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Every face in a form has the same general relationship to the crystallographic axes and therefore the same general Miller index, yet every face in a form has a different specific relationship to the crystallographic axes and therefore has a different Miller index. These statements can be clarified by using an example. Figure 4.23 shows the common eight‐faced isometric form called the octahedron.

Each face in the octahedron has the same general relationship to the three crystallographic axes in that each intersects the three crystallographic axes at unity. The Miller indices of each face are some form of (111). However, only the top, right front face intersects the positive ends of all three axes. The bottom, left back face intersects the negative ends of all three axes, and the other six faces intersect some combination of positive and negative ends of the three crystallographic axes. None of the faces are parallel to one another; each belongs to a different set of parallel planes within the crystal. The Miller indices of these eight faces and the set of planes to which each belongs are (111), ( ), ( ), ( ), ( ), ( ), ( ), and ( ). Their unique Miller indices allow us to distinguish each of the eight faces and the sets of planes to which they belong. However, they are all parts of the same form because they all have the same general relationship to the crystallographic axes. It is cumbersome and often unnecessary to have to recite the indices of every face within a form. To represent the general relationship of the form to the crystallographic axes, the indices of a single face, called the form face, are chosen and placed in brackets to indicate that they refer to the form indices. The rule for choosing the form face is generally to select the top face if there is one, or the top right face if there is one, or the top right front face if there is one. In the case of the octahedron, the top right front face is the face that intersects the positive ends of the a₁‐axis (front), the a₂‐axis (right), and the a₃‐axis (top) and has the Miller indices (111). The form indices for all octahedral crystals are the Miller indices of the form face placed between curly brackets {111}. Similarly the form indices for the cube (see Figure 4.22), in which the faces intersect one axis and are parallel to the other two are {001}, the Miller indices of the top face, whereas the form indices for the dodecahedron, in which each face intersects two axes at unity and is parallel to the third is {011}, the indices for the top, right face.

Figure 4.24 Five common forms in the isometric system: (a) cube, (b) octahedron, (c) dodecahedron, (d) tetrahedron, (e) pyritohedron.

Many other forms exist. Every crystal form has a form index, which is the Miller index of the form face placed in brackets. Each form consists of one or (generally) more faces and each face possesses a Miller index different from that of every other face in the form. Every crystal system has a characteristic suite of forms that reflect the unique characteristics of the crystal lattice of the system, especially the relative lengths of the three crystallographic axes that directly or indirectly reflect the lengths of the unit cell edges. The forms that are characteristic of each class (space point group) in each crystal system are beyond the scope of this text (see Klein and Hurlbut 1985). However, a brief review of some common forms in each crystal system is appropriate.