Sequel to Group Theory

We'll start with reminding the reader about the

Sequel to Infinite two-dimensional periodic patterns

Figure 1. *Inserting composed motifs, possessing point symmetry ***6*** , into a 2-D hexagonal point lattice, yields a periodic pattern representing Plane Group ***P6*** .
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

A unit cell is indicated (yellow).

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.*

With the Plane Groups P1, P2, P3, P4 and P6, we have discussed the 'direct' plane groups, that is groups in which only rotations and translations occur.

In all the remainig (12) groups to discuss (including the group P4mm, discussed earlier) there are (also) reflectional elements ('opposite symmetries') present, mirror lines and/or glide reflections.

Figure 6. *When we place motifs, possessing a point symmetry ***m***, on the nodes of a primitive rectangular net, we get a pattern that represents Plane Group ***Pm***.*

The

Figure 7. *The total symmetry content of the Plane Group ***Pm***.
The only symmetry elements this Group has are mirror lines parallel to the *

For clarity the nodes of the lattice are indicated (black dots).

Figure 8. *The ***Pm*** pattern can be generated by two parallel mirror lines, ***a*** and ***b*** , and a translation ***t*** .
In the present Figure each group element is represented by a motif unit, which is -- as before -- a comma. Two partly overlapping commas form a composed motif, that has as its only symmetry element a mirror line.
One motif unit is chosen as initial motif unit, representing the identity element of the group, denoted *

Three generators are needed to produce all group elements, or, correspondingly, to produce all motif units, and thus to produce the whole Pm pattern.

The motif unit

The motif unit

The motif unit

A unit cell is indicated (yellow). The pattern must be conceived as extending indefinitely in two-dimensional space.

We will now give the Pm pattern again and label some motif units such, as representing generated group elements.

Figure 9. *Generation of some group elements (represented by motif units) of the group ***Pm*** by the generators ***a, b*** and ***t** .

Figure 10. *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 11.

Figure 11. *Patterns representing Plane Group ***Pg*** have glide lines parallel to the ***y*** direction. One of them is shown.*

The

Figure 12. *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 13. *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 genrators ***g*** (a horizontal glide reflection) and ***t*** (a vertical translation).*

Figure 14. *Placing motifs with ***2mm*** point symmetry in a primitive rectangular 2-D lattice creates a periodic pattern of these motives representing the Plane Group ***P2mm***. The pattern must be conceived as extending indefinitely in two-dimensional space.*

The

Figure 15. *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 16. *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 17. *Generation of the group elements of the group P2mm by the generators ***a, b, c, d** .

The primitive Rectangular Net can also accommodate motifs possessing only 2-fold rotational symmetry, i.e. motifs having a point symmetry **2**. Figure 18 shows a regular array of such motifs, based on a primitive rectangular net. The pattern represents the Plane Group **P2gg**.

Figure 18. *Motifs possessing only a ***2*** symmetry can be accommodated in a primitive 2-D rectangular lattice, resulting in a periodic structure. This structure must be conceived as extending indefinitely in two-dimensional space.*

The

Figure 19. *The total symmetry content of the Plane Group ***P2gg***.
2-fold rotation axes perpendicular to the plane of the drawing are indicated by small solid red ellipses.
Glide lines are indicated by dashed red lines.
Mirror lines are absent.*

The next Figure ilustrates one of the glide reflections present in the P2gg pattern.

Figure 20. *A glide reflection ***g*** in the P2gg pattern is indicated. Each (composed) motif consists of two motif units (commas), such that the composed motif possesses 2-fold rotational symmetry. One motif unit is chosen to figure as the initial motif unit from which the pattern is produced by (three) generators. Each motif unit represents a group element.*

Figure 21. *A 2-fold rotational symmetry is present at the point ***R*** (there are more such points). We will choose this rotation (180 ^{0} about R ) as one of the three generators of the group P2gg.*

The next Figure indicates, in addittion to the one generator already chosen, the other two (chosen) generators. These three together can generate the whole pattern (and as such this repesents the generation of the group P2gg ).

Figure 22. *The three chosen generators of the group P2gg :
The glide reflection *

The next Figure shows how a number of group elements (represented by motif units) can be generated by means of the three chosen generators. Eventually all elements of the group can be generated.

Figure 23. *Generation of the P2gg pattern by using the generators ***g , t , r** .

(For reasons of clarity of the drawing not all motif units are provided with their corresponding symbol).

Figure 24. *Two motifs, each having point symmetry ***m*** , are placed in a mesh of a primitive rectangular lattice, as indicated.*

Figure 25. *If we place two motifs, having a symmertry of ***m*** , in each mesh of the primitive net, as indicated in Figure 24, then we will obtain a periodic pattern of motifs representing the Plane Group *

The total

Figure 26. *Total symmetry content of the Plane Group ***P2mg***.
Solid red lines indicate mirror lines.
Small solid red ellipses indicate 2-fold rotation axes perpendicular to the plane of the drawing.
The glide lines are all parallel to the *

The P2mg pattern, as realized in Figure 25, consists of composed motifs. Each such motif itself consists of two motif units -- commas -- which represent group elements (and each group element is a symmetry transformation of the pattern).

For generating the P2mg pattern, and with it the (elements of the) group P2mg, we choose the following set of generators

The element

The element

The element

Figure 27. *Three generators for the P2mg pattern are indicated.*

The next Figure shows how all the group elements can be generated from the three generators

Figure 28. *Generation of the P2mg pattern from the three generators.*

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