434 
Then (fig. 32, Plate IV.) o l G 2 G s being a face of the three- 
faced octahedron, bisect G 2 G 3 in d 6 . Join o 7 d 6 , and produce 
it to W 6 , making o 1 d 6 W 6 =o 1 d 6 W 5 (fig. 29, Plate IV.). Join 
G 2 W 5 and G s W 6 . 
Then the trapezium o 1 G 3 W 5 G 2 is a face of the twelve- faced 
trapezohedron derived from the three-faced octahedron whose 
face is o 1 0 2 (7 3 . 
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 1 0 4 0 8 0 5 (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 112, whose symbols are 2 0 Naumann, 12 2 
Miller, and a 2 Brooke ; whose symbols are -|-(1 2 2) ; 
20 
Naumann, k 1 2 2 Miller, %(a 2 ) Brooke, occurs parallel 
to faces of crystals of blende, diamond, and pharmacosiderite. 
One derived from the three - faced octahedron 1 1 -§-, 
2. 
| -0 Naumann, 2 3 3 Miller, and a 3 Brooke, whose symbols 
3.0 2 _ 
are respectively |-(1 1 f ) ; -2 — k 2 3 3; and |-(& 3 ), 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-icositetra- 
hedron, triaMs -tetrahedron, pyramidal tetrahedron, and huproid. 
It is formed by producing the three faces of the twenty-four- 
faced trapezohedron, corresponding to each face of the octa- 
hedron 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 o v o 3 , o 6 , and o 8 (fig. 4), are produced 
to meet in the points 0 2 , 0 4 , 0 5 , and 0 7 (figs. 19 and 20, 
Plate III.), making by their intersections a three-faced tetra- 
hedron, bounded by twelve equal and similar isosceles triangles, 
Ofd 2 o 7 , &c. 
If we call this the positive three-faced octahedron, the 
negative will be formed by the twelve faces of the twenty- 
four-faced trapezohedron which meet in groups of three in 
the points o 2 , o 4 , o 5 , and o 7 . 
