(Part Thirteen)

We continue our investigation concerning the generation of group elements of the Plane Group P3m1.

While generating the group elements (represented by triangular areas of the same size, that tesselate the plane completely) of the Group **P3m1** , we will use an enlargement of the Figures used in the previous document in order to have more space for the notation of group elements.

Figure 1. *Magnification of the last Figure of the previous document (Figure 24). In the partitioned *

The initial element

The generator

The generator

The generator

In the next Figure we generate the elements

Figure 2. *Generation of some new group elements of the Plane Group ***P3m1** .

Figure 3. *By means of (repeatedly) applying the generator ***t*** the group elements involved in the second row of motifs s.l. are generated.*

The next Figure shows that a rotation about the point

Figure 4. *Indication of the effect of rotation about the three-fold axis at the point ***R*** of the elements around the lattice point next to *R* to its right. The possible rotations are represented by the elements ***p*** and ***p ^{2}**

Figure 5. *Smaller scale image of the presently used pattern representing the Group ***P3m1*** , also showing the possibility of generating new group elements by rotation about the point ***R** .

The next Figure generates the group elements that result from the above mentioned rotations of the elements situated around the lattice point next right to the lattice point

Figure 6. *The first and third rows of motifs s.l. have been reached by means of the above mentioned rotations about the point ***R*** .*

The first and third rows can now be completed by (repeatedly) applying the generator

Figure 7. *Completion of the first and third rows of the motifs s.l. of the pattern according to the Plane Group ***P3m1** .

Next we will again use the rotation

Figure 8. *We can reach the fourth row of motifs s.l. of our ***P3m1*** pattern, by rotating the elements of the last motif s.l. of the second row, about the point ***R*** 240 ^{0} anticlockwise, which is equivalent to applying *

Figure 9. *Reaching the fourth row of motifs s.l. of our ***P3m1*** pattern by means of the rotation ***p ^{2}**

This fourth row can now be completed by applying translations to the lastly obtained group elements

Figure 10. *Completion of the fourth row of motifs s.l. of our ***P3m1*** pattern.*

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

In the

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