(Part Fourteen)

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

The next motif we're going to insert into a 2-D hexagonal lattice has the same point symmetry as the one we used for the exposition of Plane Group **P3m1** in the previous two documents. But with respect to the connecting lines of the net our new motif is oriented differently **:** It is rotated 30^{0} with respect to the one used earlier. See Figure 1.

Figure 1.

(1). Orientation of the **3m*** motif, compatible with the Plane Group ***P3m1*** (discussed in the previous two documents).
(2). Orientation of the *

Placing two-dimensional motifs with

Figure 2. *When motifs, having point symmetry ***3m*** (i.e. having a 3-fold rotation axis and three equivalent mirror lines), are inserted into a (primitive) hexagonal net, in the way (i.e. the orientation) shown, a pattern of repeated motifs will emerge that represents Plane Group ***P31m*** .
Each (composed) motif consists of three augmented motif units, in such a way that the symmetry of the composed motif is *3m

The symmetry of the motifs is indicated by their shape

The pattern must be conceived as extending indefinitely in two-dimensional space.

Figure 3. *A unit mesh is chosen (yellow). This unit mesh is primitive, and its point symmetry is ***m**,* i.e. the only symmetry element it possesses is a mirror line.*

If the pattern representing Plane Group

Figure 4. *The translation-free residue of the pattern -- representing Plane Group ***P31m*** -- of Figure 2. The point symmetry of this residue is ***3m*** , and as such it represents the Point Group ***3m*** to which the present Plane Group belongs. The translation-free residue is at the same time the motif s.l. and tiles the 2-D plane completely.*

The symmetry elements involved in a pattern representing Plane Group

Figure 5. *A pattern representing Plane Group ***P31m*** has mirror lines. One of them is depicted here.*

Figure 6. *A pattern representing Plane Group ***P31m*** has glide lines. One of them is depicted here.*

Figure 7. *A pattern representing Plane Group ***P31m*** has 3-fold rotation axes. One of them is depicted here.*

The

Figure 8. *Total symmetry content of Plane Group ***P31m*** .
Mirror lines are indicated by solid lines (red and black).*

Now we will show the *generation* of the group P31m by means of **basic** motif units that legitimately represent group elements (i.e. elements of the full group P31m). In Figure 2 we had motifs, each consisting of three units, making angles with each other of 120^{0}, and providing the composed motif with 3m symmetry (Which is equivalent to a D_{3} group structure). But such a motif unit is still symmetric in itself, so in fact it is composed of still more basic motif units **:** It is an *augmented* motif unit. And indeed the *basic* units, as can be obtained by dividing the augmented motif units, do not have (and ought not to have) any symmetry at all. They're going to *build up* a symmetric pattern from scratch. They can legitimately represent group elements (i.e. elements of the full group P31m). The next image shows a composed motif, equivalent to the ones in Figure 2, but partitioned into six basic motif units.

This type of motif can perhaps be more clearly expressed as follows **:**

The two basic motif units composing an augmented motif unit (of which three together make up the full composed motif) can conveniently be distinguished by **colors**, *provided we do not interpret the difference between colors as expressing an asymmetry*. The two basic motif units, red and blue in the next Figures, are symmetrically related to each other

The latter motifs we will now place in a hexagonal lattice (That lattice having the same orientation as that used for depicting the pattern of the group P3m1 ), resulting in the periodic pattern according to the group P31m. The effect is that both patterns, P31m and P3m1, have the same type of composed motif (and also of basic motif unit for that matter), but in each case those motifs are differenly orientated with respect to the lattice lines (i.e. the edges of the unit cell). This difference in orientation is 30^{0}.

Figure 9. *Pattern according to the Plane Group P31m .
The (composed) motifs consist of six basic motif units (red and blue), each representing a group element (i.e. an element of the group P31m). The difference in color should not be interpreted as an asymmetry.
The pattern must be conceived as extending indefinitely in two-dimensional space.*

The next Figure gives this same pattern. Some lattice points are marked

The basic motif unit

The basic motif unit

The basic motif unit

Figure 10. *Pattern according to the Plane Group P31m .
The initial motif unit, three generators and some lattice points are indicated.*

The next Figure depicts an enlargement of the composed motif at the lattice point

Figure 11. *Composed motif (consisting of six basic motif units) of the P31m pattern at the point R in Figure 10.*

The identity of the remaining basic motif units of the composed motif at the lattice point R can now be determined (

Figure 12. *The group elements of the composed motif at the point R .
Together they form the subgroup *

The elements

We will now produce the composed motif at the point

Figure 13. *The elements (basic motif units) of the composed motif at the lattice point ***S*** . They form the left coset of the D _{3} subgroup by the element *

Next we determine the elements of the composed motif at the lattice point

To generate those elements we must subject the elements of the composed motif at the point S to an anticlockwise rotation of 240

Figure 14. *Generation of the basic motif units of the composed motif at the lattice point *U* . The names of the newly generated elements are given at the perimeter of the image (i.e. outside the image).*

The second and third row of composed motifs (Figure 10) can now be completed by applying the translations

We will do so with respect to the composed motif at the point

Figure 15. *Generation of the elements of the composed motif at the point ***X*** . Together they form the left coset of the subgroup D_{3} by the element *

The notation of the newly generated elements is given at the perimeter of the image.

From this lastly obtained composed motif we can reach the fourth row (Figure 10) of composed motifs (to be generated), by applying

Figure 16. *Generation of the elements of the composed motif at the lattice point ***W*** . Together they form the left coset of the subgroup D_{3} by the element *

The notation of the newly generated elements is given at the perimeter of the image.

The fourth row can now be completed by translations. Continuation of this procedure will generate in principle the whole group

What follows next is the generation of that same group P31m by means of

Here an augmented motif unit means two symmetrically related

Figure 17. *Composed motif (at the lattice point ***R*** (For its position, see Figure 10)), consisting of three augmented motif units. One is chosen to be the initial motif unit ***1*** , representing the identity element of the group P31m , another such augmented motif unit is chosen to be the generator ***p*** representing an anticlockwise rotation of 120 ^{0} about the lattice point *

As a second generator we choose the horizontal translation

Because we now use

The next Figure shows the two chosen generators in the context of the lattice **:**

The 'element' (augmented motif unit) **p** , resulting from the 'element' (augmented motif unit) **1** , by an anticlockwise rotation of 120^{0} about the point **R** .

The 'element' **t** (augmented motif unit), resulting from the 'element' **1** by the translation **t** .

Figure 18. *The two chosen generators ***p*** and ***t*** for the P31m pattern.*

Figure 19. *The whole P31m pattern can be produced by the generators ***p*** and ***t** .

As can be seen from the above results, we can say the following

As

We're now going to generate our group P31m again, but now letting the group elements be represented not only by a basic unit of the motif

Figure 20. *(Choice of) Motifs ***s.l.*** (yellow hexagons) of the ***P31m*** pattern of Figure 9.*

In the next Figure we partition the motifs s.l. in such a way that the resulting areas (plus their content) can represent group elements.

Figure 21. *Division of each motif s.l. of the previous Figure into six equally sized and shaped areas (yellow and green bi-isosceles triangles) that can legitimately represent group elements. Each such area contains one basic unit of the motif s.str. PLUS corresponding background.*

Figure 22. *A partitioned motif s.l. isolated. It consists of six basic units of the motif s.l. Each such unit contains one basic unit of the motif s.str. PLUS corresponding background.*

In the next Figure we indicate explicitly the initial group element and the generators

Figure 23. *The ***P31m*** pattern of Figure 9 is partitioned in areas (bi-isosceles triangles) that can represent group elements (as was already done in Figure 21). Four such group elements are explicitly indicated :
The initial element *

The generator

The generator

The generator

In the

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