(Part Seven)

We will now investigate the generation of the Plane Group

**Square Net**

The Square Net (2-D Square Lattice) has point symmetry **4mm**, so it does support 4-fold rotation and two types of mirror line. Consequently it surely can accommodate the insertion of a motif that has **4mm** symmetry. This results in a periodic pattern that represents the Plane Group **P4mm**. See Figure 1.

Figure 1. *Placing 2-D motifs with point symmetry ***4mm*** into a square 2-D lattice yields a pattern that represents the Plane Group ***P4mm***.*

Figure 2. *A unit mesh choice is given in yellow. This unit mesh contains four 1/4 motifs, which is equivalent to it containing precisely one motif. This unit mesh, not containing a motif in its center, is primitive and is denoted by the symbol ***P***. Its point symmetry is ***4mm*** .*

The motif s.str. and motif s.l. are depicted in the next Figure.

Figure 3. *Repeated motif with point symmetry ***4mm*** in the pattern of Figure 1.
This is, in the present case, also the figure that emerges when all translations are eliminated. Its point symmetry is *

Besides simple translation the periodic pattern representing the Plane Group

Figure 4. *One of the possible glide lines in the pattern of Figure 1.*

The

Figure 5. *Total symmetry content of the Plane Group ***P4mm***.
All continuous lines, including those that represent (the lines connecting the nodes of) the net, are mirror lines. There are two types of them, diagonal and non-diagonal. Dashed lines signify glide lines, small solid ellipses signify 2-fold rotation axes perpendicular to the plane of the drawing, and small solid squares signify 4-fold rotation axes perpendicular to the plane of the drawing.*

In the realization of a **P4mm** pattern as given in Figure 1, we see that each motif **s.str.** consists of eight motif units (partly overlapping commas). Each such motif unit can represent a group element.

To generate the pattern we can choose the following **generators** **:**

An anticlockwise quarter-turn (90^{0} rotation) about some chosen point.

A reflection in some mirror line.

A horizontal translation.

The next Figure shows **:** The motif unit chosen as initial group element (identity element), the group element representing the translation (generator), the group element representing the quarter-turn (generator) and the group element representing the mirror reflection (generator). The symmetry elements associated with the latter two transformations are also given **:** The point about which the chosen quarter-turn takes place, and the line in which reflection takes place.

Figure 6. *Each motif ***s.str.*** of the ***P4mm*** pattern of Figure 1 consists of eight motif units. Each such unit can represent a group element. In the present Figure four such elements are explicitly indicated :
The initial group element *

The group element representing an anticlockwise quarter-turn

The group element representing the reflection

The group element representing the horizontal translation

The next Figure shows how two more group elements (motif units) are generated by repeatedly appling the quarter-turn

Figure 7. *Generation of the group elements (motif units) ***q ^{2}**

Figure 8. *Some more group elements (motif units) of our ***P4mm*** are generated.*

Instead of concentrating further on the units of the motifs

For this purpose we first determine the

Figure 9. *Motifs ***s.l.** (red and blue squares) of the **P4mm*** pattern of Figure 1.
( The red and blue colors do not signify qualitative differences)*

The next Figure divides each motif s.l. into eight parts, each of them representing a group element.

Figure 10. *Division of the motifs ***s.l.*** results in (red and blue) triangular areas each representing a group element.
( The colors red and blue do not signify qualitative differences)*

In the above Figure we're now going to indicate the initial group element and the three generators as established in Figure 6, and insert the relevant symmetry elements (mirror line and rotatation point) associated with the generators.

Figure 11. *The ***P4mm*** pattern of Figure 1 is partitioned (as it was already in the previous Figure) into triangular areas, each representing a group element. Four such elements are explicitly indicated :
The group element *

The group element

The group element

The group element

We will now start to generate more group elements, by applying the generators to the initial group element, and (again) to the results.

Figure 12. *Some group elements of the ***P4mm*** pattern are generated by using the defined generators.*

The next Figures depict our

Figure 13. *To reach the first row of group elements, we must use the generator ***q*** .
So we take the already generated element *

Figure 14. *In the same way we take the already generated element ***tq ^{2}m**

Figure 15. *In the same way we take the already generated element ***tq ^{3}**

Figure 16. *Again, in the same way we take the already generated element ***tqm*** and rotate it anticlockwise by 90 ^{0}, resulting in the element *

By using the translations

Figure 17. *The (displayed part of the) first row of group elements is completed by applying translations.*

The second and third rows of group elements can also be completed by applying translations

Figure 18. *More group elements are generated, completing the first three rows of the displayed part of the pattern ( P4mm ).*

To reach the fourth row of group elements, we must again apply the rotation

Figure 19. *Applying ***r ^{3}**

By appling

Figure 20. *The element ***q ^{3}tq^{2}m**

We choose the generator

Figure 21. *The fourth row of the displayed part of our ***P4mm*** pattern is completed by applying the generator ***m*** to the elements of the first row.*

To reach the fifth row of group elements we must again apply the element

Figure 22. *The group element ***q ^{3}t**

Figure 23. *In the same way as in the previous Figure some more elements are generated.*

Figure 24. *Again, in the same way as in the previous Figure some more elements are generated.*

Figure 25. *Finally, in that same way, one more element is generated.*

We can now complete the last two rows of the displayed part of our

Figure 26. *Completion of the last two rows of group elements in the displayed part of the ***P4mm*** pattern (as represented in Figure 1) (with the just mentioned "*as represented*" we mean the choice of motifs s.str. We could have chosen different motifs as long as they have ***4mm*** symmetry)*.

So we have now generated the group elements of the Plane Group

In the

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