Page:Encyclopædia Britannica, Ninth Edition, v. 16.djvu/374

Rh 356 MINERALOGY can be noticed. Many others more complicated will occur in the descriptive part of this article. Among holohedral combinations, the cube, octahedron, and rhombic dodecahedron are the predomin ant forms. In fig. 27 the cube has its angles replaced by the faces of the octahedron, which truncate the angles, and the sign of this combination is ooOo, 0. In fig. 28 this process may be regarded as having proceeded still farther, so that the faces of the octahedron nearly equal those of the cube, while in fig. 29 they now predom inate ; the sign, still of the same two elements, but in reverse order, is 0, ooOoo. It will thus be seen that, through an increase in the amount of the abstraction of the faces of the cube, the figure gradually passes over into that of the octahedron. This may occur in all cases, and is termed the passage of the cube into the octa hedron (or vice versa), or a &quot;transition by decrement.&quot; In fig. 31 the cube has its edges replaced by the faces of the rhombic dodecahedron, which truncate the edges, the sign being ooO, ooO; while in fig. 32 there is the same combination, but with the faces of the cube subordinate, and hence the sign is ooO, ooOco. The former figure, it will be seen, has more the general aspect of the cube, the latter of the dodecahedron. Here the solid angles of the latter are truncated by the faces of the cube, and we have the passage of the cube into the dodecahedron by decrement. The same transition, through truncation or decrement, could be shown in all cases of combinations, and in both directions, the last stage of the passage into one or other form always consisting of the replacement of its solid or interfacial angles by faces of the de parting figure, more or less minute. A few illustrations of this may be given, in the three most important forms. The relationship of the tetrakishexahedron to the cube has above been stated to be, that its faces form six low quadrilateral pyramids, which rest upon or spring from the edges of the cube. (From this the form derives its trivial name of four-faced cube.) Hence these faces bevel the edges of the cube. The first stage of such bevelling (or the last stage of the truncation of the tetrakis hexahedron by the faces of the cube whichever way it may be regarded) is seen in fig. 55. As the cubic face is here dominant, the sign is ooOoo, oo03. Fig. 56 shows a somewhat similar stage C*^&quot; ! S rX V&quot;! ! / X ^-^ 1 ! | ^^- ; ; / S**&quot;^ Fig. 55. Fig. 56. in the modification produced through the combination of the icosi tetrahedron with the cube. The trilateral pyramid which this form places iipon the faces of the cube rests upon its solid angles, instead of, as in the last case, upon its edges ; hence it is these solid angles which, in the process of decrement, it replaces by faces which form a low three-sided pyramid. The triakisoctahedron, Fig. 57. Fig. 58. again, modifies the solid angles of the cube, as shown in fig. 57, by a low three-sided pyramid, positioned at right angles to that considered in the last combination. As the hexakisoctahedron is merely the two-faced form of that last considered, the pyramid which modifies the solid angles is, in its combination with the eube, six-sided, as in fig. 58. As the faces of the rhombic dodeca hedron truncate the edges of the octa hedron, fig. 34 represents the first stage of such truncation or combination; while fig. 35 may be taken as representing the last, the faces of the octahedron being there nearly totally removed. Fig. 59 shows the first stage of the Fig. 59. passage of the octahedron into the icositetrahedron, in the trunca tion of the solid angles of the former form by a four-sided pyra mid formed by the (6 x 4) faces of the latter. The faces of the octahedron truncate the three-faced solid angles of the rhombic dodecahedron. Fig. 35 shows the first stage of this truncation, while fig. 34 shows an advanced amount. The faces of the icosi- Fig. 60. Fig. 61. tetrahedron truncate the edges of the rhombic dodecahedron, as in fig. 60 ; while those of the latter truncate the unequal-angled tetra gonal (or rhombic) angles of the former (fig. 61). The faces of the hexakisoctahedron bevel the edges of the rhombic dodecahedron. While such transitions may appear indefinite, yet certain minerals have either in themselves a habit, or have at certain localities a habit, of crystallizing so markedly in a certain stage of these transitions as to be absolutely capable of recognition thereby. Combinations of hemihedral or, as they have been called, semi- Combh tesseral forms are of three classes: those with holohedral forms, tions ol those in which the faces fall obliquely on one another, and those hemi- with parallel faces. Fig. 62 shows the combination of a right- hedral ,, -~ forms. Fig. 62. Fig. 63. handed tetrahedron with the cube, which truncates its edges, the tetrahedron here being dominant. Fig. 63, again, shows a com bination of the cubo-dodecahedron with a right-handed tetrahedron, the first or holohedral form being in this case markedly dominant. Fig. 64 is an illustration of the second class, combinations of Fig. 64. Fig. 65. oblique-faced semitesseral forms with each other. In it a right- handed tetrahedron has its solid angles truncated by the faces of one which is left-handed ; and so its sign is - t. Fig. 65 Li shows a combination of a right-handed tetrahedron with a left- Fig. 66. Fig. 67. handed three-faced tetrahedron. Fig. 66 shows a combination of a right-handed hemihedron of the icositetrahedron with a right- handed tetrahedron. Parallel-faced hemihedrons generally form combinations -with holohedral forms ; and the amount of relative dominance is of all degrees. Fig. 67 shows a combination, in equal amount, of the cube