APPENDIX General definition of geometric elements

The assignment in Table 1 of geometric elements to symmetry operations has been performed in such a way that symmetry elements are in accordance with crystallographic tradition. The difference from that tradition is, firstly, that the two-sided character of symmetry elements has been made explicit and unambiguous through the introduction of geometric elements and element sets. The second difference, in the present definition, is that the geometric item is defined primarily for each single operation whereas existing definitions of `symmetry element' tend to be associated with a (cyclic?) group of operations. For instance: `The geometrical locus about which a group of repeating operations act is called a symmetry element' (Buerger, 1963); or `It is imperative to distinguish between symmetry operations (group elements) and symmetry elements (cyclic groups)' (Donnay $\&$ Donnay, 1972).

The Ad-hoc Committee had previously discussed proposals to establish such a group assignment, but had found them awkward when applied to space groups in which non-cyclic site symmetries occur. Existing definitions, such as the two just cited, do not specify exactly which group belongs to a given geometric item. Therefore, it was decided to drop the group association and first define the `geometric element' for each given operation.

The obvious aim of such a definition is to provide a `space anchor' for the orientation and the location of the operation. With that aim in view, there is actually little choice. For point-group operations only the orientation is needed, whereupon the real eigenvector(s) of the operation yield the geometric elements of Table 1 almost uniquely (not quite uniquely: the eigenvector with negative eigenvalue of the reflection, for instance, can be omitted but that of a rotoinversion cannot). Extension to space-group elements, fixing their location as well, is also easy but not entirely automatic.

The question hence arises whether a general definition can be given for space groups as well. The purpose of this Appendix is to review some of the past attempts in that direction, and to present a new interpretation ( $\S A$ 3). The latter produces Table 1 directly.

In the past, a symmetry element has sometimes been described as a line, plane or point which is invariant for (that is, not displaced by) a symmetry operation. Stated in that form, such a description is useless. For example, a reflection leaves not only the mirror plane in place, but also all planes perpendicular to it, which are clearly not wanted. Hence the line etc. must not only be invariant as a whole, but every point on it must remain in place. A definition of the geometric element for a given symmetry operation may thus be `the subspace consisting of invariant points' (`SCIP'). An obvious difficulty then arises in dealing with operations such as screw rotations and glide reflections, for which no point whatever is invariant. The solution for this problem is to remove from the given operation its `intrinsic translation', the latter being parallel to an invariant direction of the operation. The reduced operation thereby results, with a fully determined position for the reflection plane, or the axis of rotation, which now becomes the geometric element.

This term also applies to operations that have zero intrinsic translation: a (roto-) inversion or a pure reflection or rotation. For these, the given operation is identical to its reduced operation. The other extreme occurs when the given operation is a mere translation. In that case the reduced operation is just the identity. The geometric element of any given operation is hence defined as the `subspace consisting of all invariant points for the reduced operation' (`SCIPRO').

The SCIPRO definition of geometric elements apparently yields the results given in Table 1, but a severe discrepancy remains for the category of rotoinversions for which it yields just one point. In Table 1, the traditional axis line has been added because otherwise the rotoinversion would not be oriented. In all other categories, the reduced operation is completely located as well as oriented by the geometric element as defined by SCIPRO. However, the fact that SCIPRO fails for rotoinversions shows that the `invariant-points' reasoning, which led to the SCIPRO definition, is basically inadequate.

A new interpretation of the geometric element assigned to a given symmetry opration W by Table 1 follows:

Consider the Euclidean normalizer of W, that is, the group of all congruence operations which `commute with W'; in other words, which transform W into itself. The ten different types of normalizer for all possible operations W are listed in Table 3 (two types each for rotations and screw rotations as shown in the footnote to Table 3, and one type for each of the remaining six kinds of operation). All groups are noncrystallographic and continuous. In the case of a reflection in a plane, for example, the normalizer contains all parallel translations, reflections in all perpendicular planes, etc. The short and rather obvious description in the second column of Table 3 is, however, sufficient to yield the invariant subspaces listed in the third column. This latter column is in accordance with the second column of Table 1. The ensuing definition hence becomes: the geometric element of a symmetry operation W consists of the subspace(s) invariant for all operations belonging to the Euclidean normalizer of W.

**Table 3:** *Euclidean normalizers of symmetry operations and their invariant subspaces*
Symmetry operation W	Euclidean normalizer N of W consists of	Invariant subspace(s) of N
Identity	All congruences	None
Translation, vector t	Those congruences which conserve vector t	None
Reflection in plane A	Those congruences which conserve plane A	Plane A
Glide reflection in plane A, glide vector v	those congruences which conserve plane A and vector v	Plane A
Rotation about line b	Those congruences which conserve line b*	Line b
Screw rotation about line b, screw vector u	Those congruences which conserve line b and vector u*	Line b
Rotoinversion with respect to line b and point P	Those congruences which conserve line b and point P, as well as the sense of rotation	Line b, point P, and plane perpendicular to b through P
Inversion	Those congruences which conserve point P	Point P

*If the rotation angle is not 180 $^{\circ}$ , then the sense of rotation must also be conserved. The normalizer then lacks mirror planes through b, for instance. However this does not change its invariant subspaces.

(i) The occurrence of three items for the rotoinversion (instead of two in Table 1) is not a discrepancy: if the point is invariant, invariance of the line follows from that of the plane, and vice versa. Hence, one of the latter two is redundant.

(ii) `Subspace' should be taken in the proper sense, because in the improper sense (`all space') it is invariant for any congruence. It would have to be added to all geometric elements but would not increase their information content.

(iii) The invariance need only obtain for the subspace as a whole, not necessarily pointwise as required in the SCIPRO definition.

(iv) It should be noted that the congruence operations referred to above are operations in point space (see IT A83, $\S$ 8.1.5), not vector space. The given definition of geometric elements hence applies to symmetry operations in point space only, not to those in vector space.