How can one find the geometric meaning of a matrix-column pair ? Large parts of the following recipe apply not only to crystallographic symmetry operations but also to general isometries.

- One must know the reference coordinate system of the matrix-column
pair. Without this knowledge a geometric evaluation is impossible.
Example. The matrix (in IT A shorthand notation) describes a 6-fold anti-clockwise rotation if referred to a hexagonal basis. If referred to an orthonormal basis it does not describe an isometry at all but contains a shearing component.

- The matrix part is evaluated first.
In general the coefficients of the matrix depend on the choice of the basis; a change of basis changes the coefficients, see Section 5.3.2. However, there are geometric quantities which are independent of the basis. Correspondingly, there exist characteristic numbers of a matrix from which the geometric features may be derived and

*vice versa*.- The preservation of the `handedness' of an object,
*i.e.*the question if the symmetry operation is a rotation or rotoinversion is a geometric property. The corresponding property of the matrix is its determinant: - The
*angle of rotation*. It does not depend on the coordinate basis. The corresponding invariant of the matrixis the**W***trace*, it is defined by. The rotation angle of the rotation or of the rotation part of a rotoinversion can be calculated from the trace by the formula

The sign is used for rotations, the sign for rotoinversions.

- The preservation of the `handedness' of an object,

One can list this correlation in a table

tr()W |
3 | 2 | 1 | 0 | 0 | 1 | ||||

type | 1 | 6 | 4 | 3 | 2 | |||||

order | 1 | 6 | 4 | 3 | 2 | 2 | 6 | 4 | 6 | 2 |

By this table the type of operation may be found, as far as it is
determined by the matrix part. For example, one takes from the table
that a specific operation is a two-fold rotation but one does not
know if the operation is a rotatation or a screw rotation, what the
direction of the rotation axis is and where it is located in space.
This characterization will be done in the following list for the
*crystallographic symmetry operations*.

- Type 1 or : no preferred direction
- All other symmetry operations have a preferred axis (the
*rotation*or*rotoinversion axis*). The directionof this axis may be determined from the equation**u**

The sign is for rotations, the sign for rotoinversions.For type , reflections or glide reflections,

is the direction of the normal of the (glide) reflection plane.**u** - If
is the matrix of a rotation of order or of a reflection (), then , and one determines the**W***intrinsic translation part*, also called*screw part*or*glide part*by*screw*or*glide vector*. This vector is invariant under the symmetry operation: : Indeed, multiplication withpermutes only the terms on the right side of equation 5.2.5. Thus, the screw vector of a screw rotation is parallel to the screw axis. The glide vector of a glide reflection is left invariant for the same reason. Therefore, it is parallel to the glide plane.**W**If

The column is called the=**t**holds, then (**o**,**W**) describes a**w***rotation*or*reflection*. For , (,**W**) describes a**w***screw rotation*or*glide reflection*. One forms the so-called*reduced operation*by subtracting the*intrinsic translation part*/k from (**t**,**W**):**w**

*location part*because it determines the position of the rotation or screw-rotation axis or of the reflection or glide-reflection plane in space.If

is a diagonal matrix,**W***i.e.*if only the coefficients are non-zero, then either is and is a screw or glide component, or and is a location component. Ifis not a diagonal matrix, then the location part has to be calculated according to equation 5.2.6.**W** - The
*fixed points*are obtained by solving the equation

Equation (5.2.7) has a unique solution for all rotoinversions (including , excluding ). There is a 1-dimensional set of solutions for rotations (the rotation axis) and a 2-dimensional set of solutions for reflections (the mirror plane). For screw rotations and glide reflections, there are no solutions: there are no fixed points. However, a solution is found for the reduced operation,*i.e.*after subtraction of the intrinsic translation part, by equation 5.2.8

The formulae of this section enable the user to find the geometric contents
of any symmetry operation. In reality, IT A have provided the necessary
information for all symmetry operations which are listed in the plane-group
or space-group tables. The entries of the General position are numbered.
The geometric meaning of these entries is listed under the same number in
the block **Symmetry operations** in the tables of IT A. The explanation
of the symbols for the symmetry operations is found in Sections 2.9 and 11.2
of IT A.

The section shall be closed with an exercise.

Problem 2B. Symmetry described by matrix-column
pairs.

For the solution, see p. .

The matrix-column pairs (* A*,

Question

**(viii)**- Determine the geometric meaning of the
matrix-column pairs
(

,**A**), (**a**,**B**), (**b**,**C**), and (**c**,**D**).**d**

**Copyright © 2002 International Union of
Crystallography**