The basics of crystallography and diffraction (3rd Edition)

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76 Two-dimensional patterns, lattices and symmetry
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(b)
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(c)
Fig. 2.15. The additional symmetry operations for the 52 layer-symmetry groups applicable to woven
fabrics (plus the counterchange symmetry operations 2 ′ and 4 ′ (Fig. 2.11). The ‘face’ and ‘back’ of the
R symbols are shown here as black and white, respectively.
(a) Operation of an in-plane diad axis (double arrow-head) (identical to counterchange symmetry element
m ′ —see Fig. 2.11) and (b) an in-plane screw diad axis (single arrow-head). (c) Operation of in-plane
glide planes (dashed lines) for three different orientations of the glide directions—along the axes and
diagonally. (Drawn by K. M. Crennell.)
with the plane; both reflection (mirror) planes (which are not applicable to woven textiles
because the back of the cloth is not identical to the front 6 ) and glide-reflection planes
(which are applicable to woven textiles). These symmetry operations are also shown
in Fig. 2.15.
We will now apply these ideas to the plain weave and twill illustrated in Figs 2.13 and
2.14. The ‘weave repeat’ is the smallest number of warp and weft threads on which the
weave interlacing can be represented; it is a 2 × 2 square for the plain weave (Fig. 2.13)
anda3×3 square for this example of a twill weave (Fig. 2.14). It is important to note that
these weave repeat squares do not correspond with the unit cells of the plane patterns.
These unit cells and the plane group symmetry elements are also shown in Figs 2.13
and 2.14. As can be seen, the plain weave has plane symmetry p4gm and the twill plane
symmetry p2 (see Fig. 2.6).
Figures 2.13 and 2.14 also show the unit cells and layer symmetry elements for these
two weaves and the standard notation (which we will not describe in detail) which goes
with them. Notice that for the plain weave that the unit cell is identical to that for the
plane group symmetry but for the twill it is different—the primitive (p) lattice becomes
a centred (c) or diamond lattice. Notice also that the layer-group symmetry of the plain
weave is much more ‘complicated’ than that of the twill. It includes tetrad rotationreflection
axes perpendicular to the plane as well as diads and screw diads within the
plane. The twill, by contrast, has no mirror lines of symmetry at all.
6 This restriction further reduces (by 12) the number of layer-symmetry groups applicable to woven textiles,
leaving a total of 80 − 16 − 12 = 52. Still a lot!