The Isometric Normal Class

The forms of the Isometric, Normal Class, are as follows:

1. Cube or Hexahedron
The cube is a form composed of axis square faces which make 90° angles with each other. Each face intersects on of the crystallographic axes and is parallel to the other two. Its symbol is (100).

2. Octahedron
The octahedron is a form composed of eight equilateral triangular faces, each of which intersects all three of the crystallographic axes equally. Its symbol is (111). When in combination the octahedron is to be recognized by its eight similar faces, each of which is equally inclined to the three crystallographic axes. It is to be noted that the faces of an octahedron truncate symmetrically the corners of a cube.

3. Dodecahedron
The dodecahedron is a form composed of twelve rhombic-shaped faces. Each face intersects two of the crystallographic axes equally and is parallel to the third. Its symbol is (110). It is to be noted that the faces of a dodecahedron truncate the edges of both the cube and the octahedron.

4. Tetrahexahedron
The tetrahexahedron is a form composed of twenty-four isosceles triangular faces, each of which intersects one at unity, the second at some multiple, and is parallel to the third. There are a number of tetrahexahedrons which differ from each other in respect to the inclination of their faces. Perhaps the one most common in occurrence has the parameter relations 1a, 2 b, c, the symbol of which would be (210). The symbols of other forms are (310), (410), (320), etc. it is helpful to note that the tetrahexahedron, as its name indicates, is like a cube, the faces of which have been replaced by four others.

5. Trapezohedron or Tetragonal Trisoctahedron
The trapezohedron is a form composed of twenty-four trapezium-shaped faces, each of which intersect one of the crystallographic axes at unity and the other two at equal multiples. There are various trapezohedrons with their faces having different angles of inclination. A common trapezohedron has for its parameters 1a, 2b, 2c, the symbol for which would be (211). The symbols for other trapezohedrons are (311), (11), (322), etc. it will be noted that a trapezohedron is an octahedral-like form and may be conceived of as an octahedron, each of the planes of which has been replaced by theree faces. Consequently it is sometimes called a tetragonal trisoctahedron. The qualifying word, tetragonal, is used to indicate that each of its faces has four edges and to distinguish it from the other trisoctahedral form, the description of which flows. Trapezohedron is the name, however, most commonly used. The following are aids to the recognition of the form when it occurs in combinations: the three similar faces to be found in each octant; the relations of each face to the axes; and the fact that the middle edges between the three faces in any one octant go toward points which are equidistant from the ends of the two adjacent crystallographic axes. It is to be noted that the faces of the common trapezohedron truncate the edges of the dodecahedron.