(Part Seventeen)

We continue our investigation concerning the generation of group elements of the Plane Groups.

Figure 1. *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 2. *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 1 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 1), 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 3. *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 4. *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 5. *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 5 are then automatically implied

Figure 6. *The composed motif and its twelve constituent basic motif units, at the point *R* of Figure 4. 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 7. *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 8. *The composed motif next to the right of the one at the point R .*

The red motif units in Figure 8 together form the

All the twelve elements (red and blue in Figure 8 ) together form the

Next we're going to generate the composed motif at the point **Q** in Figure 7. 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 9. *The newly generated composed motif at lattice point Q (as indicated in Figure 7). The names of the new elements are given at the perimeter of the image (i.e. outside the image).*

All the elements in Figure 9 (red and blue) together form the

The third row (Figure 7) 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 10. *The composed motif at the point W (Figure 7) is completed, by applying ***p ^{5}**

The six red motif units of the above Figure (Figure 10) together form the

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

It is clear that by using the rotations and translations we can generate in principle the whole group ( The generator

The same

Figure 11. *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 12. *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 13. *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 14. *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 15. *Image ***p ^{5}t^{2}** -- under (the action of)

The action of

Figure 16. *Image ***p ^{5}t^{2}p** -- under (the action of)

The action of

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

The action of

Figure 18. *Image ***p ^{5}t^{2}p^{3}** -- under (the action of)

The action of

Figure 19. *Image ***p ^{5}t^{2}p^{4}** -- under (the action of)

The action of

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

The action of

Next we will again generate our group

In order to find these surroundings as areas that -- with their content (basic unit of motif s.str.) -- can legitimately represent true group elements of the group P6mm, we must first determine an appropriate choice of

Figure 21. *Choice of motifs ***s.l.*** (yellow hexagons) of the ***P6mm*** pattern of Figure 3.
Each such motif s.l. consists of one motif s.str. PLUS corresponding background.*

Figure 22. *Motif ***s.l.*** isolated (background differently colored than in the previous Figure).*

The next Figure shows the partition of the motifs s.l. in such a way that each motif s.l. is divided into twelve equally sized and shaped areas. Each such area contains one basic unit of the motif s.str. PLUS corresponding background, and as such can represent a group element.

Figure 23. *Partition of the motifs s.l. (as given in Figure 21 and 22), and with it (a partition) of the ***P6mm*** pattern, resulting in areas (green and yellow right-angled triangles) that can (together with their content) represent group elements of the Plane Group P6mm.*

Figure 24. *One partitioned motif ***s.l.*** isolated. The areas A, B, C, D, E, F, G, H, I, J, K, L, including their content, which consists of one basic unit of the motif s.str., represent group elements.*

Figure 25. *One area, representing one group element, isolated.*

The next Figure indicates the initial element and the three generators

Figure 26. *The ***P6mm*** pattern partitioned into areas that represent group elements (as was already done in Figure 23). Four such group elements are explicitly indicated :
The initial group element *

The generator element

The generator element

The generator element

In the

**e-mail :**

To continue click HERE

`back to the Ink-in-Glycerine Model`

`back to Part I of The Crystallization process and the Implicate Order`

`back to Part II of The Crystallization process and the Implicate Order`

`back to Part III of The Crystallization process and the Implicate Order`

`back to Part IV of The Crystallization process and the Implicate Order`

`back to Part V of The Crystallization process and the Implicate Order`

`back to Part VI of The Crystallization process and the Implicate Order`

`back to Part VII of The Crystallization process and the Implicate Order`

`back to Part VIII of The Crystallization process and the Implicate Order`

`back to Part IX of The Crystallization process and the Implicate Order`

`back to Part X of The Crystallization process and the Implicate Order`

`back to Part XI of The Crystallization process and the Implicate Order`

`back to Part XII of The Crystallization process and the Implicate Order`

`back to Part XIII of The Crystallization process and the Implicate Order`

`back to Part XIV of The Crystallization process and the Implicate Order`

`back to Part XV of The Crystallization process and the Implicate Order`

`back to Part XVI of The Crystallization process and the Implicate Order`