(Part Five)

We will now investigate the generation of the Plane Group

The (primitive) Rectangular Net, which itself has **2mm** (point) symmetry, and is consequently compatible with **2mm** motifs, can also accommodate for motifs having a lesser degree of point symmetry, provided that their symmetry elements are aligned with the corresponding symmetry elements of the net. Such a motif can either have a point symmetry of **2**, i.e. having a 2-fold rotation axis as its only symmetry element, or have a point symmetry of **m** , i.e. having a mirror line as its only symmetry element. We'll consider the latter type of motif.

Two such motifs are placed in a mesh of a primitive rectangular net as follows **:**

Figure 1. *Two motifs, each having point symmetry ***m*** , are placed in a mesh of a primitive rectangular lattice, as indicated ( The Figure depicts one mesh only. The green lines indicate the exact location of the motifs in such a mesh).*

The next Figure depicts a regular periodic pattern of motifs, that have a symmetry

Figure 2. *If we place two motifs, having a symmertry of ***m*** , in each mesh of the primitive net, as indicated in Figure 1, then we will obtain a periodic pattern of motifs representing the Plane Group *

Figure 3. *For the pattern of Figure 2 a unit mesh is chosen (yellow). This unit mesh has point symmetry ***2***, and is primitive *(**P**)*.Point *

The next Figure gives the motif s.str. and the motif s.l. of the pattern of Figure 2. The motif s.l. is repeated indefinitely across the two-dimensional plane.

Figure 4. *The motif s.str. (black) and the motif s.l. (black + blue) of the pattern representing Plane Group ***P2mg*** . The motif s.l. is indefinitely repeated along the directions of the 2-D lattice.*

To determine the translation-free residue (i.e. to determine the point group symmetry) of the pattern representing the Plane Group

Figure 5. *Eliminating all translations yields a figure that has a point symmetry ***2mm*** , which represents the translation-free residue of the Plane Group ***P2mg*** .*

In the pattern, representing the Plane Group

The next Figure indicates some of the 2-fold rotation axes.

Figure 6. *Some of the 2-fold rotation axes, belonging to the symmetry content of the Plane Group ***P2mg*** , are indicated. The colored lines serve to show that there indeed are 2-fold rotation axes perpendicular to the plane of the drawing.*

Figure 7. *The pattern representing Plane Group ***P2mg*** has mirror lines parallel to the ***y*** direction. One of them is depicted.*

Figure 8. *The pattern representing Plane Group ***P2mg*** has glide lines parallel to the ***x*** direction. One of them is depicted.*

The total

Figure 9. *Total symmetry content of the Plane Group ***P2mg*** .
Solid red lines indicate mirror lines.
Small solid red ellipses indicate 2-fold rotation axes perpendicular to the plane of the drawing.
The glide lines are all parallel to the *

The P2mg pattern, as realized in Figure 2, consists of composed motifs. Each such motif itself consists of two motif units -- partly overlapping commas -- which represent group elements (and each group element is a symmetry transformation of the pattern).

For generating the P2mg pattern, and with it the (elements of the) group P2mg, we choose the following set of generators **:**

The element **g** which results from the (chosen) initial motif unit (representing the identity element) **1** as the effect of the glide reflection (glide line) **g** .

The element **m** which results from the reflection of the initial motif unit in the mirror line **m** .

The element **t** which results from the initial motif unit in virtue of a horizontal translation **t** . See next Figure.

Figure 10. *Three generators for the P2mg pattern are indicated.*

The next Figure shows how all the group elements can be generated from the three generators

Figure 11. *Generation of the P2mg pattern from the three generators.*

In accordance with Figure 4 we now tesselate the displayed part of our P2mg pattern with motifs

Figure 12. *Tiling of the plane with motifs s.l. (red and blue rectangles) of the ***P2mg*** pattern of Figure 2.
( The red and blue colors of the motifs s.l. do not indicate qualitative differences).*

We can now divide the motifs s.l. into areas which are supposed to represent group elements. The next Figure divides the motifs s.l., but closer inspection reveals that the way of division is not correct. We show it anyway because it is instructive.

Figure 13. *Division of the motifs ***s.l.*** in order to obtain areas that represent group elements and tile the plane completely. As one can see, however, the division is not correct : From Figure 11 we can see that each group element is associated with one motif unit, i.e. with one half of a motif s.str. In the present Figure we see subareas (resulting from the division) that contain such a half-motif (s.str.). However, there are also areas (supposed to be equivalent) not containing such a half-motif. So the plane is not divided into tesselating areas, such that each of them represents a group element.*

The next Figure shows the correct division of the motifs s.l.

Figure 14. *Correct division of the motifs ***s.l.*** of Figure 12.
Each resulting subarea (red and blue slender rectangles) contains one half-motif (s.str.), and consequently represents a group element. These subareas also tesselate the plane completely.*

The next Figure again shows this division but now with the initial group element

Figure 15. *The tiles (red and blue rectangles) of the plane, as indicated in this and the previous Figure, represent (with their shape and content) group elements. Three of the latter are specifically indicated :
The initial group element *

The group element representing the glide reflection

The the group element representing the reflection

The group element representing a horizontal translation

(See also Figure 10)

The next Figure shows the generation of some group elements, by (repeatedly) applying the generator

Figure 16. *Generation of the group elements ***g ^{-2} , g^{-1} , g^{2} , g^{3}** .

The next Figure shows the generation of some more group elements, by (repeatedly) applying the generator

Figure 17. *Generation of the group elements ***g ^{-2}m , g^{-1}m , gm , g^{2}m , g^{3}m** .

We can now generate all the remaining group elements (as far as the displayed part of the pattern is concerned) by repeatedly applying the horizontal translation

Figure 18. *Generation of the group elements *** t ^{-1} , t^{-1}g^{-2} , t^{-1}g^{-1} , t^{-1}g^{2} , t^{-1}g^{3} , **

tg^{-2} , tg^{-1} , tg^{2} , tg^{3} , t^{-1}g^{-2}m , t^{-1}g^{-1}m , t^{-1}g^{2}m , t^{-1}g^{3}m , tg^{-2}m , tg^{-1}m , tg^{2}m , tg^{3}m , t^{-1}gm , t^{-1}g , t^{-1}m , tgm , tg , tm .

So we have now generated all group elements of the displayed part of the

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