THE ISOMETRIC SYSTEM
The word, isometric, means "same measure", an allusion to the equal three dimensional structure. "Cubic System" is a synonym for the isometric system and is due to the cubic nature of the crystallographic structure as well as the most common isometric form, the cube.
The most symmetrical class of all is the Hexoctahedral Class. The three identical crystallographic axes serve as four fold rotational axes with a square outline when viewed down each axis. However the principle and defining axes for the isometric system are the 4 three fold axes. They dissect the three crystallographic axes and are essentially the 4 diagonal lines through a cube. If you picture a transparent cube, then draw lines from each corner, through the middle to the other corner, then you will end up with 4 lines that represent the 4 three fold axes. The three fold rotational axes can be see on a cube as the axes through the three top faces and the three bottom faces, when the cube is held from corner to opposite corner.
The octahedron shows the three fold rotations better than the cube with its triangular faces that are perpendicular to the three fold axes. The four fold rotations though are not as easy to see on the octahedron, but a square cross-section can be seen when viewed down the points of the octahedron. The other forms of the hexoctahedral class all demonstrate these symmetry elements including the 6 two fold axes and 9 mirror planes.
The Hextetrahedral Class lacks the 3 simple four fold rotational axes because they are four fold rotoinversion axes. A four fold 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 the face is back, exactly where it started. The crystal then looks as if it has only two fold axes instead of the four fold axes that it actually has. The hextetrahedral class produces a very symmetrical tetrahedron and derivatives of this form. The three fold axes are very easy to see in this class as they run through the triangular bases and out the three faced pyramidal top. Of course there are no bases or pyramids in the isometrmic system these terms are used only as a visual guide to the shape of the tetrahedron.
The Gyroidal Class has all axes of the hexoctahedral class but
none of the mirror planes. This class produces a unique form called the
The Diploidal Class lacks any four fold rotational axes or any
four fold rotoinversion axes. Remember, the isometric system doesn't require
four fold rotational axes. There is the required 4 three fold axes as well
as 3 two fold axes (parallel to the crystallographic axes), three mirrors
and a center. The class contains a few rare minerals and one very common,
well known and usually well crystallized mineral, that being
Pyrite lends its name to the classic form that comes from this class and
the form that pyrite is famous for, the
The lowest symmetry class in the isometric system is the Tetartoidal Class. It lacks any mirror planes or a center, but has the required 4 three fold axes in addition to 3 two fold axes that lie on the crystallographic axes. The tetartoid is a unique form and is a 12 sided form that peripherally resembles the shape of a tetrahedron.
The Hexoctahedral Symmetry Class
The Hextetrahedral Symmetry Class
The Gyroidal Symmetry Class
The Diploidal Symmetry Class
The Tetartoidal Symmetry Class
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