48 MINERALOGY. 
found than absolutely symmetrical octahedrons: it must nevertheless be 
observed, that such distortions are always produced by some external 
impediment, and that nature, in the entire absence of all obstructing 
influences, always exhibits perfect symmetry. This is true, not for octa- 
hedrons alone, but for all crystals. 
2. The cube (fig. 14) is produced from the octahedron, by truncating its 
solid angles by planes perpendicular to the axes. Here the axes are again 
all equal, and connect the centres of opposite faces. Fig. 15 shows the 
relation of the cube to the octahedron. 
3. The cubic octahedron (figs. 16 and 17) is obtained from the cube, by 
truncating the six corners, until the old faces again become squares. a 
and b are new faces parallel to the old faces of the obliterated octahedron. 
4. The rhombic dodecahedron (pl. 32, fig. 18) is produced from the cube 
‘by the truncation of its edges, until the original faces are obliterated. This 
solid has twelve rhombic faces, twenty-four equal edges, and fourteen solid 
angles. Of these solid angles, six are formed, each by four rhombs meeting by 
their acute angles; and eight, each by three rhombs meeting by their obtuse 
angles. The relation of the rhombic dodecahedron to the cube is shown 
in fig. 19. 
5. The pyramidal cube, or tetrahexahedron (fig. 20), is produced by 
placing a four-sided pyramid on each face of the cube. 
The cubic octahedron has already shown that a crystal may be inclosed 
by faces belonging to two different forms of crystals. This case is often 
‘repeated. Thus, in figs. 21 and 24, the cube is represented with dodeca- 
hedral faces replacing its edges. ig. 22 shows an octahedron, with 
dodecahedral faces, a, and cube faces, b; c, indicating what is left of the 
octahedron. Fig. 23 represents an octahedron passing into a dodecahedron : 
fig. 25 is a combination of cube faces and those of the pyramidal cube ; or 
a cube with its edges bevelled. fig. 30 is the trapezohedron, or tetragonal 
trisoctahedron ; a solid bounded by twenty-four equal trapezia. It can be 
derived from the octahedron by replacing its corners by four faces, or by 
replacing each corner of the cube by three faces (jig. 28). 
The figures hitherto derived from the primary forms have been produced 
by modifying all the similar parts of the primary simultaneously. Such 
forms are called holohedral. Hemihedral forms of crystals occur in equal 
number. These are forms in which half of the similar parts of the crystals 
are modified alike, independently of the other half. Some of these 
forms are: 
6. The tetrahedron (fig. 26), a solid, inclosed by four equilateral triangles. 
Fig. 27 represents an octahedron passing into a tetrahedron, in which the 
faces, a, indicate what is left of the octahedron faces. A form of frequent 
occurrence, and likewise belonging to this place is: 
7%. The pentagonal dodecahedron, or hemi-tetrahexahedron, (fig. 29). 
This is a hemihedral form derived from the pyramidal cube, and bounded 
by twelve equal pentagons. 
We may remark, in reference to this as well as other systems, that all the 
different forms belonging to one and the same system, may occur in the 
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