Polyhedron

“ When the crystallization of matter takes place under conditions free from outside influences the peculiarities of internal structure are expressed in the external form of the mass, and there results a solid body bounded by plane surfaces intersecting in straight edges, the directions of which bear an intimate relation to the internal structure. Such a polyhedron (πολύς, many, ἕδρα, base or face) is known as a crystal. ”

“ This being so, the glands of the investing fleshy tissue secrete lime from the sea-water, which holds it in solution, and constantly deposits it, after a determinate and orderly pattern, on every part of the surface of each shelly piece; the inner face, the outer face, and each of the sides and angles of the polyhedron grow together, and all so evenly, that while the dimensions [Pg 179] increase, both of thickness and superficies, the form characteristic of that individual piece is maintained with immutable mathematical precision. ”

“ A four-dimensional pyramid may be defined as a figure bounded by a polyhedron of any number of sides, and the same number of pyramids whose bases are the sides of the polyhedron, and whose apices meet in a point not in the space of the base. ”

“ 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. ”

“ 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. ”

“ But if at last it rotate far enough for its centre of gravity to pass beyond surface A altogether, the body will fall over, on surface B, say, and abide there permanently. The pulls of gravity towards A have vanished, and may now be disregarded. The polyhedron has become immune against farther attraction from their direction. ”

“ 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. ”

“ These are cases in which (as in some of those already described) the skeleton consists (1) of radiating spicular rods, definite in number and position, and (2) of interconnecting rods or plates, tangential to the more or less spherical body of the organism, whose form becomes, accordingly, that of a geometric, polyhedral solid. It may be that there is no math­e­mat­i­cal difference, save one of degree, between such a hexagonal polyhedron as we have seen in Aulacantha, and those which we are about to describe ”

“ Six of these squares joined together formed eight solid angles, each produced by three plane right angles: and the shape of the body thus formed was cubical, having six square planes for its surfaces.” In his cosmology Plato assigned this solid to “earth,” for “‘earth’ is the least mobile of the four (elements—‘fire,’ ‘water,’ ‘air’ and ‘earth’) and most plastic of bodies: and that substance must possess this nature in the highest degree which has its bases most stable.” The mensuration of the cube, and its relations to other geometrical solids are treated in the article Polyhedron ”

“ Just as we may produce the sides of a regular polygon and form a regular cross polygon or stellar polygon, so we may have stellar polyhedrons. Kepler, the great astronomer, constructed some of these solids in 1619, and Poinsot, a French mathematician, carried the constructions so far in 1801 that several of these stellar polyhedrons are known as Poinsot solids. ”

“ This theorem has a very practical application in the study of crystals, since it offers a convenient check on the count of faces, edges, and vertices. Some use of crystals, or even of polyhedrons cut from a piece of crayon, is desirable when studying Euler's proposition. The following illustrations of common forms of crystals may be used in this connection:
The first represents two truncated pyramids placed base to base. ”

“ 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 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. ”

“ 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. ”

“ They may form the covering of any rectilinear figure at all approaching to regularity of form; as the triangle, the square, the canted square, polygons, either regular or elongated, oblongs or parallelograms of any kind; but the usual form is the octagon or other polygon, and for our general purpose it may be best to limit them to figures capable of being inscribed in a circle or an ellipse.
I have introduced this variety of dome as occurring in those which are raised high above their pendentives or corbels. They occur, however, in numerous positions. ”

“ It will be noted that these are the two dimensional correlatives of the sphere, the tetrahedron and the cube, mentioned as being among the determining forms in molecular structure. The question naturally arises, why the circle, the equilateral triangle and the square? Because, aside from the fact that they are of all plane figures the most elementary, they are intimately related to the body of man, as has been shown (Illustration 45) , and the body of man is as it were the architectural archetype. ”

“ 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. ”

“ 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) . ”

“ 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 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. ”

“ In the illustrations of the Gothic windows given in Chapter XV only the square and circle were generally involved. Teachers who feel it necessary or advisable to go outside the regular work of geometry for the purpose of increasing the pupil's interest or of training his hand in the drawing of figures will find plenty of designs given in any pictures of Gothic cathedrals. For example, this picture of the noble window in the choir of Lincoln Cathedral shows the use of the square, hexagon, and pentagon. ”

“ With twenty-seven blocks built into a cube, they found that structure to have a surface of fifty-four square inches, nine times that of each component block. As the blocks underwent the building process, a portion of their surfaces came into contact, and thus hidden could not count in the outer surfaces of the large cubes: Observation and comparison brought the boys to the rule which told exactly what proportion of surface remained exposed. They wrote, "Like solids vary in surface as the square, and in contents as the cube of their like dimensions." ”

“ When we represent a four-dimensional body our solid cubes are to us in the same position that the slabs are to the plane being. You should also notice that each of these slabs represents only the very thinnest slice of a cube. The set of nine slabs first set up represents the side surface of the block. It is, as it were, a kind of tray—a beginning from which the solid cube goes off. The slabs as we use them have thickness, but this thickness is a necessity of construction. They are to be thought of as merely of the thickness of a line. ”

“ It is a curious but significant fact that such proportions as those of an exact cube, or of two cubes placed side by side—dimensions increasing by one-half (e.g., 20 ft. high, 30 wide and 45 long) —or the ratios of the base, perpendicular and hypotenuse of a right-angled triangle (e.g. 3, 4, 5, or their multiples) —please the eye more than dimensions taken at random. No defect is more glaring or more unpleasant than want of proportion. The Gothic architects appear to have been guided in their designs by proportions based on the equilateral triangle. ”

“ 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) . ”

“ The inscription of a pentagon in a circle is effected by inscribing an isosceles triangle similar to that constructed in prop. IO, bisecting the angles at the base and producing the bisectors to meet the circle. Euclid then proves that these intersections and the three vertices of the triangle are the vertices of the required pentagon. The circumscription of a pentagon is effected by constructing an inscribed pentagon, and drawing tangents to the circle at the vertices. This supplies a general method for circumscribing a polygon if the inscribed be iven, and conversely. ”