Sequel to Group Theory

We'll start with reminding the reader about the

Sequel to Infinite two-dimensional periodic patterns

Figure 1. *Placing symmetric motifs in a centered rectangular net, such that each lattice node is associated with such a motif, yields a pattern representing Plane Group ***Cm***. The pattern must be conceived as extending indefinitely in two-dimensional space.*

In the pattern representing Plane Group

The **total symmetry content** of the Plane Group **Cm** is shown in the next Figure.

Figure 2. *Total symmetry content of the Plane Group ***Cm***.
Glide lines are indicated by red dashed lines.
Mirror lines are indicated by *

For clarity the nodes of the net are indicated (black dots).

The next Figure shows a set of two chosen generators

The element

The element

Figure 3. *The two chosen generators of the group Cm .*

These two generators together can generate the whole pattern

Figure 4. *For a number of motif units their generation is indicated.
Continuing this process will yield the whole Cm pattern.*

Figure 5. *Placing motifs with point symmetry ***2mm*** in a centered rectangular net, yields a periodic array of motifs representing Plane Group ***C2mm***.
In our representation each (composed) motif consists of four motif units (commas), such that the composed motif has *

As in all cases the pattern must be imagined to be extended indefinitely across the two-dimensional plane.

The

Figure 6. *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

The element

The element

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

Figure 7. *The three chosen generators for the C2mm pattern.*

Figure 8. *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 *

Figure 9. *A periodic pattern of motifs based on a 2-D square lattice (net). This particular pattern represents the Plane Group ***P4gm***. It must be imagined to extend indefinitely over the 2-D plane. In the present representation each (composed) motif consists of four motif units (commas), such that the symmetry of the composed motif is ***2mm*** . Each motif unit represents a group element. *

In the next Figure we depict the

Plane Group

Figure 10. *Total symmetry content of the Plane Group ***P4gm***.
Glide lines are indicated by red dashed lines.
Mirror lines are indicated by red solid lines.
4-fold rotation axes are indicated by small red solid squares.
2-fold rotation axes are indicated by small red solid ellipses.*

The next Figure depicts two chosen generators

The element (motif unit)

The element (motif unit)

Figure 11. *The two chosen generators of the group P4gm (and with it of the corresponding pattern, where the generated group elements are represented by motif units).*

Figure 12. *The P4gm pattern can be generated by the two elements ***m*** and ***p**.

Figure 13. *When motifs, having point symmetry ***6mm*** (i.e. having a 6-fold rotation axis and two types of mirror lines), are inserted into a (primitive) hexagonal net, a pattern of repeated motifs will emerge that represents Plane Group ***P6mm***.
Also here the unit mesh is rhomb-shaped with angles of 60 ^{0} and 120^{0}.
In the present representation each (composed) motif consists of six motif units, together making up a composed motif with *6mm

The

Figure 14. *Total symmetry content of the Plane Group ***P6mm***.
All rotation axes are perpendicular to the plane of the drawing.
6-fold rotation axes are indicated by small blue solid hexagons.
3-fold rotation axes are indicated by small blue solid triangles.
2-fold rotation axes are indicated by small blue solid ellipses.
Glide lines are indicated by red dashed lines.
Mirror lines are indicated by solid lines (red and black).*
(

Figure 13 gives an accurate representation of the Plane Group P6mm. All the symmetries of the group are present in the image of that Figure. So for analytic purposes (analytic group theoretic approach) the representation is appropriate.

However, for a

But we also see that each motif unit itself is still composed, namely (composed) out of two symmetrical halves. So such a symmetrical motif unit cannot represent a group element, it represents already two group elements. The genuine motif unit, representing a group element, is therefore half the motif unit as set initially. This means that each whole composed motif, as depicted above (and in Figure 13), consists not of six, but of twelve genuine motif units. And these newly conceived motif units are indeed asymmetric, that is to say they cannot be further divided anymore. And they indeed can represent the group elements **:**

To highlight these genuine motif units more clearly, we can use colors, *provided we do not interpret the difference in color to represent an asymmetry* **:** one motif unit (red or blue) is asymmetric. Two of them together (one red and one blue) make up a *symmetric* entity (representing not a group element but a subgroup of the full group), and twelve of them (six red, six blue) make up the full composed hexagonal motif **:**

The next Figure has such motifs be placed in a hexagonal net (point lattice) **:**

Figure 14a. *A representation of a P6mm pattern, in terms of genuine basic motif units (red and blue). Here the difference in color should not be interpreted as asymmetry, it only serves to indicate the basic motif units (making up the hexagonal composed motifs). Each such composed motif is supposed to have 6mm symmetry, i.e. it has D_{6} structure (not C_{6} structure). It has therefore six rotations and six reflections. Each (genuine) motif unit (either red or blue) represents a group element of the Plane Group P6mm .*

In order to

A generator element **m** , resulting from a reflection of the initial motif unit in a mirror line **m** .

A generator element **p** , resulting from a 60^{0} anticlockwise rotation of the initial motif unit about the point **R** (See next Figure).

A generator element **t** , resulting from a horizontal translation **t** of the initial motif unit.

Figure 14b. *Three generators, ***m, p*** and ***t*** , represented by motif units, and one initial motif unit, representing the identity element, are chosen. The motif unit ***p*** results from a 60 ^{0} anticlockwise rotation of the initial element *

For clarity we enlarge the region around the point R

Figure 14c. *The composed motif (enlarged) from the P6mm pattern of the previous Figure at the point ***R*** . It consists of twelve true motif units (red and blue), representing group elements. The difference in color should (here) not be interpreted as representing asymmetry.
The motif units *

The other motif units of the composed motif of Figure 14c are then automatically implied

Figure 14d. *The composed motif and its twelve constituent basic motif units, at the point *R* of Figure 14b. These basic motif units represent the group elements ***1, m, p, mp, mp ^{2}, mp^{3}, mp^{4}, mp^{5}, p^{2}, p^{3}, p^{4}**

In order to further generate the pattern (and with it the group elements) we can subject the elements of the subgroup D

Figure 14e. *Some new group elements resulting from the translation ***t*** are indicated. Also some points, bearing composed motifs, of the lattice are indicated (R , Q , U, W).*

The next Figure shows the composed motif

Figure 14f. *The composed motif next to the right of the one at the point R .*

The red motif units in Figure 14f together form the

All the twelve elements (red and blue in Figure 14f ) together form the

Next we're going to generate the composed motif at the point **Q** in Figure 14e. It can be obtained by rotating the previously generated composed motif (the motif next to the one at point R) 300^{0} anticlockwise (or, which is the same, 60^{0} clockwise) about the point R, which means that we subject all elements of the previously generated composed motif to the action of **p ^{5} **.

The next Figure gives this new composed motif. Inside the image we have left the notations for the elements as they were in the motif next to R, while the identities of the

Figure 14g. *The newly generated composed motif at lattice point Q (as indicated in Figure 14e). The names of the new elements are given at the perimeter of the image (i.e. outside the image).*

All the elements in Figure 14g (red and blue) together form the

The third row (Figure 14e) of composed motifs can now be completed by means of applying to this lastly obtained composed motif the translations **. . . t ^{-3}, t^{-2}, t^{-1}, t, t^{2}, t^{3} . . .** . And along the same lines we can complete row 2 of the pattern.

In order to reach the **fourth row** we first determine the composed motif at the point **U** by applying **t ^{2}** ( = 2 times applying the translation

and then rotate it 300^{0} anticlockwise about the point R, i.e. applying **p ^{5}**

Figure 14h. *The composed motif at the point W (Figure 14e) is completed, by applying ***p ^{5}**

The six red motif units of the above Figure (Figure 14h) together form the

This fourth row can now be completed by means of translations.

The same

Figure 14i. *Composed motif (of the P6mm pattern) consisting of six symmetric motif units. Two of them are provided with a notation. Each such motif unit in fact consists of two symmetrically related basic motif units. These non-basic motif units cannot represent group elements. The one denoted by *

We will now generate the

Figure 14j. *By the generator ***p*** the remaining elements of the subgroup D _{6} are generated : Given *

The next Figure summarizes the two chosen generators,

The element (augmented motif unit)

The element (augmented motif unit)

We must realize that, although speaking about "

If we were to insist that they are true group

Figure 15. *Two chosen generators, ***p*** and ***t*** for the P6mm pattern.
In one of the composed motifs (which are of course all identical) its composition out of six (augmented) motif units is indicated by colors.*

Figure 16. *Generation of the group P6mm by the generators ***p*** and ***t*** . See also next Figures.*

The next Figures explain some of the generative relations between some group 'elements' (augmented motif units) established by applying the generator

Figure 17. *Image ***p ^{5}t^{2}** -- under (the action of)

The action of

Figure 18. *Image ***p ^{5}t^{2}p** -- under (the action of)

The action of

Figure 19. *Image ***p ^{5}t^{2}p^{2}** -- under (the action of)

The action of

Figure 20. *Image ***p ^{5}t^{2}p^{3}** -- under (the action of)

The action of

Figure 21. *Image ***p ^{5}t^{2}p^{4}** -- under (the action of)

The action of

Figure 22. *Image ***p ^{5}t^{2}p^{5}** -- under (the action of)

The action of

In the

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