364 MINERALOGY Hemi- Hemitrope crystals we may imagine as having been formed from tropes, a single crystal, which has been cut into two halves in a particular direction, and one half turned round 180, or 90, or 60. The line about which the revolution is supposed to take place is called the "axis of revolution." From the amount of turn usually being 180, Haiiy gave the name hemitrope. The position of the two Fig. 154. Laws of Fig. 157. Fig. 156. halves in this case resembles that of an object and its image in a mirror, whose surface then would represent the plane of reunion. The following are the laws of hemitropes. The axis of revolution hemi- is always a possible crystallographic line, either an axis, a line tropism. parallel to an axis, or a normal to a possible crystalline plane. The plane normal to the axis of revolution is called the twin plane ; it is either an occurring or a possible plane, and usually one of the more frequently re curring planes. Both the axis and the twin plane Fig. 158. Fig. 159. bear the same relation to both halves of the crystal in their re versed positions ; consequently the parts of hemitrope crystals are symmetrical with reference to the twin plane (except in triclinic forms and some hemihedral crystals). The face of composition very frequently coincides with the twin plane ; when not coinciding, the twin plane and the face of com position are generally at right angles to each other, so that the composition face is parallel to the axis of re volution. But in twins of incorporation the surfaces of composition have exercised a disturbing influence on one another, so that the surface of union is exceedingly ir regular. Still in these cases the axis and the plane of twinning retain a definite position ; but the face of composition, being no longer defined, is useless as a determinant. Modes of There are three modes in which the composition may take place union. in hemitropes. These may be explained by dividing a crystal into halves, with the plane of division vertical, and then turning one of the halves round. L One of the halves may be inverted, as if by revolution through 180 on a horizontal axis at right angles to the plane of section, and the two faces again united by the surfaces which were separated. Here the surfaces of union are the original ones, but the base of one of the halves has taken the place of its summit. Examples : selenite (fig. 161) and orthoclase. 2. One of the halves may be turned round through 180, as if by revolution on a horizontal axis, parallel to the plane of section, and the face opposite and parallel to that of the plane of section an originally external face may then be applied to the other half. Here, not only has the base of one-half become a summit, but a lateral and external face of the original crystal has been thrust to its centre so as to become a face of internal union. Example: labradorite (fig. 162). 3. One of the halves may be turned round through 180, as if by revolution on a vertical axis, parallel to the plane of section, the external face ppposite and parallel to the plane of section becoming a face of union. Here, however, both the original summits retain their position as summits. Example: orthoclase. The first of these modes of composition may occur in each of the systems, but it is not always apparent until disclosed by optical properties. The second is rare, and the third still more so. In hemitrope crystals (less frequently in true twins) the halves of the crystal are frequently reduced in thickness in the direction of the ordinary twin axis ; and when there is a parallel repetition of hemitropes, which frequently occurs, they are often reduced to very thin plates, not the thickness of paper, giving to the surface of the aggregate a striated structure and appearance. In the cubic system the faces of composition, both of twinning Twins of and of hemitropic revolution, are those of the cube, the dodecahed- cubic ron, and the octahedron. system. In the first case we have the axes of the two crystals necessarily in some cases parallel, or, more correctly, falling into one ; but, as in this system all the axes are alike, or all the cubic faces similar, composition may occur along or parallel to all alike, and double or triple twins occur. We have examples in twins of the pentagonal dodecahedron (fig. 163) made up by the iuterpenetratiou of a right and a left (+ and - ), and of the tetrahedron, as seen in pyrite and fahlerz respectively. In virtue of the position required by law 2, it will be seen that the position of the solid which is common to both intersecting crystals is in the twin of pyrite the four-faced cube, which is the holohedral form of the pentagonal dodecahedron, while in the case of the fahlerz twin (fig. 164), the common por tion is an octahedron, the holohedral form of the tetrahedron. Fig. 165. Fig. 166. Twinning on an octahedral face is seen in the apposition twin of spinel (fig. 143), the tetrahedral twin of blende (fig. 165), the inter penetrative octahedral twin of blende seen in fig. 166, and the inter secting cubes of fluor (fig. 167).
This is also the usual twin face for hemitropcs of the cubic system.