry, and this may be done in three ways. (1) We may take all 

 the faces in half the octants, or half the faces in each octant, and in the 

 second case we may begin in the second octant (2) with the face adja- 

 cent to the initial face, or (3) with a face not adjacent. In accordance 

 with the first law the half of the faces of the octahedron forms the 

 tetrahedron which we naturally place in the figure, as (h) directly 

 beneath the octahedron from which it is derived. 



In the same way the tetragonal dodecahedron (i), the half form of 

 the three-faced octahedron, and the trigonal dodecahedron (/), that of 



