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If matrix *A* represents a type I symmetry operation, we can calculate the
rotation angle from the value of the *A* matrix trace. We must
remember that the trace of *A* is invariant with respect to a base system
transformation.

In a lattice base system the trace is an integer number, since the elements of
the matrix are integers. In an orthonormal base system, the counter-clockwise
rotation of an angle , for example, around the *z* axis is given by:

and then the trace is equal to .

We have then: an integer, from which it is seen that the values of compatible with a lattice base system are: 60,90, 120, 180, 240, 270,300, 360.

**Copyright © 1980, 1998 International Union of
Crystallography**