132. Hemihedral or Half-symmetrical Forms of the Cubic System.

In the holohedral or perfectly symmetrical forms of the cubical system, the solid form of the crystal is bounded by the lines where any one plane or face is intersected by the adjacent planes or faces. There are, however, symmetrical forms where half the number of the holohedral faces are omitted, the planes of the remaining faces forming a solid by the intersection of the adjacent planes.

These, called hemihedral or half-symmetrical faced forms, are of two kinds,-the inclined, in which no one face is parallel to the other; and the parallel, in which the faces are parallel in pairs.

133. The inclined hemihedral forms are the tetrahedron (figs. 15 and 16, Plate III.), the twelve-faced trapezohedron (figs. 17 and 18), the four-faced tetrahedron (figs. 19 and 20), and the six-faced tetrahedron (figs. 21 and 22); these being the hemihedral forms respectively derived from the octahedron, three-faced octahedron, twenty-four-faced trapezohedron, and six-faced octahedron, half of whose faces are produced to meet each other.

There are two hemihedral forms with parallel faces,—the twelve-faced pentagon, derived from the four-faced cube (figs. 23 and 24), and the irregular twenty-four-faced trapezohedron, derived from the six-faced octahedron.

The cube and rhombic dodecahedron do not produce hemihedral forms, according to the laws of symmetry by which the preceding are formed.

134. The tetrahedron (figs. 15 and 16, Plate III.) is formed by taking half the faces of the octahedron (fig. 7, Plate I.), in the following order,-CCC, CCC, CCC, and CCC, and producing these planes to intersect in the lines 0402, 0205 0207, 0405 0407, and 0,05. Referring to (fig. 14, Plate II.), we see that these edges are diagonals of the square faces of the cube in which the octahedron is inscribed, one edge for each face of the cube.

59

2

The tetrahedron is therefore geometrically inscribed in the same cube in which the octahedron, from which it is derived, is also inscribed. (Fig. 16, Plate III.) shows the face of the octahedron shaded on the corresponding face of the tetrahedron. Since 0204 005, and 0.0, are diagonals of equal squares, each face of the tetrahedron is an equilateral triangle, 020405 (fig. 33, Plate IV.). If we bisect the three sides of this equilateral triangle in the points C1, C2, and C, and join these points, the equilateral triangle CCC will be a face of the octahedron.

23

If, therefore, we describe an equilateral triangle (fig. 33, Plate IV.), having each of its sides equal 0,05, (fig. 27, Plate IV.), four such triangles joined together will form the net of a tetrahedron which may be inscribed in the cube, each of whose faces equal the square 0,0,0,0, (fig. 27, Plate IV.).

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Besides the tetrahedron just described, another in all respects similar and equal to the former, except as regards its position in the cube, may be formed by producing the four faces of the octahedron CCC5, C1C3 C4, C2C; C, and CCC (omitted in the former case), to meet each other. It is customary to call one of these tetrahedrons the positive, and the other the negative. Crystals of the following minerals have faces parallel to those of the tetrahedron:

Blende (sulphuret of zinc), boracite, diamond, eulytine (bismuth blende), fahlerz (grey copper), pharmacosiderite (arseniate of iron), rhodizite, tennantite, and tritonite.

Naumann's symbol for the tetrahedron is, Miller's « 111.

135. The twelve-faced trapezohedron is a half-symmetrical form with inclined faces derived from the three-faced octahedron, bounded by twelve equal and similar trapezohedrons (figs. 17 and 18, Plate III.). It is also called the deltoidal dodecahedron, the trapezoidal dodecahedron, and the hemitri-octahedron.

It is formed by producing the three faces of the three-faced octahedron corresponding to each face of the octahedron which are produced to form the tetrahedron, to form a solid by their intersection with each other.

Thus, comparing (figs. 17 and 18, Plate III.), with (fig. 6, Plate I.), the three faces meeting respectively in 01, 03, 069 and og of the three-faced octahedron, are produced to meet in the points W, W, W, and W, making, by their intersections, a twelve-faced trapezohedron bounded by twelve equal and similar trapeziums, WC,o,C3, WC10, C2, &c.

If we call this the positive twelve-faced trapezohedron, the negative will be formed by the twelve faces of the three-faced octahedron which meet in groups of three in the points 02, 04, 05, and 07.

To obtain a face of the twelve-faced trapezohedron geometrically from the three-faced octahedron from which it is derived.

Describe the (fig. 29, Plate IV.), as previously shown in § 35, for determining the face of the three-faced octahedron. Produce CA to C, and OD to 0. Take AC=D¿0=C1A. Join CO, and AO.

Produce M, to meet AO in W. Join CW.

5

Then (fig. 32, Plate IV.) o,C,C, being a face of the threefaced octahedron, bisect CC, in d. Join od, and produce it to W, making o,,W=od,W (fig. 29, Plate IV.). Join C2W, and CW5.

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Then the trapezium o,C,WC, is a face of the twelve-faced trapezohedron derived from the three-faced octahedron whose face is oCC3.

Twelve of these trapeziums form a net for the twelve-faced trapezohedron which can be inscribed in the cube whose faces are equal to the square 0,0,0,0, (fig. 27, Plate IV.).

The faces of the three-faced octahedron are shaded on those of the twelve-faced trapezohedron (fig. 18, Plate III.).

The twelve-faced trapezohedron derived from the three-faced octahedron 11 2, whose symbols are 20 Naumann, 1 2 2 Miller, and a Brooke; whose symbols are (122); 20 Naumann, « 1 2 2 Miller, (a) Brooke, occurs parallel 2 to faces of crystals of blende, diamond, and pharmacosiderite. One derived from the three-faced octahedron 1 1 3, 10 Naumann, 2 3 3 Miller, and a Brooke, whose symbols are respectively 1(1 1 1⁄2); K 233; and (a3), occurs

parallel to faces of crystals of fahlerz.

136. The three-faced tetrahedron is a half-symmetrical form, with inclined faces derived from the twenty-four-faced trape-· zohedron. It is bounded by twelve equal and similar isosceles triangles (figs. 19 and 20, Plate III.)."

It is also called the trigonal dodecahedron, hemi-icositetrahedron, triakis-tetrahedron, pyramidal tetrahedron, and kuproid.

It is formed by producing the three faces of the twenty-fourfaced trapezohedron, corresponding to each face of the octahedron which are produced to form the tetrahedron, to form a solid by their intersection.

Thus, comparing (figs. 19 and 20, Plate III.) with (fig. 4, Plate I.), the three faces of the twenty-four-faced trapezohedron, meeting respectively in 01, 03, 0%, and og (fig. 4), are produced to meet in the points O2, 04, О5, and 0, (figs. 19 and 20, Plate III.), making by their intersections a three-faced tetrahedron, bounded by twelve equal and similar isosceles triangles, 040201, 040501, &c.

If we call this the positive three-faced octahedron, the negative will be formed by the twelve faces of the twentyfour-faced trapezohedron which meet in groups of three in the points 02, 04, 05, and 07.

To obtain a face of the three-faced tetrahedron geometrically from the twenty-four-faced trapezohedron from which it is derived. Describe the (fig. 31, Plate IV.) as previously constructed, § 61, for determining a face of the twenty-four-faced trapezohedron. Produce CA to C, O1D to 05; make AC =DO=AC1 Join CO 40. Then it will be found that Od produced will cut ČŎ in 05.

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5

Let C1dod2 (fig. 39) be the face of the twenty-four-faced trapezohedron derived from (fig. 31, Plate IV.).

Produce od, to 02, and 0d, to 0, making od2 and odo, equal to od0 (fig. 31). Join 0.02; this line will pass through C1.

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Then 0,020, is a face of the three-faced octahedron derived from that of the twenty-four-faced trapezohedron whose face is C1dio,d2.

Twelve of these isosceles triangles form a net for the threefaced tetrahedron which can be inscribed in the cube whose faces are equal to the square 0,0,0,0, (fig. 27, Plate IV.).

The faces of the twenty-four-faced trapezohedron are shaded on those of the three-faced tetrahedron (fig. 20, Plate IV.). The following curious reciprocal relations may be observed between the perfectly symmetrical and half-symmetrical forms of the three-faced octahedron and the twenty-four-faced trapezohedron.

The hemihedral form of the three-faced octahedron is bounded by trapeziums similar to the faces of the twenty-four-faced trapezohedron.

The hemihedral form of the twenty-four-faced trapezohedron is bounded by isosceles triangles like the faces of the threefaced cube.

The three-faced octahedron is formed by placing a threefaced pyramid of equal isosceles triangles on each of the equilateral triangular faces of the regular octahedron as bases. The three-faced tetrahedron is formed in like manner by placing a three-faced pyramid of equal isosceles triangles on each of the equilateral triangular faces of the regular tetrahedron.

The following three-faced tetrahedrons, having faces of crystals parallel to them, have been observed in nature:

0 2

(1); Naumann, « 2 3 3 Miller, a Brooke; in tennantite.

202

Naumann, κ 112 Miller, a Brooke; in

(1 2 2);

2

(1 3 3);

boracite, eulytine, fahlerz, and tennantite.

303

; k 113; a3; in blende and fahlerz.