Tetrahedron

“ 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) . ”

“ The only geometrical form in three dimensions in space fulfilling the condition of a central object surrounded symmetrically by four things of the same kind, is the regular tetrahedron. Thus arose the theory of the "tetrahedral carbon atom."
Some compounds of carbon rotate the beam of polarized light to the right, others to the left—are "optically active," as they are termed. ”

“ A common form, shown in fig. 2, is a combination of the rhombic dodecahedron with a three-faced tetrahedron y (311) ; the six faces meeting in each triad axis are often rounded together into low conical forms. ”

“ 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. ”

“ 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. ”

“ Today, no less than in the past, the tetrahedral form of the earth and the relation of the tetrahedron to the poles and to the equator preserve the conditions that favor rapid evolution. ”

“ They form a closed figure, a tetrahedron, enclosing a light brown portion which is the portion of the cutting space which lies inside the tesseract.
We cannot expect to see this light brown portion, any more than a plane being could expect to see the inside of a cube if an angle of it were pushed through his plane. All he can do is to come upon the boundaries of it in a different way to that in which he would if it passed straight through his plane.
Thus in this solid section; the whole interior lies perfectly open in the fourth dimension. ”

“ These give the solid angles of the tetrahedron, the octahedron and the icosahedron. The angle in a square (the regular quadrilateral) equals one right angle. Hence three will form a solid angle, that of the cube, and four will not. The angle in the regular pentagon equals 6⁄5 of a right angle. Hence three of them equal 18⁄5 (i.e. less than 4) right angles, and form the solid angle of the dodecahedron. Three regular polygons of six or more sides cannot form a solid angle. Therefore no other regular solids are possible. ”

“ If we make a wire cage, in the form of a regular tetrahedron, and dip it into soap-solution, then when we withdraw it we see that to each one of the six edges of the tetrahedron, i.e. to each one of the six wires which constitute the little cage, a film has attached itself ”

“ 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. ”

“ In short, the system of films we have hereby automatically produced is precisely the system of partition-walls which exist in our tetrahedral aggregation of four spherical bubbles:—precisely the same, that is to say, in the neighbourhood of the meeting-point, and only differing in that we have made the wires of our tetrahedron straight, instead of imitating the circular arcs which actually form the intersections of our bubbles. ”

“ This theorem has been generalized for any tetrahedron; a sphere can be drawn through the four feet of the perpendiculars, and consequently through the mid-points of the lines from the vertices to the centre of the hyperboloid having these perpendiculars as generators, and through the orthogonal projections of these points on the opposite faces. ”

“ Cubes of pyrites (fig. 89) are striated parallel to one edge, the striae on adjacent faces being at right angles, and due to oscillatory combination of the cube and the pentagonal dodecahedron (compare fig. 36) ; whilst cubes of blende (fig. 90) are striated parallel to one diagonal of each face, i.e. parallel to the tetrahedron faces (compare fig. 31) . These striated cubes thus possess different degrees of symmetry and belong to different symmetry-classes. Oscillatory combination of faces gives rise also to curved surfaces. ”

“ 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. ”

“ In this assemblage we have what I call ‘close order’ or ‘close packing.’ For closest of close packings the volume of the tetrahedron (§ 24) of corresponding points of S1, S2, S3, and S4 must be a minimum, and the least of minimums if, as generally will be the case, there are two more different configurations for each of which the volume is a minimum. There will in general also be configurations of minimax volume and of maximum volume, subject to the condition that each body is touched by twelve similarly oriented neighbours. ”

“ The singular surface is the “tetrahedroid” discussed by Cayley. As a particular case, from a metrical point of view, we have L.F. Painvin’s complex generated by the lines of intersection of perpendicular tangent planes of a quadric, the singular surface now being Fresnel’s wave surface. The tetrahedral or Reye complex is the simplest and best known of proper quadratic complexes. It is generated by the lines which cut the faces of a tetrahedron in a constant cross ratio, and therefore by those subtending the same cross ratio at the four vertices. ”

“ Pertaining or related to a tetrahedron, or to the system of hemihedral forms to which the tetrahedron belongs.
Tetrahedral angle (Geom.) , a solid angle bounded or inclosed by four plane angles.
Tet`ra*he"dral*ly, adv. In a tetrahedral manner.
Tet`ra*he"drite (?) , n. [So called because the crystals of the species are commonly tetrahedrons.] (Min.) A sulphide of antimony and copper, with small quantities of other metals. It is a very common ore of copper, and some varieties yield a considerable presentage of silver. Called also gray copper ore, fahlore, and panabase. ”

“ In its crystalline form the tetrahedrite occurs in tetrahedrons, which generally have faces formed by beveling the edges and by cutting the corners, as in the two figures of plate 10. Chalcopyrite may also occur in tetrahedrons, but its golden yellow color is entirely different from the gray-black of the tetrahedrite. ”

“ Triakis-tetrahedron (fig. 28) . The hemihedral form {hkk} of the icositetrahedron; it is bounded by twelve isosceles triangles arranged in threes over the tetrahedron faces.
Hexakis-tetrahedron (fig. 29) . The hemihedral form {hkl} of the hexakis-octahedron; it is bounded by twenty-four scalene triangles and is the general form of the class. ”

“ As the nucleus of a nebula condenses and cools into a solid mass, the structure of the matter at the center becomes extremely dense and the atomic weights are high; these weights taper off to the surface, which is then of a somewhat irregular and slag-like nature like the crust of the earth and brings to mind many parallel cases in metallurgy. Many scientists hold that the earth has solidified more or less in the shape of a modified tetrahedron; if this be true, it encourages the view of a fairly homogeneous core which may be composed of elements even heavier than radium, thorium and uranium. ”

“ 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. ”

“ Of regular polyhedra, properly so-called, it may be shown geometrically that only five can exist, namely, the regular tetrahedron, cube, octahedron, dodecahedron, and icosahedron. All these are actually manifested among the Radiolaria, although but seldom. Much more common are the subregular endospherical polyhedra, e.g., spherical lattice-shells with regular hexagonal meshes of equal size; they are never exactly equal nor perfectly regular, but the divergences are so insignificant that they escape superficial observation (Pl. 20, figs. 3, 4; Pl. 26, figs. 1-3) . ”

“ And as the diamond is brought down from the heavens in the meteorites and brought up from the depths of the earth with the deepest rocks, and as it is endowed with the greatest power over light and over all solid bodies, so it presents in its almost spherical hexatetrahedron a mean around which the earth seems many times to have oscillated, as Arldt has shown, [2] now varying slightly toward the tetrahedron ”

“ This cluster of thirteen, S1 and twelve neighbours, is shown for the case of spherical bodies in the stereoscopic photograph of § 4 above. We might of course, if we pleased, have begun with the plane layer of which S1, S2, S4 are members, or with that of which S1, S3, S4 are members, or with the plane layer parallel to the fourth side S2 S3 S4 of the tetrahedron: and thus we have four different ways of grouping the twelve points of33 contact on S1 into one set of six and two sets of three. ”

“ The numbers 1, 4, 10, 20 are polyhedral numbers, and from their association with the tetrahedron are termed “ tetrahedral numbers.”
This illustration may serve for a definition of polyhedral numbers: a polyhedral number represents the number of equal spheres which can be placed within a polyhedron so that the spheres touch one another or the sides of the polyhedron. In the case of the tetrahedron we have seen the numbers to be I, 4, IO, 20; the general formula for the nth tetrahedral number is § n (n+1) (n-|-2) . Cubic numbers are 1, 8, 27, 64, 125, &c; or generally na. ”

“ 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. ”

“ Its hardness is 8. Specific gravity 3·523. Its fundamental form is the hexahedron, but it occurs crystallized in many secondary forms: octahedrons, tetrahedrons and rhombohedrons. Fracture conchoidal; lustre vitreous; colour red, passing into blue and green, yellow, brown and black; and sometimes it is nearly white. Red spinelle consists of, alumina 74·5; silica 15·5; magnesia 8·25; oxide of iron 1·5; lime 0·75. Vauquelin discovered 6·18 per cent. of chromic acid in the red spinelle. The red varieties exposed to heat, become black and opaque ”

“ 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. ”

“ 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. ”

“ 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. ”