401 
Then G Y ofL 2 (fig. 36) is a face on a plane surface of the six- 
faced octahedron which can be inscribed in a cube, each of 
whose faces are equal O-fifi 8 0 5 , fig. 27. 
Forty-eight triangles, similar and equal to G 1 o 1 d 2 , arranged 
as a net and cut out of cardboard, will fold up into a solid 
model of the six-faced octahedron. 
/0. Each face of the six-faced octahedron, if produced, cuts 
one axis of the cube at the distance = 1, another at the distance 
= m, and the third at a distance n from the centre of the cube. 
The three quantities, 1, m, and n are termed the three 
indices of the six-faced octahedron. 
Its symbol, therefore, is 1, m,n; Naumann's symbol is nOm. 
If . the three fractions 1, l be brought to a common de- 
nominator, and the three numerators divided, if they possess 
any common factor, by that factor, be represented by h , 1c, 1 , 
these being whole numbers, then li, ft, Z is Miller's symbol, and 
b b b is that of Brooke, Levy, and Des Cloizeau. 
71. The form 1, £|-, 64 occurs in garnet ; 1, f, in pyrite 
and gold; 1, f, 2 in linneite ; 1, -f-, 4 in garnet ; 1, if, if in 
linneite ; 1, ^ fr amalgam, cobaltine, cuprite, diamond, 
fahlerz, garnet, hauerite, magnetite, and pyrite; 1, |, 8 in 
pyrite; 1, 5 in boracite and pyrite ; 1, 10 in pyrite; 1, 
2, 4 in fluor, gold, and pyrite; 1, 2, 10 in pyrite; 1, -U, if in 
fluor ; 1, if, 4 in fluor ; 1, •§-, 7 in fluor; 1, 3, -^f in magnetite; 
1, 4, 8 in galena. 
72. To find the ratios of the rhombohedral and octahedral 
axes of the six-faced octahedron to those of the circum- 
scribing cube. 
In fig. 35, Plate IV., the sides of the square AG 1 D 1 G 2 are 
by construction equal to unity. Hence AD 1 = X / 2, and angle 
B Y AG 2 =4h° O'. Also AM=m by construction. Let angle 
AGzd^ — a. Then AdfJ^lSO 0 — (a + 45). 
Then cos a=^=l, 
m m 
and in triangle Ad 1 G 2 , 
Ad x _ sin a 
sin a 
sin (a + 45) 
AG 2 sin {180— (a + 45)} 
Ad, = 55ii = 1 
sin a cos 45 + cos a sin 45 V 7 J- + v /-,} cos o 
- V^2 _ _J_ 
i 1 +1 
m i/i 
AD V 
