Dodecahedron

“ Each vertex is doubly enclosed by the succession of co vertical faces, while the centre of the solid is triply enclosed by the faces. The great stellated dodecahedron is formed by stellating the faces of a great dodecahedron. It has X2 faces, which meet in 30 edges; these intersect in threes to form 20 vertices. ”

“ The point is that the dodecahedron approaches more nearly to the sphere than any other of the regular solids. The twelve pieces of leather used to make a ball would all be regular pentagons; and, if the material were not flexible like leather, we should have a dodecahedron instead of a sphere. ”

“ 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. ”

“ Describe a Sphere about an Icosahedron; let perpendiculars be drawn from the centre of the Sphere on its faces and produced to meet the surface of the Sphere. Now, if the points of intersection be joined, a Dodecahedron is formed within the Sphere. By a similar process an Icosahedron may be constructed from a Dodecahedron. (See Todhunter's "Spherical Trigonometry," p. 141, art. 193) . The figure constructed as above described will represent the universe of matter and the universe of Astral Light as they actually exist. ”

“ 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. ”

“ An assemblage of equal and similar rhombic dodecahedra goes far to meet the case: it completely fills up space; all its surfaces or interfaces are planes, that is to say, surfaces of constant curvature throughout; and these surfaces all meet together at angles of 120°. Nevertheless, the proof that our rhombic dodecahedron (such as we find exemplified in the bee’s cell) is a surface of minimal area, is not a comprehensive proof ”

“ 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 ”

“ Modern science gives the stones more sides, and, while seeming to round them into more adaptable forms, only produces a number of small facets, making a somewhat polished dodecahedron, which rolls about in any position and is of no particular service, except, maybe, to stick upon a gate-post at a girl's school, though it has given a great deal of trouble in the cutting. ”

“ Now it is very significant that Plato starts from fire and earth, [802] and in the construction of the elements proceeds in such a way that the octahedron and the icosahedron can easily be transformed into pyramids, while the cube and the dodecahedron cannot. From this it follows that, while air and water pass readily into fire, earth cannot do so, [803] and the dodecahedron 341is reserved for another purpose, which we shall consider presently. This would exactly suit the Pythagorean system ”

“ In this class there is no centre of symmetry nor cubic planes of symmetry; the three tetrad axes become dyad axes of symmetry, and the four triad axes are polar, i.e. they are associated with different faces at their two ends. The other elements of symmetry (six dodecahedral planes and six dyad axes) are the same as in the last class.
Of the seven simple forms, the cube, rhombic dodecahedron and tetrakis-hexahedron are geometrically the same as before, though on actual crystals the faces will have different surface characters. ”

“ 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. ”

“ This is the icosahedron. That the triangle could give rise to no other solid followed from the fact that six covertically placed triangles formed a plane. The pentagon is the basis of the dodecahedron Three pentagons may be placed at a common vertex to forni a solid angle, and by forming several such sets and placing them in juxtaposition. a solid is obtained having 12 pentagonal faces, 30 edges, and 20 vertices. ”

“ 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. ”

“ The crystals belong to the cubic system, generally assuming the form of the octahedron (fig. 1) , but they may, in accordance with the principles of crystallography, also occur in other forms symmetrically derived from the octahedron,—for example, the cube, the 12-faced figure known as the rhombic dodecahedron (fig. 2) , or the 48-faced figure known as the hexakis-octahedron (fig. 3) , or in combinations of these. The octahedron faces are usually smooth; most of the other faces are rounded (fig. 4) . The cube faces are rough with protruding points. ”

“ 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. ”

“ Thus the Cube is clearly a body congenial to Rest, because of the stability and firmness of its surfaces. 152No one can fail to detect the fire-like, active character of the Pyramid in the fineness of its sides and the sharpness of its angles. The nature of the Dodecahedron, which embraces |E| the other figures, might well be taken for an image of Being in relation to all that is corporeal. ”

“ It represents therefore the regular "pentagonal dodecahedron," the same remarkable form which is found in some Astrosphærida, and in the pollen-grains of some plants, e.g., Bucholzia maritima, Fumaria spicata, Polygonum amphibium, &c. (Gener. Morphol., vol. i. p. 412, Taf. ii. fig. 18) .
The three genera of Circoporida mentioned therefore represent three different forms of regular polyhedrons, in the exact mathematical sense, viz., Circoporus, the regular octahedron, Circogonia, the regular icosahedron, and Circorrhegma, the regular dodecahedron. ”

“ In the intervals of applying algebra to the second book of Euclid, we experimented with circles; we stuck bamboos into the parched earth, measured their shadows at different hours of the day, and drew exciting conclusions from our observations. Sometimes, for fun, we [327] cut and folded sheets of paper so as to make cubes and pyramids. One afternoon Guido arrived carrying carefully between his small and rather grubby hands a flimsy dodecahedron. ”

“ The most common form of rough Diamonds is the octahedron, variously modified on the edges with planes, and low pyramids on each face of the triangles. The next is the dodecahedron, which occurs also variously modified, both on its edges and planes: others are flat and triangular (heliotrope) ,these are termed veiny, and being thin, are generally split and cut into roses. The rarest form is the cube. ”

“ 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) . ”

“ 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. ”

“ Theorem. There cannot be more than five regular convex polyhedrons.
Eudemus of Rhodes, one of the principal pupils of Aristotle, in his history of geometry of which Proclus preserves some fragments, tells us that Pythagoras discovered the construction of the "mundane figures," meaning the five regular polyhedrons. Iamblichus speaks of the discovery of the dodecahedron in these words:
As to Hippasus, who was a Pythagorean, they say that he perished in the sea on account of his impiety, inasmuch as he boasted that he first divulged the knowledge of the sphere with the twelve pentagons. ”

“ The Sphere would willingly have continued his lessons by indoctrinating me in the conformation of all regular Solids, Cylinders, Cones, Pyramids, Pentahedrons, Hexahedrons, Dodecahedrons, and Spheres: but I ventured to interrupt him. Not that I was wearied of knowledge. On the contrary, I thirsted for yet deeper and fuller draughts than he was offering to me. ”

“ 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 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. ”

“ 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.” ”

“ They are terminated by an apparently hexagonal pyramid, but which really consists of a pair of complementary rhombohedra, which are purely trigonal forms; three upper faces of each rhombohedron are developed at one end of the prism which may be regarded as the upper, and the three lower faces of each of the two individual rhombohedra likewise at the lower end of a fully developed doubly terminated crystal. The rhombohedron is the characteristic form of the trigonal system of crystal symmetry, the systematic crystallographic axes being parallel to its edges. ”

“ 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.) ”

“ 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. ”