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Further Examples Including More Complicated Space Groups

Once a bit of experience has been acquired in using these patterns to learn about space groups it is tempting to consider some more complicated cases. The examples in Plates (vii) to (xv) provide more opportunities to `discover' some of the idiosyncrasies of space groups. To get the full benefit of such studies, aids, if at all necessary, should be limited to a simple list of the 230 space groups in standard notation. (The International Tables ,⁵ however useful they are in general, should not be used in this connection except perhaps for the occasional checking of results.)

In the patterns based on feet the tacit assumption is made that they are all looking the same way up. When using hands for patterns it is somewhat simpler to indicate which side of the object is looking upwards. As has already been shown in Fig. 2 to 4 a coin (white dot) in a hand is used as a simple indication that the palm of the hand is oriented upwards (otherwise it is oriented downwards). Thus, patterns (vii) and (ix) could also be composed of feet (as before) but not the other designs based on hands.

The space groups of the remaining patterns, from (vii) onwards, are not directly disclosed in the text portion of this account in consideration of readers who prefer to find out themselves. (See References and Notes for solutions.) Patterns (x) to (xv) can also serve for more advanced exercises. Particularly in case of such patterns of higher symmetry it is not so simple to draw up complete diagrams of symmetry elements (i.e. without missing any of the symmetry). When this appears possible the following procedure should be incorporated in the exercise. After having decided upon a unit cell and space group it is a simple matter to determine p, the number of equipoints for a general position, since this equals the number of hands per unit cell. Hands marking equivalent (but not identical) positions are then numbered from 1 to p. The main part of the exercise is then to determine the space group operations for $1 \rightarrow 2$, $1 \rightarrow 3$ $\dots$ $1 \rightarrow p$. In this way, symmetry elements which may have been overlooked before are likely to emerge.

Other questions which arise naturally in exercises based on space group patterns pertain to

• the point symmetry of special positions,
• the choice of the origin (cf. patterns (x) and (xii)),
• alternate settings of the space group, etc.

Pattern (viii) provides another example of a space group in a non-standard setting. Two of the patterns have been included to show that space groups like $P\overline{4}2c$ and $P\overline{4}c2$ or $P\overline{4}2m$ and $P\overline{4}m2$ do not differ merely with respect to the setting as one might be inclined to think on the basis of the corresponding crystal class $\overline{4}2m$, but are truly different space groups.

Plate (vii) to (xv). Worksheets for practical exercises.

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Plate vii	Plate viii	Plate ix
Plate x	Plate xi	Plate xii
Plate xiii	Plate xiv	Plate xv

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