436 
•§»(! 4 4); ^9^ ; k 1 1 4) a 4 ; in blende. 
2 
1(1 5 5); 22^; K 1 1 5; a 5 ; in blende. 
2 
137. Tbe 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-h ex-octahedron } liexalcis-tetrahe- 
dron, 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 octa- 
hedron, meeting respectively in o v o 3 , o 6 , and o 8 (fig. 3, Plate I.), 
are produced to meet in the points TF 2 , [F 4 , W 5 , and W 7 (figs. 21 
and 22, Plate III.), making by their intersections a six-faced 
tetrahedron, bounded by 24 equal and similar scalene triangles, 
°iCiW 2 , o 1 C 3 W 2) &c. 
If we call this the positive six-faced tetrahedron, the nega- 
tive will be formed by the twenty-four faces of the six-faced 
octahedron which meet in groups of six in the points o 2 , o 4 , o 5 , 
and o 7 (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 C ± A to C Q , OfD. to 0 5 ; make AC 6 = 
D 5 0 5 =C 1 A. Join C 6 0 5 and A0 5 . Produce No x d 5 to meet A0 5 
in W 5 , and join C G W 5 . 
Then (fig. 36, Plate IV.) let C 1 o 1 d 2 be a face of the six-faced 
octahedron constructed as in § 69. 
Produce o x d 2 to W 2 and make o ± d 2 W 2 = o 1 d 5 W~, fig. 35. 
Join G 1 JV 2 . Then the scalene triangle o 1 W 2 C 1 is a face of 
the six-faced tetrahedron derived from the six-faced octahedron 
whose face is (7 1 o 1 c7 2 . 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 (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 : 
i(lf3); 121 Naumann; K 3 2 1 Miller ; 
u 
Brooke ; in crystals of the diamond. 
