(Part Three)

We will now investigate the generation of the Plane Group

We're now going to generate the group elements of the Plane Group **Pg**.

We first let the motif units composing the motifs **s.str.** represent the group elements. Then we determine the motif units of the motif **s.l.** and let them represent the group elements, and generate the latter again.

Figure 1. *Assymmetric motifs (in fact motif units) placed in a primitive rectangular net, such that they alternate along the ***y*** edges (horizontal) of the meshes, produce a pattern that represents the Plane Group ***Pg*** .*

The only symmetry elements the Plane Group

See Figure 2.

Figure 2. *Patterns representing Plane Group ***Pg*** have glide lines parallel to the ***y*** direction. One of them is shown.*

The

Figure 3. *Total symmetry content of the Plane Group ***Pg***.
Glide lines are indicated by red dashed lines. For clarity the nodes of the net are indicated (black dots).*

Figure 4. *The pattern representing Plane Group ***Pg*** consists of motif units (commas). They can be generated from a given initial motif unit, denoted ***1*** , by two generators ***g*** (a horizontal glide reflection) and ***t*** (a vertical translation).*

When we remove all translations, the individuals of the same type of motif unit will be superimposed upon each other, resulting in only one individual per type. In the present case this means that only two individuals remain, one of each type (= the two types of comma in our Figures), and these two individuals become aligned to each other because of the disappearance of translational distances, and in this way together form a figure that has point symmetry

Figure 5. *The translation-free residue of the pattern representing Plane Group ***Pg*** has ***m*** symmetry (i.e. the only symmetry element it has is one mirror line), and represents the Point Group to which the Plane Group ***Pg*** belongs.*

The next Figure shows what the very motif (s.str.) actually is, and that it is constituted out of two motif units. This motif, together with its surroundings (such that together they make up the motif

Figure 6. *The real motif, constituting the pattern of Figure 1.*

Figure 7. *The real motif s.l. tiles the entire 2-D plane in a periodic fashion.*

The next Figure depicts a more extensive tiling (tesselation) of the same motifs

Figure 8. *Tesselation of the 2-D plane with motifs ***s.l.*** of the ***Pg*** pattern of Figure 1 . *

Our

As can be seen in Figure 4, each motif unit represents a group element. The next Figure takes the surroundings of those motif units also into account, resulting in the representation of each group element by the complete corresponding area of the pattern, which here means the comma PLUS its corresponding background.

Figure 9. *The red and blue elongate rectangles represent the group elements of the ***Pg*** pattern of Figure 1. Each such a rectangle contains one motif unit (of the motif s.str.).*

The next Figure is the same as the previous Figure, but now with the initial element

The two generator elements are

A glide reflection

A vertical translation

Figure 10. *Areas of the ***Pg*** pattern of Figure 1 corresponding to the group elements. The initial element ***1*** and the two generators ***g*** and ***t*** indicated and highlighted (purple).*

All elements of the Group

Figure 11. *The group elements ***(g ^{-1})^{2} **(= g

The rest of the group elements (of the depicted part) of our

Figure 12. *The group elements ***t ^{2}(g^{-1})^{3} , t(g^{-1})^{2} , , t^{-1}(g^{-1})^{2} , **

t^{-1}(g^{-1})^{3} , t^{-2}(g^{-1})^{2} , t^{2}g^{-1} , tg^{-1} , t^{-1} , t^{-1}g^{-1} , t^{-2} , t^{2}g , tg^{2} , tg , t^{-1}g^{2} , t^{-1}g ,

t^{-2}g^{2} , t^{2}g^{3} , tg^{4} , tg^{3} , t^{-1}g^{4} , t^{-1}g^{3} , t^{-2}g^{4}

The colors blue and red do not signify qualitative differences.

We have thus generated the group elements of the Group

We're now going to generate the group elements of the Plane Group **P2mm**.

As before we first let the motif units composing the motifs **s.str.** represent the group elements. Then we determine the motif units of the motif **s.l.** and let them represent the group elements, and generate the latter again.

Figure 13. *Placing motifs with ***2mm*** point symmetry in a primitive rectangular 2-D lattice creates a periodic pattern of these motifs representing the Plane Group ***P2mm***. The pattern must be conceived as extending indefinitely in two-dimensional space.*

The

Figure 14. *The total symmetry content of the Plane Group ***P2mm***.
Solid lines (black and red) indicate mirror lines.
Small red solid ellipses indicate 2-fold rotation axes perpendicular to the plane of the drawing.*

The next Figure again gives the P2mm patttern, and prepares for letting it be generated.

Figure 15. *Each (composed) motif of our version of a P2mm pattern consists of four motif units (commas) partially overlapping, such that the symmetry of the resulting composed motif is *2mm* . As before, each motif unit represents a group element.
One such motif unit is chosen to be the initial motif unit, representing the identity element of the group, and denoted *

A second motif unit, denoted

A third motif unit, denoted

A fourth motif unit, denoted

Finally, a fifth motif unit, denoted

The next Figure shows how the motif units -- representing group elements of the group P2mm -- are generated by the generators

Figure 16. *Generation of the group elements of the group P2mm by the generators ***a, b, c, d .**

The next Figure shows the motifs

Figure 17. *Motifs ***s.l.*** (red and blue rectangles) of the ***P2mm*** pattern of Figure 13. Each such motif s.l. consists of one motif ***s.str.*** (itself consisting of four partly overlapping motif units (commas)), PLUS its corresponding background area. And each motif ***s.l.*** is associated with a lattice point.
(The colors red and blue do not signify qualitative differences).*

The next Figure depicts the areas (of our pattern) representing group elements.

Figure 18. *The motifs ***s.l.*** of our ***P2mm*** pattern consist of four smaller rectangular areas each representing a different group element.*

The next Figure is the same as Figure 18, but now with the symmetry elements of the four generators (four mirror reflections) inserted.

Figure 19. *Group elements represented by (smaller) rectangular areas of the ***P2mm*** pattern of Figure 13 . Each such a *__group element representative__* contains one quarter of a motif ***s.str.*** PLUS a corresponding background area. The symmetry elements of the four generators are indicated : four mirror lines, *

( The different colors of the rectangles do not signify qualitative differences).

The next Figure shows some elements generated from the given elements by applying the horizontal reflections.

Figure 20. *The group elements ***dc, cd, dcd, cdc, cdcd*** , are generated by subjecting the already given elements to the horizontal reflections ***c*** and ***d** .

We can now obtain new group elements by applying the vertical reflection

Figure 21. *The group elements ***adcd, adc, ad, ac, acd, acdc, acdcd*** are generated by applying the reflection ***a*** to some elements already generated.*

We can generate yet more group elements by applying the reflection

Figure 22. *The group elements ***bdcd, bdc, bd, bc, bcd, bcdc, bcdcd, badcd, badc, bad, ba, bac, bacd, bacdc, bacdcd*** are generated by applying the reflection ***b*** to elements already generated.*

To obtain still further group elements we can apply the reflection

Figure 23. *The group elements ***abdcd, abdc, abd, ab, abc, abcd, abcdc, abcdcd*** are generated by applying the reflection ***a*** to the elements of the last column but one.*

When we subject the last obtained group elements to the reflection

Figure 24. *The group elements ***babdcd, babdc, babd, bab, babc, babcd, babcdc, babcdcd*** are generated by applying the reflection ***b*** to the elements of the extreme left column.*

So we have now generated the group elements of the Group

In the

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