[1 2 ∞]; ∞02; 2 ; 210; b3, in cobaltine, cubane, fal lerz, gersdorfitte, and pyrite. ∞ 0 3 [130]; ; 310; b3, in hauerite, pyrite, and sal ammoniac. [1 4 ∞]; 2 ∞ 0 4 141. Platonic bodies.-There are five solid bodies described by the ancient geometers as regular solids. From their mathematical properties having been investigated by Plato and his followers, they are called the Platonic bodies. They have all their faces, edges, and angles, whether plane or solid, equal for each body. They are the tetrahedron, bounded by four equal faces, each being an equilateral triangle; the cube, bounded by six equal squares; the octahedron, bounded by eight equal faces, each being an equilateral triangle; the pentagonal dodecahedron, bounded by twelve equal and equilateral pentagons; and the icosahedron, by twenty equal faces, each being an equilateral triangle. The first three, described by Plato himself, have been observed in natural crystals. The last two, described after his death, have not been observed in nature. The regular pentagonal dodecahedron is that particular case of the pentagonal dodecahedron, where the unequal edge, such as (fig. 23, Plate IIF.), is equal to the other four 801, 9181, 8104, and 0484. 4 In this case m=cot λ3: 1+ √3 = 1·618031, = 2 but cot 31° 43′=1.618085. Hence λ=31° 43′ true to minutes. The value of m is generally determined by continued. fractions. Thus m=341.619046 and cot 31° 42′ 1.61914 13 m=18=1.625 m= &=1·6 The regular icosahedron is derived from the particular pentagonal dodecahedron in which the edge &a line joining the points 8, and S. In this case m=cot λ ̧=3±√5–2·61803=cot 20° 54′, 2 where the ratio for m expressed in its lowest terms is m=33. In this particular pentagonal dodecahedron each solid angle at 0, 0, &c., o, is cut off through the lines 88, 85, and 86, &c., forming a solid bounded by twenty equilateral triangles,-eight being parallel to the faces of the octahedron inscribed in the dodecahedron, and the remaining twelve faces of the pentagonal dodecahedron. Ozonam, in his Mathematical Recreations, remarks that "The ancient geometricians made a great many geometrical speculations respecting these bodies; and they form almost the whole subject of the last books of Euclid's Elements. They were suggested to the ancients by their believing that these bodies were endowed with mysterious properties, on which the explanation of the most secret phenomena of nature depended." 142. The irregular twenty-four-faced trapezohedron is a halfsymmetrical form with parallel faces derived from the six-faced octahedron. It is called the irregular twenty-four-faced trapezohedron because its trapezoidal faces have only two equal edges, and to distinguish it from the twenty-four-faced trapezohedron, which is a holohedral form and has the four edges of its trapezoidal faces equal in pairs. It is bounded by twenty-four irregular trapeziums (figs. 25 and 26, Plate II.). It is also called the hemi-octakis-hexahedron, the trapezoidal icosi-tetrahedron, the dyakis dodecahedron, the diploid, and the diplopyritoid. It is formed from the six-faced octahedron by taking three out of the six faces which meet in 01, 02, &c., og (fig. 31, Plate I.), and producing them to meet each other and form a solid bounded by twenty-four irregular trapeziums. Thus (fig. 8, Plate I.) the twenty-four faces Cod1, C2ods, C301d2, C204d8, C104d1, C304d4, &c., are produced to meet in the points d1, d, &c., 12 (fig. 25, Plate III.), to form the positive irregular twenty-four-faced trapezohedron. The remaining twenty-four-faces if produced will form the negative trapezohedron. To obtain a face of the irregular twenty-four-faced trapezohedron geometrically from that of the six-faced octahedron from which it is derived.-Describe (fig. 35, Plate IV.), as previously constructed for finding a face of the six-faced octahedron, § 68 and § 137. Join CN cutting C1d, produced in d. Let Cod, (fig. 38, Plate IV.) be a face of the six-faced octahedron. Produce Cd to 8, and make Cd, fig. 38, =С1d11 (fig. 35). Join od, on base C201, describe the triangle C2010, having CC, fig. 35, and 0800 fig. 38. 08C8 will be a face of the irregular twenty-four-faced trapezohedron, and twenty-four such faces will form a net for the same, which can be inscribed in a cube whose faces are equal to the square 0,0,0,01 (fig. 27, Plate IV.). The faces of the six-faced octahedron are shaded on those of the irregular twenty-four-faced trapezohedron in (fig. 26, Plate III.). The following irregular twenty-four-faced trapezohedrons, having faces of crystals parallel to them, have been observed in nature. [13]; 10 Naumann; 5 4 3 Miller; b b b 2 Brooke, in crystals of pyrite. [112]; 20; 4 3 2; bb b, in linnéite. 20/0 2 [13]; 30 ; # 8 2 1; ¿¿ b', in cobaltine, hauerite, and pyrite. b b bbb, + [145]; 50; ; π 5 3 1; b3 b3 b1, in pyrite. 2 402 b1, in pyrite. [124]; ; π 4 2 1; b‡ bằ b1, in pyrite. [1510]; 2 143. Let u be the supplement of the angle of adjacent faces over the edges, such as CS2, C81, C385, &c. ע that over the edges 0,81, 0,85, 0182, &c. Then μ is the inclination of normal of face Co,d, to that of C204ds, fig. 26, Plate III., but indices of Cod, are m 1n, and of С2d ̧ ì 1 n (fig. 31*, Plate IV.*). Also v is the inclination of normal of face C2do, to that of Cdo (fig. 26 Plate III.), but indices of C2do, are m 1 n, and of C1d11, n m 1 (fig. 31*, Plate IV.*). Or, expressing μ and v in terms of the polar distances C2od =P2P1P3 and Cod-P2P1P3 = 5 And cos u=cos Pi-cos P2+cos3 P3, cos v=cos p2 cos P3+cos P1P2+COS P1P3; formulæ calculable at once by Byrne's dual logarithims, or easily adapted to logarithmic computation by subsidiary angles. All the formula for the pentagonal dodecahedrons are immediately derivable from those of the irregular twenty-fourfaced trapezohedron. 144. Limits of the Form of the Irregular Twenty-four-faced Trapezohedron. As m and n approach in magnitude to unity, the irregular twenty-four-faced trapezohedron approximates to the octahedron; and when mand n both equal unity, it becomes the octahedron. In this case the three planes meeting in the points 01, 02, &c., 0 (fig. 25, Plate III.), lie in the same plane, and the edges, such as C11, C2d, lie in the same line. Asm and n both increase in magnitude and become infinitely great, this form approximates to and becomes the cube. In this case the four planes meeting in C1, C2, &c., C, become the same plane, and the edges, such as 08101, 01850%, &c., the same straight line. As m approaches to unity while n increases in magnitude and becomes infinitely great, the form approaches the rhombic dodecahedron. When m equals unity, while n remains finite, the form becomes the three-faced octahedron. When m and n equal each other and are both finite and greater than unity, the form becomes that of the regular twenty-four-faced trapezohedron. Finally, when m remains finite and greater than unity and n becomes infinite, the form becomes that of the pentagonal dodecahedron. 145. As yet the half-symmetrical forms with parallel faces, the pentagonal dodecahedron and the irregular twenty-fourfaced trapezohedron have only been found in combination with those of the full symmetrical forms of the cubical system, and never with those of the half-symmetrical forms with inclined faces. 146. For the pentagonal dodecahedrons the following are the values of the angles μ and v. E F G H M N μ=77° 19′ v=60° 48′. v=61° 19′. 88 μ=67° 23′ v=62° 31'. μ=53° 8′ v=66° 25′. μ=36° 52′ ∞] μ=28° 4' For the irregular twenty-four-faced trapezohedrons the following are the values of u and v. Sub 147. Some crystals have a tendency to split in directions parallel to a certain form. This is called a cleavage-plane. If they split readily, the cleavage is called a perfect one. stances which crystallize in the cubical system have only been observed to split or cleave parallel to the planes of the cube, octahedron, and rhombic dodecahedron. Minerals whose crystals cleave parallel to the faces of the cube, those printed in italics indicating that the cleavage is easy and perfect : Minerals whose crystals cleave parallel to the faces of the octahedron: Minerals whose crystals cleave parallel to the faces of the rhombic dodecahedron : |