Symmetry

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Revision as of 05:53, 2 December 2006 by Doos (talk | contribs) (Center of symmetry)
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Basic

Axes of symmetry

Axes of symmetry have to do with a crystal's balance of shape when rotated around these imaginary axes.
Every crystal belongs to a particular crystal system (cubic, tetragonal, hexagonal, trigonal, orthorhombic, monoclinic or triclinic) and the symmetry for each of these systems is defined by ideal shapes.

Following is an illustration of symmetry axes in the orthorhombic system.
When determining the axes of symmetry, it is important to rotate (or spin) the crystal around that axis through a 360° rotation and judge how many times the exact image is repeated during the rotation.

The basic form that makes up the orthorhombic system looks like a matchbox.
Here the match box is represented as 3 pinacoids (3 parallel faces).
An imaginary needle (axis) is pierced through the center of the top plane.

We take an arbitrary plane as our starter for the rotation (the front plane in this case).
During a 360° rotation of the prism around the axis, the exact same image is shown twice.
The same process is repeated but now with the needle (axis) pierced through the side faces.

We take another arbitrary plane as our starter for rotation (the top plane).
Again during a 360° rotation of the prism around the axis, the exact same image is shown twice.
The final axis of symmetry (in the orthorhombic prism) is through the front plane.

We now take a pinacoidal face (the front plane) as the start of our rotation.
And again, during a 360° rotation of the prism around the axis, the exact same image is shown twice.
If one would place the matchbox on a different pinacoidal face, one would get identical results.

As can be seen in the above images, there are 3 axes of symmetry in the orthorhombic system and each axis produces the same image twice during a 360° spin around that axis.
When an axis shows the same image twice, we say it has a 2-fold axis of symmetry (or better: a "digonal axis of symmetry"). So the orthorhombic system is characterized by 3 2-fold axes of symmetry.

Other crystal systems will have fewer or more axes of symmetry. A 3-fold axis of symmetry means that the image is repeated 3 times (named a "trigonal axis of symmetry"), etc.

Planes of symmetry

Planes of symmetry can be regarded as mirror planes. They divide a crystal in two. Each side of the division is the mirror of the other while the total image is not altered by the mirror plane (the symmetry stays in tact).
As with the axes of symmetry, the orthorhombic system is used for illustration and there are 3 planes of symmetry in this crystal system.

First plane of symmetry
Second plane of symmetry
Third plane of symmetry

In all the above images, the dividing plane acts as a mirror plane. In other crystal systems, there may be fewer or more planes of symmetry.

Not a plane of symmetry


To illustrate that not all divisions by a plane create a symmetry plane, the illustration on the left shows a mirror that transforms the crystal into a kite form instead of into its original prismatic shape.

Center of symmetry

Center of symmetry


A center of symmetry is the central point from which crystal faces appear the same on either end of the center.
In this image, the center of symmetry is where the green, the blue and the red axis of symmetry meet.

Center of symmetry
Center of symmetry
Center of symmetry
Center of symmetry
Center of symmetry
Center of symmetry

Sources