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.

**Procedure 1**

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.

- (1)
- The origin Let with
coordinates
be the image of the origin with
coordinates
,**o***i.e.*. Examination of the equations (4.1.1) shows that =,**w***i.e.*the columncan be determined separately from the coefficients of the matrix**w**without any effort.**W** - (2)
- The coordinate points We consider the
point . Inserting 1,0,0 in the equations (4.1.1) one obtains
or
, The
first column of
is separated from the others, and for the solution only the known coefficients have to be subtracted from the coordinates of the image point of . Analogously one calculates the coefficients from the image of point : 0,1,0 and from the image of point : 0,0,1.**W**

Example 1

What is the pair (* W*,

- (a)
- Image of the origin : The origin is left invariant by the
reflection part of the mapping; it is shifted by the glide part to
1/2,1/2,0 which are the coordinates of . Therefore,
= 1/2,1/2,0.**w** - (b)
- Images of the coordinate points. Both the points and
are not affected by the reflection part but is then shifted to
3/2,1/2,0 and to 1/2,3/2,0. This results in the equations
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 , .

- (c)
- The matrix-column pair is thus
= and**W**= .**w**

Example 2 [Draw a diagram !]

What is the pair (* W*,

- (a)
- The anti-clockwise 4-fold rotation maps the origin onto the point 1,0,0; the following inversion in 1/2,1/2,1/2 maps this intermediate point onto the point 0,1,1, such that .
- (b)
- For the other points:
The equations are

- (c)
- The result is
= ;**W**= .**w**

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.

**Procedure 2**

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

Example 3

What is the pair (* W*,

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 [111] and other distinguished directions. The direction [111] 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 [10], [01], and [01] are perpendicular to [111] 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

,

, and

.

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.

**(v)**- Calculate the matrix-column pairs of the symmetry operations of the square.
**(vi)**- Construct the multiplication table of the group of
the square. [The multiplication table of a group
of
order is a table with rows and columns. The elements of
the group are written on top of the table and on the left side,
preferably in the same sequence and starting with the unit element.
In the intersection of the th row and the th column the product
is listed for any pair of indices
. The complete table is the
*multiplication table*].Are there remarkable properties of the multiplication table ?

**Copyright © 2002 International Union of
Crystallography**