1 S CKYS1 ALLOGEAPHT. 



edges, six in number (because the number of edges is twelve), 



and called the dodecahedral axes. 



Fig. 2 represents the octahedron, a solid contained under 

 eight equal triangular faces (whence the name from the Greek 

 eight and face), and having the three axes like those in the cube. 

 Its plane angles are 60° ; its interfacial angles, that is the incli- 

 nation of planes 1 and 1 over an intervening edge (usually written 

 1 A 1) = 109° 28' ; and 1 on 1 over a solid angle, 70° '62'. 



Fig. 3 is the dodecahedron, a solid contained under twelve 

 equal rhombic faces (whence the name from the Greek for twelve 

 and face). The position of the cubic axes is shown in the fig- 

 ure. It has fourteen solid angles ; six formed by the meeting of 

 four planes, and eight formed by the meeting of three. The 

 interfacial angles (or i on an adjoining i) are 120° ; i on i over 

 a four-faced solid angle = 90°. 



Fig. 4 is a trapezohedron, a solid contained under 24 equal 

 trapezoidal faces. There are several different trapezohedrons 

 among isometric crystalline forms. The one here figured, which 

 is the common one, has the angle over the edge _Z?, 131° 49', 

 and that over the edge (?, 146° 27'. A trapezohedron is also 

 called a tetragonal trisoctahedron, the faces being tetragonal 

 or four-sided, and the number of faces being 3 times 8 (tris t 

 octo, in Greek). 



Fig. 5 is another trisoctahedron, one having trigonal or three- 

 sided faces, and hence called a trigonal trisoctahedron. Com- 

 paring it w T ith the octahedron, fig. 2, it will be seen that three 

 of its planes correspond to one of the octahedron. The same is 

 true also of the trapezohedron. 



Fig. 6 is a tetrahexahedron, that is a 4 x 6-faced solid, the 

 faces being 24 in number, and four corresponding to each face 

 of the cube or hexahedron (fig. 1). 



Fig. 7 is a hexoctahedron, that is a 6 X 8-faced solid, a pyramid 

 of six planes corresponding to each face in the octahedron, as is 

 apparent on comparison. There are different kinds of hexocta- 

 hedrons known among crystallized isometric species, as well as 

 of the two preceding forms. In each case the difference is not 

 in number or general arrangement of planes, but in the angles 

 between the planes, as explained beyond. 



But these simple forms very commonly occur in combination 

 frith one another ; a cube with the planes of an octahedron and 

 the reverse, or with the planes of any or all of the other kinds 

 above figured, and many others besides. Moreover, all stages 

 between the different forms are often represented among the 

 crystals of a species. Thus between the cube and octahedron, 



