(Part Sixteen)

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

Figure 1. *Inserting composed motifs, possessing point symmetry ***6*** , into a 2-D hexagonal point lattice, yields a periodic pattern representing Plane Group ***P6*** .
Also here the unit mesh (yellow) is rhomb-shaped with angles of 60 ^{0} and 120^{0}.
Each motif consists of six motif units. Each motif unit is in itself asymmetric and thus can represent a group element (wherever that unit occurs in the pattern). The point *

The motif unit labelled

The motif unit labelled

The motif unit

The motif unit

The motif unit

The motif unit

The motif unit

The motif unit labelled

The group

The pattern must be conceived as indefinitely extended in 2-D space.

The next Diagram depicts the total symmetry content of the Plane Group P6.

Figure 2. *Total symmetry content of the Plane Group ***P6*** .
A small solid ellipse indicates the position of a 2-fold rotation axis.
A small solid triangle indicates the position of a 3-fold rotation axis.
A small solid hexagon indicates the position of a 6-fold rotation axis.*

The next Figure again gives the P6 pattern, and demonstrates the six-fold symmetry at the point R (as indicated in Figure 1).

Figure 3. *Six-fold symmetry of the infinite P6 pattern at the point ***R*** (There are of course many such points in the pattern).
The motif unit denoted *

The motif unit denoted

The transformation

As before, these elements can be represented by the position and orientation of motif units.

The next Figure shows how some group elements -- represented by motif units -- are generated from the elements together representing the initial composed motif. In that composed motif the elements **p ^{2} , p^{3} , p^{4}** , and

Figure 4. *Generation of several composed motifs from the initial composed motif.*

From the third composed motif in the second row we can produce the fourth composed motif in that same row by applying the translation

Directly from the third composed motif in the second row we can produce a composed motif in the third row by applying

Figure 5. *Generation of two more composed motifs (in addition to one obtained by a translation) of the pattern according to the Plane Group P6.*

We will now generate the group elements of the group P6 again, but now letting them be represented not only by a motif unit of the motif

Figure 6. *Motifs ***s.l.*** (yellow hexagons) of the ***P6*** pattern of Figure 1. Each motif s.l. consists of one motif s.str. PLUS corresponding background. Partition of such motifs s.l. will yield areas that can represent group elements.*

The next Figure shows the partition of the motifs s.l. into equally sized and shaped areas that can represent group elements of our

Figure 7. *The motifs s.l. are each divided into six areas (bi-isosceles red and yellow triangles). Each such area can represent a group element. It consists of one basic unit of the motif s.str. (or at least it represents it) PLUS corresponding background. The colors red and yellow, in the partitioned motifs s.l. do not necessarily signify difference in symmetry. So at the center of each motif s.l. there is a ***six**-*fold rotation axis (and not just a three-fold rotation axis). *

Figure 8. *One partitioned motif ***s.l.*** isolated. The areas A, B, C, D, E, F represent group elements. They are situated around a six-fold rotation axis.*

The next Figure indicates the areas that represent the initial element (identity element) and the two generators.

Figure 9. *The ***P6*** pattern of Figure 1 is partitioned into areas representing group elements (as it was already in Figure 7). Three such group elements are explicitly indicated :
The initial group element *

The generator element

The generator element

By repeatedly applying the generator

Figure 10. *By applying repeatedly the generator ***p*** to the initial element, i.e. rotating the latter anticlockwise through angles of 120 ^{0} (=2x60^{0}), 180^{0}, 240^{0} and 300^{0} about the point *

When we now subject all the group elements that are situated around the point

Figure 11. *Completion of the filling-in of the second row of motifs s.l. of the present ***P6*** pattern.*

The next Figure indicates how we can reach the first and third rows of motifs s.l., namely the first row by the transformation

Figure 12. *Two rotations can bring us to the first and third rows of the ***P6*** pattern. See next Figure.*

Figure 13. *The first and third rows have been reached. For two motifs s.l. the group elements are generated.*

These rows can now be completed by means of translations

Figure 14. *The first and third rows are completed.*

The next Figure indicates how the fourth row can be reached by applying an anticlockwise rotation of 300

Figure 15. *Indication how to reach the fourth row of the ***P6*** pattern.*

Figure 16. *The fourth row has been reached, and the corresponding group elements generated.*

This row can now be completed by means of translations

Figure 17. *The fourth row of the ***P6*** pattern is now completed.*

We have now generated all group elements of the displayed part of our

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