433 
If, therefore, we describe an equilateral triangle (fig. 33, 
Plate IV.), having each of its sides equal 0 4 0 5 , (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 Of) fif) 5 (fig. 27, Plate IV.). 
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 G X G 2 G 5 , Gf 2 G 4 , G 2 G s G g , and G 5 G 4 G 6 (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 *c 1 1 1. 
135. The tivelve-faced trapezohedron is a half-symmetrical 
form with inclined faces derived from the three-faced octahe- 
dron, 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 hemi- 
tri -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 o v o 3 , o 6 , 
and o 8 of the three-faced octahedron, are produced to meet in 
the points W 2 , W 4 , W 5 , and W 7 , making, by their intersections, 
a twelve-faced trapezohedron bounded by twelve equal and 
similar trapeziums, W 2 0 1 o 1 G s , WfJpfJ^ &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 o 2 , o 4 , 
o 5 , and o 7 . 
To obtain a face of the twelve-faced trapezohedron geo- 
metrically 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. Pro- 
duce OfL to G 6 , and 0 4 D 5 to 0 5 . Take AG^J)fi^G x A. 
Join G 6 0 5 and A0 5 . 
Produce Md 5 to meet A0 5 in W 5 . Join G 6 W 5 . 
