*THE POPULAR SCIENCE MONTHLY*

*g* on the diagram. As the space has two dimensions this has two variables, 1:*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 adjacent 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 (*j*), that of