In this section it is assumed that not only the kind of symmetry operation is known but also its details, e.g. it is not enough to know that there is a 2-fold rotation, but one should also know the orientation and position of the rotation axis. At first one tries to find for some points their images under the symmetry operation. This knowledge is then exploited to determine the matrix-column pair which decribes the symmetry operation.
Examples will illustrate the procedures. In all of them the point coordinates are referred to a Cartesian coordinate system, see Section 1.2. The reader is recommended to make small sketches in order to see visually what happens.
In the system (4.1.1) of equations there are 12 coefficients to be determined, 9 and 3 . If the image point of one point is known from geometric considerations, one can write down the 3 linear equations of (4.1.1) for this pair of points. Therefore, writing down the equations (4.1.1) for 4 pairs (point image point) is sufficient for the determination of all coefficients, provided the points are independent, i.e. are not lying in a plane. One obtains a system of 12 inhomogeneous linear equations with 12 undetermined parameters and . This may be difficult to solve without a computer. However, if one restricts to crystallographic symmetry operations, the solution is easy more often than not because of the special form of the matrix-column pairs.
In many cases it may be possible to apply the following strategy, which avoids all calculations. It requires knowledge of the image points of the origin and of the 3 `coordinate points' : 1,0,0; : 0,1,0; and : 0,0,1.
What is the pair (W,w) for a glide reflection with the plane through the origin, the normal of the glide plane parallel to c, and with the glide vector g = 1/2,1/2,0 ?
3/2 = + 1/2, 1/2 = + 1/2, 0 = + 0 for and
1/2 = + 1/2, 3/2 = + 1/2, 0 = + 0 for .
One obtains , and .
Point : 0,0,1 is reflected to 0,0, and then shifted to 1/2,1/2,.
This means or , .
W = and w = .
Example 2 [Draw a diagram !]
What is the pair (W,w) for an anti-clockwise 4-fold rotoinversion if the rotoinversion axis is parallel to c, and 1/2,1/2,1/2 is the inversion point ?
The equations are
The resulting matrix-column pair is checked by mapping the fixed point 1/2,1/2,1/2 and the point 1/2,1/2,0. Their images are 1/2,1/2,1/2 and 1/2,1/2,1 in agreement with the geometric meaning of the operation.
If the images of the origin and/or the coordinate points are not known, other pairs `point-image point' must be used. It is difficult to give general rules but often fixed points are appropriate in such a case. In addition, one may exploit the different transformation behaviour of point coordinates and vector coefficients, see Section 4.4. Vector coefficients `see' only the matrix W and not the column w, and that may facilitate the solution. Nevertheless, the calculations may now become more involved. The next example is not crystallographic in the usual sense, but related to twinning in `spinel' mineral.
What is the pair (W,w) for a 2-fold rotation about the space diagonal  with the point 1/2,0,0 lying on the rotation axis ?
It is not particularly easy to find the coordinates of the image of the origin . Therefore, another procedure seems to be more promising. One can use the transformation behaviour of the vector coefficients of the direction  and other distinguished directions. The direction  is invariant under the 2-fold rotation, and the latter is described by the matrix part only, see Section 4.4. Therefore, the following equations hold
On the other hand, the directions , , and  are perpendicular to  and thus are mapped onto their negative directions. This means
From the equations (5.1.2) one concludes
Together with equations (5.1.1) one obtains
Thus, W =
The point 1/2 0 0 is a fixed point, thus
The coefficients of w are then:
There are different tests for the matrix: It is orthogonal, its order is 2 (because it is orthogonal and symmetric), its determinant is , it leaves the vector invariant, and maps the vectors , and onto their negatives (as was used for its construction). The matrix-column pair can be tested with the fixed points, e.g. with ; with ; or other points on the rotation axis.
Problem 1B. Symmetry of the square. For the solution, see p. .
Problem 1A, p. , dealt with the symmetry of the square, see Fig. 3.4.1.
There are 2 more questions concerning this problem.
Are there remarkable properties of the multiplication table ?
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