Hexagonal System, Crystallographic Axes and Normal Class

Crystallographic Axes
The crsyatllographic axes of the hexagonal system are four in number. Three of these lie in the horizontal plane, while the fourth is vertical. The three horizontal axes are of equal length and interchangeable. They make angles of 60° and 120° with each other. The vertical axis varies in its relative length for each hexagonal mineral, and this is expressed in terms of the length of the horizontal axes, which is taken as unity. Thus in the case of beryl, the vertical axis, designated as c, has a length which in relation to the length of the horizontal axes can be expressed as c=0.499.

When properly orientated, one of the horizontal crystallographic axes is parallel to the observer, and the other two make 30° angles on either side of a line perpendicular to him. The proper position of the horizontal axes when viewed in the direction of the vertical axis. As the three horizontal axes are interchangeable with each other, they are usually designated a1, a2 and a3. Note that a1 is parallel to the observer and its positive end is at the right, while a3 is to the right of the observer and its positive end is at the back. In giving the indices of any face upon a hexagonal crystal four numbers must be given, since there are four axes. The numbers referring to the intercepts of the face with the three horizontal axes are given first in their proper order, while the number referring to the intercept on the vertical axis is given last.

Normal Class

Symmetry and Forms
The symmetry of the Normal Class of the Hexagonal System is as follows: The vertical crystallographic axis is an axis of hexagonal symmetry. There are six horizontal axes of binary symmetry, three of them being coincident with the crystallographic axes and the other three lying midway between them. There is a horizontal plane of symmetry and six vertical planes of symmetry. The forms of the Normal Class are as follows:

1. Prism of First Order.
This is a form consisting of six rectangular vertical faces each of which intersect two of the horizontal crystallographic axes equally and is parallel to the third. The symbol for the form is (1010).

2. Prism of Second Order
This is a form consisting of six rectangular vertical faces, each of which intersect two of the horizontal axes equally and the intermediate horizontal axis at one-half this distance. Fig. 109 shows the prism of the second order. The symbol for the form is (1120). As in the Tretragonal System, the prisms of the first and second order are geometrically identical forms, the distinction between them lying only in their orientation.

3. Dihexagonal Prism.
The dihexagonal prism has twelve rectangular vertical faces, each of which intersects all three of the horizontal crystallographic axes at different lengths. There are various dihexagonal prisms, depending upon their differing relations to the horizontal axes. The symbol of a common dihexagonal prism is (2130).

4. Pyramid of First Order
This form consists of twelve isosceles triangular faces, each of which intersects two of the horizontal crystallographic axes equally, is parallel to the third horizontal axis and intersects the vertical axis. There are various pyramids of the first order possible, depending upon the inclination of their faces. The unit-form would have the symbol (1011).

5. Pyramid of the Seconds Order
This is a form composed of twelve isosceles triangular faces, each of which intersects two of the horizontal axes equally, the third and intermediate horizontal axis at one-half this distance, and also intersects the vertical axis. There are various pyramids of the second order possible, depending upon the inclination of their faces. A common form would have for its symbol (1122). The relations between the pyramids of the first and second order is the same as between the corresponding prisms; see above.

6. Dihexagonal Pyramid
The dihexagonal pyramid is a form of twenty-four isosceles triangular faces, each of which intersects all three of the horizontal axes differently and intersects also the vertical axis. There are different dihexagonal pyramids which vary in their intercepts, one of the most common having for its symbol (2131)

7. Basal Pinacoid
The basal pinacoid is a form composed of two horizontal faces. It is shown in combination with the different prisms in Figs. 108, 109 and 110. its symbol is (0001).