(Part Nine)

We will continue our investigation concerning the generation of the Plane Group

In the partitioned

Figure 1. *The pattern representing Plane Group ***P4gm*** is partitioned (as it had been already in the previous document) into triangular areas representing group elements. Three such group elements are explicitly indicated and also the relevant symmetry elements (rotation axis and mirror line) associated with the generators.*

In the following -- where we generate the group elements -- we shall use an enlarged version of the above Figure in order to have enough space for the markings of group elements.

First we generate the elements **p ^{2}** and

Figure 2. *Generation of the group elements ***p ^{2}**

Figure 3. *Rotating the element ***m*** anticlockwise by resp. 90 ^{0}, 180^{0} and 270^{0} we obtain the elements *

Figure 4. *New group elements are generated by applying the mirror reflection ***m*** to already existing elements.*

Figure 5. *More group elements are generated by (repeatedly) applying ***p*** (anticlockwise rotation of 90 ^{0} about the point *

Figure 6. *Still more group elements are generated by (repeatedly) applying ***p*** (anticlockwise rotation of 90 ^{0} about the point *

Figure 7. *One new element is generated by applying the reflection ***m*** *

Figure 8. *Two more group elements are generated by (repeatedly) applying ***p*** (anticlockwise rotation of 90 ^{0} about the point *

Figure 9. *Some new elements are generated by applying the reflection ***m*** to already existing elements.*

Figure 10. *Two more group elements are generated by (repeatedly) applying ***p*** (anticlockwise rotation of 90 ^{0} about the point *

Figure 11. *Some new elements are generated by applying the reflection ***m*** to already existing elements.*

Figure 12. *Three more group elements are generated by (repeatedly) applying ***p*** (anticlockwise rotation of 90 ^{0} about the point *

Figure 13. *Again, three more group elements are generated by (repeatedly) applying ***p*** to an already existing element.*

Figure 14. *Three new elements are generated by applying the reflection ***m*** to already existing elements.*

Figure 15. *Again, two more group elements are generated by (repeatedly) applying ***p*** to an already existing element.*

In the

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