435 
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 IY.) as previously con- 
structed,, § 61, for determining a face of the twenty-four-faced 
trapezohedron. Produce G X A to 0 6 , 0 X D 5 to 0 5 ; make AG 6 
=D 5 0 5 =AG v Join G 6 0 6 , A0 5 . Then it will be found that 
0 1 d 5 produced will cut 0 6 0 5 in 0 5 . 
Let C x d x o x d 2 (fig. 39) be the face of the twenty-four-faced 
trapezohedron derived from (fig. 31, Plate IV.). 
Produce o x d 2 to 0 2 , and O x d x to 0 4 , making o x d 2 0 2 and 
o x d x 0 4 equal to" o x d 5 0 5 (fig. 31). Join 0 4 0 2 ; this line will pass 
through G x . 
Then 0 4 0 2 o 1 is a face of the three-faced octahedron derived 
from that of the twenty -four -faced trapezohedron whose face 
is G x d x o x d 2 . 
Twelve of these isosceles triangles form a net for the three- 
faced tetrahedron 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 IY.). 
The faces of the twenty-four-faced trapezohedron are shaded 
on those of the three-faced tetrahedron (fig. 20, Plate IY.). 
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 trape- 
zohedron. 
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 three- 
faced cube* 
The three-faced octahedron is formed by placing a three- 
faced 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 tetra- 
hedron. 
The following three-faced tetrahedrons , having faces of 
crystals parallel to them, have been observed in nature : — 
1.(1 ll); i9A Naumann, k 2 3 3 Miller, aA Brooke; in 
2 
tennantite. 
1.(1 2 2); ^ Naumann, k 1 1 2 Miller, a 2 Brooke; in 
2 
boracite, eulytine, fahlerz, and tennantite. 
|(1 3 3)- ; k 1 1 3; a 3 ; in blende and fahlerz, 
2 
