(Part Six)

We will now investigate the generation of the Plane Group

Placing motifs with **2mm** symmetry in a *centered* rectangular net (= centered rectangular 2-D lattice) yields the Plane Group **C2mm**. See Figure 1.

Figure 1. *Placing motifs with point symmetry ***2mm*** in a centered rectangular net, yields a periodic array of motifs representing Plane Group ***C2mm***.
As in all cases, the pattern must be imagined to be extended indefinitely across the two-dimensional plane.*

Figure 2. *A unit mesh choice is given (yellow). This unit mesh has point symmetry ***2mm***, and is centered.*

If we eliminate all translations from the pattern representing Plane Group

Figure 3. *The translation-free residue of the pattern representing Plane Group ***C2mm***. The resulting figure represents Point Group ***2mm***. *

The unit mesh, as given in Figure 2, contains two motifs s.str. (1 + 1/4 + 1/4 + 1/4 + 1/4). The whole unit mesh with its content can be considered as the motif s.l. that tiles the 2-D plane completely.

Patterns representing the Plane Group **C2mm** have 2-fold rotation axes, mirror lines and glide lines. See Figures 4---7.

Figure 4. *A pattern representing Plane Group ***C2mm*** has mirror lines in the ***x*** and ***y*** directions. Some of them are indicated *(**m**)*.*

Figure 5. *A pattern representing Plane Group ***C2mm*** has 2-fold rotation axes perpendicular to the plane of the drawing. Some of them are indicated (red dots).*

Figure 6. *A pattern representing Plane Group ***C2mm*** has glide lines.
One of them is indicated *(

Figure 7. *A pattern representing Plane Group ***C2mm*** has glide lines. Some of them are indicated *(**g**)*.*

The

Figure 8. *Total symmetry content of Plane Group ***C2mm***.
2-fold rotation axes perpendicular to the plane of the drawing are indicated by small solid red ellipses.
Mirror lines are indicated by solid lines (*

The next Figure shows three generators that can generate the whole pattern **:**

The element **a** that results from the element **1** (chosen to represent the identity element) by a reflection in the line **a** .

The element **b** that results from the element **1** by a reflection in the line **b** .

The element **s** that results from the element **1** by a half-turn about the point **S** .

These three generators together can produce all other group elements (each such element may be represented by a motif unit).

Figure 9. *The three chosen generators for the C2mm pattern.*

Figure 10. *The C2mm pattern can be generated by the elements ***a, b*** and ***s*** .
The right part of the pattern can be reached as follows :
From the generated element *

As has been said earlier, we can consider each mesh plus content as a motif

Figure 11. *The motifs ***s.l.** (red and blue rectangles) of the **C2mm*** pattern, as depicted in Figure 1, tesselate the whole plane.*

The next Figure depicts one motif

Figure 12. *A motif ***s.l.*** of the ***C2mm*** pattern of Figure 1.*

The following two Figures are about dividing the motifs

The next Figure shows a partition of the motifs s.l. that does not yet satisfy this demand.

Figure 13. *Partition of the motifs ***s.l.*** in order to obtain areas that represent group elements. Each area resulting from this partition (red and blue rectangles) contains two motif units (s.str.), and so does not represent a group element. The areas must be divided again, as is done in the next Figure.*

Figure 14. *Correct partition of the ***C2mm*** pattern of Figure 1, such that each resulting area contains one motif unit (s.str.) and so can -- as area + contents -- represent a group element.*

We will now explicitly indicate the initial group element and the three generator elements.

Figure 15. *Areas of the ***C2mm*** pattern of Figure 1 representing group elements. Some elements are explicitly indicated :
The initial group element *

The group element representing the first generator

The group element representing the second generator

The group element representing the third generator

We will now successively generate group elements of the displayed part of our

Figure 16. *Generation of the group elements ***ba*** (by reflecting the element ***a*** in the mirror line ***b*** ), and ***sba, sa, sb*** , by means of the generator ***s*** , which is a half-turn about the point ***S** .

Figure 17. *Generation of the group elements ***bs, bsba, bsa, bsb*** , by means of the generator ***b*** , which is a reflection in the mirror line ***b** .

Figure 18. *Generation of the group elements ***as, asa, asb, asba, absb, absba, abs, absa*** , by means of the generator ***a*** , which is a reflection in the mirror line ***a** .

Figure 19. *Generation of the group elements ***sabsba, asabsba, bsabsba, absabsba*** , by means of the generators ***s*** , ***a*** and ***b** .

Figure 20. *Generation of the group elements ***sasba, bsasba, asasba, basasba*** , by means of the generators ***s*** , ***a*** and ***b** .

Figure 21. *Generation of the group elements ***sbsba, asbsba, basbsba, bsbsba*** , by means of the generators ***s*** , ***a*** and ***b** .

The generation process can and must be continued indefinitely. To reach the right-hand part of the displayed pattern, we can, for example subject the generated group element

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