Any other inclination than 35 1/4 degrees produces an oblique hexahedron, or rhomboid, with six equal rhombic faces. For instance, the very dissimilar figures 764, 765, and 766, were cut from hexagonal prisms of the same size, and respectively as large as the prisms would permit. In fig. 764, which is an acute or elongated rhomboid, the angle at which the prism met the saw was 10 degrees; and in fig. 766, an obtuse or compressed rhomboid, the angle was 80 degrees. Viewed along the dotted line or through their common axis, the three figures all appear as equal hexagons, and show the three pyramidal planes of each solid as equal rhombuses, as in the figure 767; but the axis of fig. 764 is about four times as long as that of the cube, 765, the axis of 766 is only about one eighth as long as the cube, and its edge is acute like a knife.
• Mathematically, 19°. 28'. 17".
† Mathematically, 35°. 15'. 52".
The octahedron, with 8 planes, each an equilateral triangle, may be viewed as a double square pyramid, cut off at an angle of 35 1/4 degrees,* and is produced in that manner with very little difficulty from a square prism. When the prism meets the saw at a smaller angle than 35 1/4 degrees, the octahedron is said to be acute or elongated; and when the angle is greater, the octahedron is obtuse or compressed, as recently explained in regard to the rhomboids figs. 764 and 766.
It has been considered unnecessary to represent the regular tetrahedron, hexahedron, and octahedron, which are simple, and familiarly known; and the subsequent figures 76S to 771, of the dodecahedron, the icosahedron, and trapezohedron, are to be viewed as explanatory diagrams, and not as faithful representations of these respective polyhedra.
The dodecahedron, fig. 768, with 12 planes each an equilateral pentagon, may be viewed as frusta of two pentagonal pyramids, the sides of which are interposed or macled, and the pyramids being truncated form the two remaining pentagons. The double 5-sided pyramids, are first cut at the angle of 26 1/2 degrees, † and discontinuously, by means of the positions shown in figs. 756 and 757, the sides of the pyramids will then be found to meet at 36°, the angle made by the first and third sides of a pentagon. The outer plane is obtained by cutting off the point of the pyramid at right angles to the prism, and extending it by trial, until the terminal pentngon itself, and the 5 pentagons near it, become equilateral. The second pyramid, not having been cut so far as the center, the solid is now removed from its matrix or prism, by one cut at right angles to the prism, and so far removed from the angles of the zig-zag line on which the pyramids join, as the corresponding pentagon, at the outer end of the solid.
• Mathematically 35o. 15'. 52"., or half the supplement to 109o. 28'. 16"., the angle at which the pyramidal planes of the octahedron meet. See Brooke's Crystallography, page 116.
† Mathematically 26o. 33'. 54".
The above, or the pentagonal dodecahedron, is also called the Platonic dodecahedron; but there is another kind named the rhombic dodecahedron, which is more referred to by mineralogists. The rhombic dodecahedron, fig. 769, has 12 faces, each an equilateral rhombus, and may be viewed as a hexagonal prism with a shallow triangular pyramid at each end.
The rhombic dodecahedron may be therefore sawn from the hexagonal prism, provided, that first three pyramidical planes are cut at the angle of 54 3/4 degrees,* and that the solid is then released from the prism, by three similar but inverted cuts on the intermediate angles of the hexagon, so much of the central prism being left, as will make six rhombuses equal to those terminating the original prism.
The rhombic dodecahedron may be also viewed as a square prism terminating in two square pyramids cut off at an angle of 45°; but as these planes run on to the angles of the prism, it is needful the bed should be inclined 45° horizontally, for the pyramids, and also 45° vertically, for their displacement.
The icosahedron, fig. 770, with 20 planes each an equilateral triangle, may be viewed as two obtuse pentagonal pyramids, united by frusta of two other pentagonal pyramids a to b, the sides of which are very acute and interposed. The icosahedron may be sawn from the pentagonal prism nearly in the manner of the last; the first guide is the angle of 10 3/4 degrees,* and suitable to cutting the two central frusta. This guide is first employed as in fig.756, and then shifted as in fig. 757, the 10 cuts produce the 10 angles, each of 60°, constituting the central zone of the figure. The extreme end of the is then sawn at five cuts on a bed of 52 1/2 degrees,† so that the five planes of the outer pyramids constitute equilateral triangles exactly terminating on the line a, or on the sides of one series of five triangles, and the points of the other series, constituting the central zone of the solid. The icosahedron is removed from the prism by placing the guide block as in 757, and cutting the second pentagonal pyramid, which similarly to the first, falls on the line b, and just meets both the sides and angles of the 10 central triangular faces; when the work is accurately performed, every point is the center of a group of five equilateral triangles.
• Mathematically, 54°. 44'. 8".
The solid fig. 771, with 24 equal trapezoidal planes, may be viewed as two frusta of octagonal pyramids, joined base to base with continuous edges, and surmounted by two obtuse four-sided pyramids. This solid belongs rather to mineralogy than geometry, and occurs with various angles; its usual name is an icositessera-hedron; but it has been sometimes termed a trapezohedron, from the shape of its faces: three of its varieties will be noticed. In the first, the three quadrantal sections, namely, through A o E, through C o G, and through A B C D E F G H,are all regular octagons, and the angles of the solid are throughout alike; this variety may be therefore called the regular trapczohedron. In others the three sections arc irregular octagons, and the alternate angles dissimilar; these may be called irregular trapezohedra, and two of these varieties that occur in mineralogy are referred to in the annexed table.