Rhombohedron

“ They are inclined to one another at equal oblique angles, and they are all equally inclined to the principal axis; further, they are all of equal length and are interchangeable. With such a set of axes there can be no statement of an axial ratio, but the angle between the axes (or some other angle which may be calculated from this) may be given as a constant of the substance. Thus in calcite the rhombohedral angle (the angle between two faces of the fundamental rhombohedron) is 74° 55′, or the angle between the normal to a face of this rhombohedron and the principal axis is 44° 361⁄2′. ”

“ If a knife blade be pressed into the edge of a cleavage rhombohedron of calcite (at b, fig. 91) the portion abcde of the crystal will take up the position a′b′cde. The obtuse solid angle at a becomes acute (a′) , whilst the acute angle at b becomes obtuse (b′) ; and the new surface a′ce is as bright and smooth as before. This result has been effected by the particles in successive layers gliding or rotating over each other, without separation, along planes parallel to cde. This plane, which truncates the edge of the rhombohedron and has the indices (110) , is called a “glide-plane.” ”

“ Crystals are usually rhombohedral in habit, and the primitive rhombohedron r{100} is a common form, the faces being often curved as represented in the figure. Acute rhombohedra in combination with the basal pinacoid are also frequent, giving crystals of octahedral aspect. ”

“ This lower degree of symmetry, which is the same as that of dioptase and phenacite, is occasionally shown by the presence of an obliquely placed rhombohedron, and also by the want of symmetry in the etching and elasticity figures on the faces of the primitive rhombohedron. ”

“ In this system it is quite common to have forms which result from the development of each alternate face of either the prism or the double pyramid. In the case of the prism, if every alternate face is developed (and the others omitted) a three-sided prism results, as in tourmaline. In the case of the double pyramid if the three alternate faces above are united with the three alternate faces below, a six-sided figure is formed, which is known as the rhombohedron, as all the faces are rhombohedral in out-line and all equal. ”

“ Now, I present the rhombohedron so that none of its faces are [Pg 129] parallel or perpendicular to the plane of vision,—thus (vide C.) Well, I shall still see three faces, but foreshortened—thrown out of shape by perspective; and the three others will be indicated by the dotted lines. In the evenings, therefore, make as many little models as you can, representing the stones you have seen at the works, and copy these little models in every position. ”

“ Water crystallises like quartz, in the trigonal system, its normal forms being the hexagonal prism and the rhombohedron. A slab of lake ice is generally a huge crystal plate perpendicular to the trigonal axis, or in the case of disturbed growth an interlacing mass of such crystals, all perpendicular to the optic axis, the axis of the hexagonal prism and of trigonal symmetry. ”

“ It is already evident that, with a definite change of composition, certain angles remain constant, notwithstanding that others are subject to alteration. Such an instance of the relation of forms was observed by Laurent, and named by him hemimorphism (an anomalous term) when the analogy is limited to certain angles, and paramorphism when the forms in general approach each other, but belong to different systems. So, for example, the angle of the planes of a rhombohedron may be greater or less than 90°, and therefore such acute and obtuse rhombohedra may closely approximate to the cube. ”

“ The inspection of fig. 7, bis, shows then that the branches of the nodal hyperbola of No. 3, parallel to , consequently to the plane , are straighter than those of the plates which precede or follow it; and admitting that this maximum of recession occurs equally in quartz for the corresponding diagonal plane of its rhombohedron, as this plane forms with the cleavable face of the pyramid an angle of 96° 0′ 13′′, the plate in question will be inclined 57° 40′ 13′′ to the axis of the crystal, the face of the pyramid forming with this axis an angle of 38° 20′ ”

“ A “form” is the technical term for a set of faces having an equal value with respect to the symmetry. Thus the prismatic form in Fig. 6 is the hexagonal prism, a form which is common to the hexagonal and trigonal systems of symmetry, and the form “indices” (numbers [1] inversely proportional to the intercepts cut off from the crystal axes by the face typifying the form) of which are {2̄1̄1}; the large development of this form confers the elongated prismatic habit on the crystal. The terminations are faces of the flat rhombohedron {110}. The pyramidal form of the dog-tooth spar shown in Fig. ”

“ The cube is the most stable of them because resting on a quadrangular plane surface, and composed of isosceles triangles. To the earth then, which is the most stable of bodies and the most easily modelled of them, may be assigned the form of a cube; and the remaining forms to the other elements,—to fire the pyramid, to air the octahedron, and to water the icosahedron,—according to their degrees of lightness or heaviness or power, or want of power, of penetration. The single particles of any of the elements are not seen by reason of their smallness; they only become visible when collected. ”

“ In fig. 10 is shown the stereographic projection, or stereogram, of a cubic crystal; a1, a2, &c. are the poles of the faces of the cube. o1, o2, &c. those of the octahedron, and d1, d2, &c. those of the rhombic dodecahedron. The straight lines and circular arcs are the projections on the equatorial plane of the great circles in which the nine planes of symmetry intersect the sphere. A drawing of a crystal showing a combination of the cube, octahedron and rhombic dodecahedron is shown in fig. 11, in which the faces are lettered the same as the corresponding poles in the projection. ”

“ The “small stellated dodecahedron,” the “great dodecahedron” and the “great stellated dodecahedron” are Kepler-Poinsot solids; and the “truncated” and “snub dodecahedra” are Archimedean solids (see Polyhedron) . In crystallography, the regular or ordinary dodecahedron is an impossible form since the faces cut the axes in irrational ratios; the “pentagonal dodecahedron” of crystallographers has irregular pentagons for faces, while the geometrical solid, on the other hand, has regular ones. The “rhombic dodecahedron,” one of the geometrical semiregular solids, is an important crystal form. ”

“ The rhombic dodecahedron has six tetrahedral angles, and eight trihedral angles; and it is obvious, on consideration, that at each of the former six dodecahedra meet in a point, and that, where the four tetrahedral facets of each coalesce with their neighbours, we have twelve plane films, or interfaces, meeting in a point. In a precisely similar fashion, we may imagine twelve plane films, drawn inwards from the twelve edges of a cube, to meet at a point in the centre of the cube. ”

“ Aristæus, probably a little older than Euclid, wrote a book upon these solids.
As an interesting amplification of this proposition, the centers of the faces (squares) of a cube may be connected, an inscribed octahedron being thereby formed. Furthermore, if the vertices of the cube are A, B, C, D, A', B', C', D', then by drawing AC, CD', D'A, D'B', B'A, and B'C, a regular tetrahedron will be formed. Since the construction of the cube is a simple matter, this shows how three of the five regular solids may be constructed. The actual construction of the solids is not suited to elementary geometry. ”

“ If the plane faces of a polyhedron be equal and similar rectilineal figures, it is called a regular polyhedron.
iii. A pyramid is a polyhedron of which all the faces but one meet in a point. This point is called the vertex; and the opposite face, the base.
iv. A prism is a polyhedron having a pair of parallel faces which are equal and similar rectilineal figures, and are called its ends. The others, called its side faces, are parallelograms.
v. A prism whose ends are perpendicular to its sides is called a right prism; any other is called an oblique prism. ”

“ This modification consists in a decrease on the six solid angles of the primitive crystal, so that each is replaced by an equilateral quadrangular plane, perpendicular to the axis that passes through the angle, and forms the passage of the octahedron into the cube. The angle caused by the meeting of the faces P and 1, Fig. 16, is, according to Haüy, 125°, 15′, 52′′ Fig. 24 shews the direction of the laminæ of the cube. Fig. 25 may rather be considered an accidental circumstance, than as forming an important part of this series. ”

“ The Pyramid is the finest and smallest, the largest and the one of most parts is the Dodecahedron; of the remaining two the Eicosahedron is more than double the Octahedron in number of triangles. [171] It follows that it is impossible for all to take their origin at once from one and the same matter. For those which are fine and small, and more simple in their structures, must be the first to obey him who stirs and fashions the mass ”

“ It was done through a simple generalization which had become indispensable in the study of geometry, when she passed from plane to solid figures. By means of wooden models she learned, in addition to the cube—the sphere, ovoid, oblate, cylinder, prism, tetrahedron, octahedron, and dodecahedron. She then was led to make parallel lines of plane and solid figures with a corresponding number of sides or angles, then to abstract the Greek numerals tri, tetra, penta, hexa, etc., found to belong to both columns, and set this in the center, with the syllable gon on one side, and hedron on the other. ”

“ Some of our readers may be glad to have a drawing of the Platonic solids, since they play so large a part in the building up of elements. The regular solids are five, and five only; in each:
(1) The lines are equal.
(2) The angles are equal.
(3) The surfaces are equal.
It will be seen that the tetrahedron is the fundamental form, the three-sided pyramid on a triangular base, i.e., a solid figure formed from four triangles. Two of these generate the cube and the octahedron; five of these generate the dodecahedron and the icosahedron. ”

“ He gave to each kind of particle one of the five regular solids, about which the geometrical speculations of himself and his pupils had been employed. The particles of fire were pyramids, because they are sharp, and tend upwards; those of earth are cubes, because they are stable, and fill space; the particles of air are octahedral, as most nearly resembling those of fire; those of water are the icositetrahedron, as most nearly spherical. The dodecahedron is the figure of the element of the heavens, and shows its influence in other things, as in the twelve signs of the zodiac. ”

“ The second solid is composed of the same triangles, which unite as eight equilateral triangles, and make one solid angle out of four plane angles—six of these angles form a regular octahedron. The third solid is a regular icosahedron, having twenty triangular equilateral bases, and therefore 120 rectangular scalene triangles. The fourth regular solid, or cube, is formed by the combination of four isosceles triangles into one square and of six squares into a cube. ”

“ All forms are referred to four axes, three equal axes inclined to each other at angles 60 degrees in a common horizontal plane, and a fourth vertical axis at right angles, and longer or shorter. The rhombohedral division comprises crystals having but three planes of symmetry, intersecting at angles of 120 degrees in the vertical axis. ”

“ The fracture of the crystals, particularly of the more translucent ones, is very smooth and inclines to conchoidal; but is frequently uneven, and inclining to shattery, in those of a darker hue. I have met with some octohedrons that admitted a fracture in the direction of their faces, but have not found any of a cubical form in which a division parallel to the circumscribing planes was practicable.
The form of the primitive crystal, according to Haüy, is the regular octahedron (fig. 1.) ”

“ There, before my ravished eye, a Cube, moving in some altogether new direction, but strictly according to Analogy, so as to make every particle of his interior pass through a new kind of Space with a wake of its own—shall create a still more perfect perfection than himself, with sixteen terminal Extra-solid angles, and Eight solid Cubes for his Perimeter. And once there, shall we stay our upward course? In that blessed region of Four Dimensions, shall we linger on the threshold of the Fifth, and not enter therein? ”

“ Shapes can easily be cut out of cardboard which, when folded together, form not only the tetrahedron and the octahedron, but also samples of all the sections of the tesseract taken as it passes cornerwise through our space. ”

“ He who has watched the organization of crystalline forms under electric currents has seen the operation of the same law which has formed the solid part of the earth we live on. In its early and plastic condition it was a sphere like the dew-drop, but, with the constant currents of organizing magnetism, it has assumed a crystalline form, and to-day its solid exterior, were its oceans emptied and removed, would present the polyhedron. If we carefully examine the almost universal features of all land known to us, we find a prevailing form wherever we turn. ”