Octahedron

“ Examples of simple forms amongst crystallized substances are octahedra of alum and spinel and cubes of salt and fluorspar. More usually, however, two or more forms are present on a crystal, and we then have a combination of forms, or simply a "combination." Figs. 6, 7 and 8 represent combinations of the octahedron and the cube; in the first the faces of the cube predominate, and in the third those of the octahedron; fig. 7 with the two forms equally developed is called a cubo-octahedron. Each of these combined forms has all the elements of symmetry proper to the simple forms. ”

“ The incidences subjoined, are, for the most part, the result of many perfect agreements of each, on different crystals. In no instance has an average result been noticed. All are not to be relied on with equal confidence. The plane forming the 9th modification of the primitive octahedron is always so striated, and those of the 3d and 10th, are always so dull, that the incidences of those planes with any other in the subsequent series can only be considered as approximations. ”

“ Such regular interpenetrations are known in crystallography as “twins.” There are also twins of diamond in which two octahedra (fig. 6) are united by contact along a surface parallel to an octahedron face without interpenetration. On account of their resemblance to the twins of the mineral spinel (which crystallizes in octahedra) these are known as “spinel twins.” They are generally flattened along the plane of union. The crystals often display triangular markings, either elevations or pits, upon the octahedron faces ”

“ 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.) ”

“ 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. ”

“ The icosahedron, for instance, approaches the spherical form more nearly than the octahedron does; it is also important to observe that the most precious crystals, and especially the diamond, (which being pure carbon, is therefore, from its composition, most closely allied to the organized bodies,) are those wherein we observe the most compact crystallization, at least that which approaches most nearly to the sphere and is therefore in nowise columnar ”

“ These have forms such as the cube, octahedron, etc., which belong to the isometric system, and it is in this system that the Diamond crystallizes. The crystals do not possess, however, the highest isometric symmetry, but belong to the class designated by Groth as hexakistetrahedral, being tetrahedral with inclined face hemihedrism. It is very common for the faces to be curved instead of flat and to show etching figures of various kinds. ”

“ 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 ”

“ 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. ”

“ We may refer the planes of all crystals to three equal rectangular axes and only three permutations can be made from the only non-variable parameters 1 and x, viz., for the octahedron, for the dodecahedron, for the cube—and so we reach the same result algebraically. ”

“ 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. ”

“ This salt is also particularly suitable for screen demonstrations. The beautiful skeletal ramifications follow the axial directions of the cubic axes, ammonium chloride crystallising in the pentagonal-icositetrahedral class of the cubic system. Good crystals may, however, be very slowly grown from metastable solutions, and they usually exhibit as the principal forms the icositetrahedron (predominating) , cube, octahedron, rhombic dodecahedron, and the class-distinguishing pentagonal icositetrahedron. ”

“ 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. ”

“ All garnets crystallize in the cubic system, usually in rhombic dodecahedra or in icositetrahedra, or in a combination of the two forms (see fig.) . Octahedra and cubes are rare, but the six-faced octahedron occurs in some of the combinations. ”

“ One of the faces, even in cases such as alum or a tetragonal crystal, where three or two might have equal values as regards the symmetry, generally predominates, and affords a very much more brilliant image of the goniometer signal than the others in the bundle, so that an unwary observer might easily come to the conclusion that this was the really valid image corresponding to the octahedron face or to the simple primary prism face, or whatever particular face was expected in the neighbourhood indicated by the bundle of images. ”

“ It is bounded by 4 triangular and 4 hexagonal faces; there are 18 edges, and 12 vertices, at each of which two hexagons and one triangle are co vertical.
2. The cub octahedron is a tesserescae-decahedron (Gr. re¢nrapes-uat-Baca, fourteen) formed by truncating the vertices of a cube so as to leave the original faces squares. It is enclosed by 6 square and 8 triangular faces, the latter belonging to a coaxial octahedron. It is a common crystal form. ”

“ Fluor-spar crystallizes in the cubic system, commonly in cubes, either alone or combined with the octahedron, rhombic dodecahedron, four-faced cube, &c. The four-faced cube has been called the fluoroid. In fig. 1, a is the cube (100) , d the rhombic dodecahedron (110) , and f the four-faced cube (310) . Fig. 2 shows a characteristic twin of interpenetrant cubes. The crystals are sometimes polysynthetic, a large octahedron, e.g., being built up of small cubes. The faces are often etched or corroded. Cleavage is nearly always perfect, parallel to the octahedron. ”

“ Again, in some instances, we have more than three different planes of cleavage, as in Fluor Spar, where we have four. The solid, bounded by four planes, is a tetrahedron; or if we take four pairs of parallel faces, an octahedron. But if we attempt to take either of these forms for our integrant molecule, we are met by this difficulty: that a collection of such forms will not fill space. Perhaps this difficulty will be more readily conceived by the general reader if it be contemplated with reference to plane figures. ”

“ A form, if of general character, that is, if composed of faces each of which is inclined to all three axes, will comprise more faces the higher the symmetry. Thus, in the cubic system, the form shown in Fig. 21, the hexakis octahedron, comprises as many as forty-eight faces, all covered by the form symbol {321}; while in the rhombic system the highest number of faces in a form is eight, in the monoclinic only four, and in the triclinic system two. It will also have become clear that the law of rational indices limits the number of forms possible of any one type. ”

“ Both hypotheses are doubtful or false: yet both these methods are good as modes of description: nor is Haüy’s terminology vitiated, if we suppose (as in fact we must suppose in many instances,) that crystalline bodies are not really made up of such small solids. The mode of describing an octahedron of fluor spar, as derived from the cube, by decrements of one row on all the edges, would still be proper and useful as a description, whatever judgment we should form of the material structure of the body. ”

“ This solid has 4 faces, 4 vertices and 6 edges. In a similar manner, four co vertical equilateral triangles stand on a square base. Two such sets placed base to base form the octahedron, which consequently has 8 faces, 6 vertices and 12 edges. Five equilateral triangles co vertically placed would stand on a pentagonal base, and it was found that, by forming several sets of such pyramids, a solid could be obtained which had 20 triangular faces, which met in pairs to form 30 edges, and in fives to form 12 vertices. ”

“ 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. ”

“ 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. ”

“ It will readily be seen that a number of equal squares may be put together so as to fill the space which they occupy; but if we take a number of equal regular octagons, we may easily convince ourselves that no possible arrangement can make them cover a flat space without leaving blank spots between. In like manner octahedrons or tetrahedrons cannot be arranged in solid space so as to fill it. They necessarily leave vacancies. Hence the structure of Fluor Spar, and similar crystals, was a serious obstacle in the way of the theory of integrant molecules. ”

“ A nitrate formed from three molecules of nitric acid; also, less properly, applied to certain basic nitrates; as, trisnitrate of bismuth.
Tris*oc`ta*he"dron (?) , n. [Gr. &?; thrice + FE. octahedron.] (Crystallog.) A solid of the isometric system bounded by twenty-four equal faces, three corresponding to each face of an octahedron.
Tetragonal trisoctahedron, a trisoctahedron each face of which is a quadrilateral; called also trapezohedron and icositetrahedron. — Trigonal trisoctahedron, a trisoctahedron each face of which is an isosceles triangle. ”

“ We will exhibit the gradations. Suppose we cut fig. 437; we will obtain (fig. 438) the cube. The next is merely the cube with angles and edges cut off; and if we proceed regularly we shall arrive at fig. 442, the rhombic dodecahedron, or twelve-sided figure, whose equal planes are rhombs.
We can, by taking away alternate angles or edges situated opposite, arrive at other secondary crystals. From the original octohedron we can [Pg 429] thus obtain figs. 443 and 444. These are known as tetrahedron. The pentagonal dodecahedron is another secondary form (fig. 445) . ”