Isometric | Hexagonal | Trigonal | Orthorhombic | Monoclinic | Triclinic |
The tetragonal system is the least populated by natural crystals
of all the crystallographic systems. The basic setup is similar to the
in that all angles between the crystallographic axes are 90
degrees. The difference between the isometric system and the tetragonal
system is that one of the three axes is longer or shorter than the other
two. This produces a unique direction in the tetragonal crystals giving
rise to the reference of the tetragonal system as uniaxial. This unique
direction in the crystal becomes the major axis, also called the c
axis. The other two axes are identical and are referred to as a
The basic form of the isometric system can be thought of as a square
box; however, in the tetragonal system the basic form can be thought of
as either an elongated or a flattened box. The box will still have a square
cross-section, but side profiles will show rectangles. The direction through
the square cross-section defines the four fold rotational axis, or the
c axis, that is required for this system. Tetragonal crystals that
have a box like form or a pyramidal form can be pseudo-
The system's most symmetrical class, the Ditetragonal Dipyramidal Class, has a mirror plane perpendicular to the major axis. Each of the two identical crystallographic axes (a and a') serve as the 2 two fold rotational axes. Two mirror planes are also found running parallel to the major axis and are perpendicular to the two fold axes. With all this symmetry, more symmetry is actually produced from the position of the mirrors and axes. Two more two fold axes are found between the a and a' axes in the plane perpendicular to the major axis. Also two more mirror planes are found parallel to the major axis, again between the other two mirror planes. The four separate mirrors divide the crystal into 45 degree slices just as four cuts through a pizza divide it into eight slices. The eight slices are an analogous representation of the ditetragonal (or 2 X 4 sided) aspect of this class and the Ditetragonal Pyramidal Class which lacks the perpendicular mirror and the two fold axes.
The other lower symmetry classes have few minerals as members and fewer minerals that crystallize well enough to show their true symmetry. The Tetragonal Scalahedron Class and the Tetragonal Disphenoidal Class do not even appear to belong to the tetragonal system as they seem to lack a four fold axis. They don't really lack a four fold axis, it's just a four fold rotoinversion axis. A tetragonal rotoinversion axis takes a face, rotates it 90 degrees (one fourth of a rotation) and then inverts it (up to down & right to left) through the crystal to the other side. Then it rotates it again 90 degrees and inverts it again through the crystal. Another rotoinversion operation and finally another (four in all) and it is back exactly where it started. The result is two major faces on the top and two on the bottom of the crystal but in perpendicular orientation. The axis then looks like a two fold axis instead of the four fold axis that it actually is. The tetragonal scalahedron has eight faces (four pairs) instead of the four faces of the tetragonal diphenoid, but the symmetry operation is the same.
The other classes include the Tetragonal Trapezohedral Class which lacks any mirror planes and is analogous to other trapezohedral classes of other systems such as the Trigonal Trapezohedral Class, 12th, (3 2) and the Hexagonal Trapezohedral Class, 19th, (6 2 2).
The Tetragonal Dipyramidal Class and the Tetragonal Pyramidal Class simply have a four fold axis with no parallel mirrors or any two fold axes. The difference between these last two classes is the presence, or lack there of, of a perpendicular mirror plane. A dipyramid is just a pyramid on top that is then reflected on the bottom.
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