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137. The six-faced tetrahedron is a half-symmetrical form with inclined faces derived from the six-faced octahedron. It is bounded by twenty-four equal and similar scalene triangles (figs. 21 and 22, Plate III.).

It is also called the hemi-hex-octahedron, hexakis-tetrahedron, and boracitoid.

It is formed by producing the six faces of the six-faced octahedron, corresponding to each face of the octahedron which are produced to form the tetrahedron, to form a solid by their intersection. Thus, comparing (figs. 21 and 22, Plate III.) with (fig. 3, Plate I.), the six faces of the six-faced octahedron, meeting respectively in 01, 03, 0, and og (fig. 3, Plate I.), are produced to meet in the points W, WA, W., and W, (figs. 21 and 22, Plate III.), making by their intersections a six-faced tetrahedron, bounded by 24 equal and similar scalene triangles, 0CW2, 0C3W, &c.

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If we call this the positive six-faced tetrahedron, the negative will be formed by the twenty-four faces of the six-faced octahedron which meet in groups of six in the points 02, 04, 05, and o, (fig. 3, Plate I.). To obtain geometrically a face of the six-faced tetrahedron from the six-faced octahedron from which it is derived, describe the (fig. 35, Plate IV.), as previously constructed, § 68, for determining a face of the six-faced octahedron. Produce CA to C, OD to 0; make AC= DO=C4. Join CO, and 40. Produce No,d, to meet ÅO in W, and join CW

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Then (fig. 36, Plate IV.) let Cod, be a face of the six-faced octahedron constructed as in $ 69.

Produce od, to W2 and make o1d2W2=o,dW, fig. 35.

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2

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Join CW2. Then the scalene triangle o,WC, is a face of the six-faced tetrahedron derived from the six-faced octahedron whose face is Cod. Twenty-four such scalene triangles form a net for the six-faced 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 six-faced octahedron are shaded on those of the six-faced tetrahedron (fig. 22, Plate III.). The following six-faced tetrahedrons, having faces of crystals parallel to them, have been observed in nature:

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Naumann; 3 2 1 Miller; (1)

Brooke; in crystals of the diamond,

(15) Naumann;

50; 531 Miller; 4(b 16* v*)
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Brooke; in crystals of boracite.

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By the construction fig. 35, the ratio may be readily

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determined by plain trigonometry, just as the ratio was

in § 73.

It can also be readily determined by geometry of three dimensions. For (fig. 22, Plate III.) W2 is a point in each of the three planes Cod2, Cod2, C1033.

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Now the equation to the plane Cod, referred to rectangular co-crdinates, AC1, AС2, AC3,

To the plane C112 is

To the plane Cod is

is

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+

= 1

(C). (See fig. 31*,

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and fig. 32*, Plate IV.*)

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x

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And since x, y, z will be the same for the point IV2 where these planes meet,

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Again, let w be the angle which the normals of the faces Code, Cod, make with each other, or 180°- be the anglo of inclination of the two faces of the six-faced tetrahedron (fig. 21, Plate III.), over the edge CW.

Then since m n 1 is the symbol of C10d2,

and -n-m 1 that of

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COS W=

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Or by $ 110,

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-cos P2 COS P3-COS P2 COS P3+ cos P1 cos Pi cos2 P1-2 cos P2 cos P3.

Which may be computed at once by Byrne's dual logarithms, or thus adapted for ordinary logarithmic computation.

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cos P2 COS

P3

cos 60 cos2 P1
cos2 p1 cos (a +45)°

cos a sin 45°

138. Limits of the Form of the Six-faced Tetrahedron. As m and n approach in magnitude to unity, the six-faced tetrahedron approximates to the tetrahedron. When m=n=1, the six-faced tetrahedron becomes the tetrahedron, the points. W1, W2, W, and W, (fig. 21, Plate III.) coincide with the points 01, 02, 05, and 0, (fig. 15). CW, and C1W2 become the straight line 0,04, &c., and the six faces round each point 01, 03, 0, and 08. lie in the same plane.

As m and n increase in magnitude greater than unity, and also in equality to each other, the six-faced octahedron approximates to the cube. When m and n are both infinitely great, it coincides with it. In this case each of the four faces which meet in the six points C1, C2, C3, &c., Ce, lie in the same plane. As m approaches to unity, while n increases in magnitude, the six-faced tetrahedron approximates to the rhombic dodecahedron. When m=1 and n=∞ it becomes the rhombic dodecahedron, and the two faces which lie on each side of the twelve lines W201, W401, W501, &c., lie in the same plane, and the Co and CW become equal.

When m equals unity, while n remains finite, the six-faced tetrahedron becomes the twelve-faced trapezohedron, and the faces on each side of the twelve edges W2O, lie in the same plane, but the edges Co and CW are not equal.

When m and n are equal to each other, both finite and greater than unity, the six-faced tetrahedron becomes the three-faced tetrahedron, and the faces on each side the twelve lines Co1, C301, C201, &c., lie in the same plane. W coincides with O and WOW becomes a straight line. When m remains finite, and n becomes infinite, the six-faced octahedron becomes the fourfaced cube, and its scalene triangles become isosceles.

From the above it follows that the cube, rhombic dodeca

hedron, and four-faced cube, which have no hemihedral forms with inclined faces, are limiting forms of the six-faced tetrahedron.

Also that all the formulæ of the tetrahedron, three-faced tetrahedron, and twelve-faced trapezohedron may be derived from those of the six-faced octahedron by giving the proper values to m and n.

139. Table showing the symbols and formulæ of the halfsymmetrical forms which are not included in the table § 131, for the holohedral forms. The letters refer to holohedral forms, $131.

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140. The pentagonal dodecahedron is a half-symmetrical form with parallel faces derived from the four-faced cube. It

is bounded by twelve equal and similar pentagons. These pentagons are, except in one species of the pentagonal dodecahedron, irregular (figs. 23 and 24, Plate III.); four edges or sides of the pentagon being equal, and the fifth unequal. When the five edges are equal, the pentagonal dodecahedron is called the regular pentagonal dodecahedron, and is one of the five Platonic bodies.

It is also called the hemi-hexa-tetrahedron and pyritoid.

It is formed from the four-faced cube by taking three out of the six faces (fig. 2, Plate I.) which meet in the points 1, 02, &c., o; taking the faces alternately and producing them to form by their intersections a solid by twelve pentagonal faces.

Thus the faces C10104 C10203, C20105, C2040, C30102, C305069 C40200 C40307, C50403, C5070, C00, and Coo, are produced to form the positive pentagonal dodecahedron; the twelve remaining faces to form the negative pentagonal dodecahedron. The faces so produced meet in twenty-four equal edges o11, 082, &c. (figs. 23 and 24, Plate III.); and six other edges, but unequal to the former 88, 88, &c.

To obtain a face of the pentagonal dodecahedron geometrically from that of the four-faced cube from which it is derived (fig. 37, Plate IV.), being described as in § 53. Produce Cd, to meet DC, in 8.

Describe Coo, as in § 54, a face of the four-faced cube (fig. 34, Plate IV.). Bisect 0,04 in d. Produce C1d, to d1, inaking Cd=C1d11 (fig. 37). Join o,, and . Through C, draw &C, parallel to 0104.

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Then (fig. 34) take C, and C, each equal C2d, (fig. 37). Join 0,8 and 0,82.

Then 200 is a face of the pentagonal dodecahedron derived from the four-faced cube whose face is C10401

Twelve such pentagonal faces form a net for the pentagonal dodecahedron 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 four-faced cube are shaded on those of the pentagonal dodecahedron (fig. 24, Plate IV.).

The following pentagonal dodecahedrons, having faces of crystals parallel to them, have been observed in nature:

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