The Isometric, Trisoctahedron
and Hexoctahedron

6. Trisoctahedron or Trigonal Trisoctahedron
The trisoctahedron is a form composed of twuenty-four isosceles triangular faces, each of which intersects two of the crystallographic axes at unity and the third axis at some multiple. There are various trisosctahedrons the faces of which have different inclinations. A common trisoctahedron has for its parameters 1a, 1b, 2c, its symbol being (221). Other trisoctahedrons have the symbols (331), (441), (332), etc. It is to be noted that a tripezohedron has for its parameters 1a, 1b, 2c, its symbol being (221). Other tisoctahedrons have the symbols (331), (441), (332), etc. it is to be noted that the trisoctahedron, like the trapezohedron, is a form that may be conceived of as an octahedron, each face of which has been replaced by three others. Frequently it is spoken of as the trigonal trisoctahedron, the modyfin word indicating that its faces have each three edges and so differ from those of the trapezohedron. But when the word “trisoctahedron” is used alone it refers to this form. The following points would aid in its identification when it is found occurring in combination: the three similar faces in each octant; their relations to the axes, and the fact that the middle edges between them go toward the ends of the crystallographic axes. Fig. 43 shows the simple trosctahedron and Fig. 47 a combination of a trisoctahedron and an octahedron. It will be noted that the faces of the triscotahedron bevel the edges of the octahedron.

7. Hexoctahedron
The hexoctahedron is a form composed of forty-eight triangular faces, each of which cuts differently on all three crystallographic axes. There are several hexoctahedrons, which have varying ratios of intersection with the axes. A common hexocatahedron has for its parameter relations 1a, 3/2b, 3c, its symbol being (321). Other hexoctahedrons have the symbols (421), (531), (432), etc. it is to be noted the hexoctahedron is a form that may be considered as an octahedron, each face of which has been replaced by six others. It is to be recognized when in combination by the facts that there are six similar faces in each octant and that each face intercepts the three axes differently. Fig. 48 shows a simple hexoctahedron, Fig. 49 a combination of cube and hexoctahedron, and Fig. 50 a combination of dodecahedron and hexoctahedron, and Fig. 51 a combination of dodecahedron, trapezohedron and hexoctahedron.

Zonal Relations of Isometric Forms
A crystal zone consists of a series of faces, all of which lie parallel to some one crystal direction and whose intersections with each other are all parallel to this direction. For instance in a cube and dodecahedron planes, marked a and d, fall into three zones, each zone being parallel to one of the crystal axes; the cube and octahedron faces fall into six similar zones with the edges between the faces lying parallel in each case to one of the diagonal axes of binary symmetry. It gives in the form of a diagram the positions in relation to each other of all the possible form in the Normal Class. It will be noticed that any face falling in the zone between cube and dodecahedron (or in other words truncating the edge between them) must belong to a tetrahexahedron; similarly any face between cube and octahedron belongs to a trapezohedron; between dodecahedron and octahedron will fall the trisoctahedron; while fall the trisoctahedron; while lastly a face that does not occur in any one of these zones must belong to a hexoctahedron. Therefore, to determine the forms present on any isometric crystal, it is only necessary to discover the crystal; recognize the cube, octahedron, and dodecahedron faces that may be present, or, if they fail, to realize where they would properly occur, and them by applying the principle of zonal relations determine the character of the other forms upon the crystal.

Occurrence of Isometric Forms
The cube, octahedron and dodecahedron are the most common of the isometric forms. The trapezohedron is also frequently observed on a few minerals. The other forms, the tetrahexahedron, trisoctahedron and hexoctahedron, are rare and are ordinarily to be observed only as small truncations in combinations.

The following is a list of the commoner minerals upon the crystals of which each form is prominent.

• Cube: Galena, halite, sylvite, fluorite, cuprite.
• Octahedron: spinel, manietite, franklinite, cromite.
• Dodecahedron: Magnetite, franklinite, chromite.
• Dodecahedron: Magnetite, garnet
• Trapezohedron: Leucite, garnet, analcite.