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In
the isometric system the axes are of equal length, and at right angles
to each other. In the tetragonal system one axis, usually taken as the
vertical, is longer or shorter than the other two, which are equal in
length. The axes are all at right angles to each other, however. In the
hexagonal system one axis, usually taken as the vertical, is longer or
shorter than the lateral axes and at right angles to them. The lateral
axes are three in number, of equal length, and meet at angles of 60°.
In the orthorhombic system the three axes are of unequal length, but
meet at right angles. In the monoclinic system the three axes are of
unequal length. Two of them meet at right angles, while
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reflection
in a mirror; and another way of stating the previous observation would
be to say that an isometric crystal can be held before a mirror in more
positions in which the crystal and its reflection present the same
appearance, than one of any other system, while with a triclinic
crystal no such position can be found. Besides the division into six
systems, each system is itself subdivided into groups of varying kinds
of symmetry. There are thirty-two of these groups, characterized by a
particular kind of symmetry, and a substance crystallizing in a certain
group will invariably show that symmetry.
A
few simple forms peculiar to different systems may be mentioned here,
since the terms will be often employed in the text. Four common forms,
exhibited by minerals crystallizing in the isometric system, are the
cube, octahedron, dodecahedron, and trapezohedron. The cube is a solid
bounded by six similar faces, each parallel to two of the axes. Each
face is a square, and the interfacial angles are all 90°. Crystals of
this form are exhibited by pyrite, fluorspar, and rarely by diamond.
The octahedron is bounded by eight similar faces, meeting the axes
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