**The symmetry of regular 2-D arrays of repeated motifs ** (continued)

**Rectangular Net and Centered Rectangular Net.**

The highest symmetrical motif that the Rectangular Net (and also the Centered Rectangular Net) can accommodate for is a motif with the same point symmetry as that of the net (lattice). The Rectangular Net as well as the Centered Rectangular Net has point symmetry **2mm**. So the highest symmetrical motif, that can be placed in it, will have that same point symmetry. It then represents the highest symmetrical Class of the 2-D Rectangular Crystal System, the Class **2mm**. The next Figure depicts an example of such a motif with **2mm** symmetry.

Figure 1. *Two motif units (black), together forming a motif with point symmetry ***2mm***.
Red solid lines signify mirror lines.
The small red solid ellipse signifies a 2-fold rotation axis perpendicular to the plane of the drawing. The two mirror lines are not equivalent.*

Also the motif in the next Figure has **2mm** symmetry.

Figure 2. *A motif with ***2mm*** symmetry.*

This Class (

The Class (= planar Point Group)

- P2mm
- P2mg
- P2gg
- C2mm

So if we eliminate all translations, then the above Plane Groups will turn into their corresponding Point Groups, and one can clearly see that only

In the Plane Groups

In the Plane Groups

The Rectangular Net and the Centered Rectangular Net, can also accommodate for motifs having a lower symmetry than the net itself, provided their symmetry elements are aligned with the corresponding symmetry elements of the net. Only one such lower symmetry is possible, namely **m**. It represents the lowest symmetrical Class (planar Point Group), **m**, of the 2-D Rectangular Crystal System. So this System has two Classes, **2mm** and **m**.

The Class **m** can be based on three possible internal structures. So it can be supported by one of three possible Plane Groups, and these are **:**
**
**

- Pm
- Pg
- Cm

Figure 3. *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***.*

Figure 3a. *A unit mesh choice is given in yellow. Its point symmetry is ***2mm*** , and it is a primitive mesh *( **P** )* . *

If we eliminate all translations, and thus telescope the pattern into itself, we end up with the motif depicted in the next Figure.

Figure 3b. *Removing all translations from the pattern representing Plane Group ***P2mm*** (Figure 3) gives, in the present case, the motif s.str. and the motif s.l. ( The motif s.l. is the motif in the usual sense + its proper surroundings -- here given in blue). This motif has point symmetry *

As can be seen from the pattern in Figure 3, the symmetry of the Plane Group

The

Figure 3c. *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.*

So we now have the first Plane Group on which the Class

The (primitive) Rectangular Net can also accommodate for motifs having a lesser degree of point symmetry, provided that their symmetry elements are aligned with the corresponding symmetry elements of the net. Such a motif can either have a point symmetry of **2**, i.e. having a 2-fold rotation axis as its only symmetry element, or have a point symmetry of **m** , i.e. having a mirror line as its only symmetry element. We'll start with the latter type of motif.

Two such motifs are placed in a mesh of a primitive rectangular net as follows **:**

Figure 4. *Two motifs, each having point symmetry ***m*** , are placed in a mesh of a primitive rectangular lattice, as indicated.*

The next Figure depicts a regular periodic pattern of motifs, that have a symmetry

Figure 4a. *If we place two motifs, having a symmertry of ***m*** , in each mesh of the primitive net, as indicated in Figure 4, then we will obtain a periodic pattern of motifs representing the Plane Group *

Figure 4b. *For the pattern of Figure 4a a unit mesh is chosen ( yellow). This unit mesh has point symmetry ***2***, and is primitive *(**P**)*. Point *

The next Figure gives the motif s.str. and the motif s.l. of the pattern of Figure 4a. The motif s.l. is repeated indefinitely across the two-dimensional plane.

Figure 4c. *The motif s.str. (black) and the motif s.l. (black + blue) of the pattern representing Plane Group ***P2mg***. It is indefinitely repeated along the directions of the 2-D lattice.*

To determine the translation-free residue (i.e. to determine the point group symmetry) of the pattern representing the Plane Group

Figure 4d. *Eliminating all translations yields a figure that has a point symmetry ***2mm*** , which represents the translation-free residue of the Plane Group ***P2mg***.*

In the pattern, representing the Plane Group

The next Figure indicates some of the 2-fold rotation axes.

Figure 4e. *Some of the 2-fold rotation axes, belonging to the symmetry content of the Plane Group ***P2mg***, are indicated. The colored lines serve to show that there indeed are 2-fold rotation axes perpendicular to the plane of the drawing.*

Figure 4f. *The pattern representing Plane Group ***P2mg*** has mirror lines parallel to the ***y*** direction. One of them is depicted.*

Figure 4g. *The pattern representing Plane Group ***P2mg*** has glide lines parallel to the ***x*** direction. One of them is depicted.*

The total

Figure 4h. *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 *

So now we have determined the second possible Plane Group on which the Class

The primitive Rectangular Net can also accommodate motifs possessing only 2-fold rotational symmetry, i.e. motifs having a point symmetry **2**. Figure 5 shows a regular array of such motifs, based on a primitive rectangular net. The pattern represents the Plane Group **P2gg**.

Figure 5. *Motifs possessing only a ***2*** symmetry can be accommodated in a primitive 2-D rectangular lattice, resulting in a periodic structure.*

Figure 5a. *A unit mesh choice (yellow). This unit mesh is primitive and has a point symmetry of ***2***.
Point *

As can be seen in the patterns of the above Figures the "motifs" are

In the next Figure we have considered the motif in the strict sense (motif s.str.) + its surroundings, making up the motif in the broad sense (motif s.l.). This latter motif is indeed repeated and tiles the 2-D plane completely.

Figure 5b. *The motif s.str. (black) and the motif s.l. (black + blue) in the pattern representing Plane Group ***P2gg***.*

Figure 5c. *The motif s.l. is indeed repeated over and over again (four repeats are indicated), and tiles the entire 2-D plane.*

If all translations in the pattern representing Plane Group

Figure. 5d. *The translation-free residue of the Plane Group ***P2gg*** is its Point Group. As can be seen, the figure (shape), representing this translation-free residue, has ***2mm*** symmetry.*

The pattern representing Plane Group

Figure 5e. *The center of each mesh, as well as the centers of the edges of each mesh, and also the nodes of the net, contain a 2-fold rotation axis (indicated by a red point) perpendicular to the plane of the drawing. Some examples of these axes (clarified by means of green lines) are indicated.*

The pattern representing Plane Group

Figure 5f. *The pattern representing Plane Group ***P2gg*** has glide lines parallel to the ***y*** direction. One of them is indicated.*

Figure 5fa. *The pattern representing Plane Group ***P2gg*** has glide lines parallel to the ***x*** direction. One of them is indicated.*

The

Figure 5g. *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.*

So now we have determined the third possible Plane Group on which the Class

In the *centered* Rectangular Net we can place motifs with **2mm** symmetry to obtain yet another Plane Group associated with the Point Group **2mm**. Placing the other possible motifs in the centered Rectangular Net does not yield new and different Plane Groups anymore.

In the next Part we will continue our investigation of periodic planar patterns, and place motifs with **2mm** symmetry in a centered rectangular net.

To continue, click HERE for Part Four.