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

Placing two-dimensional motifs with **6mm** symmetry into a (primitive) hexagonal net (2-D hexagonal lattice), yields a periodic pattern representing Plane Group **P6mm**. See Figure 1.

Figure 1. *When motifs, having point symmetry ***6mm*** (i.e. having a 6-fold rotation axis and two types of mirror lines), are inserted into a (primitive) hexagonal net, a pattern of repeated motifs will emerge that represents Plane Group ***P6mm***.*

Figure 1a. *A unit mesh is chosen (yellow). This unit mesh is primitive, and its point symmetry is ***2mm***.*

If the pattern representing Plane Group

Figure 1b. *The translation-free residue of the Plane Group ***P6mm***.
The symmetry of this residue is *

The translation-free residue is, in the present case, also the motif s.l., which means that as such it tiles the 2-D plane completely, as the next Figure shows.

Figure 1c. *Tiling of the two-dimensional plane by the motif s.l. of the pattern of Figure 1.*

The symmetries involved in patterns representing Plane Group

Figure 1d. *A pattern representing Plane Group ***P6mm*** has 3-fold rotation axes. One of them is shown.*

Figure 1e. *A pattern representing Plane Group ***P6mm*** has 2-fold rotation axes. Two of them are shown.*

Figure 1f. *A pattern representing Plane Group ***P6mm*** has glide lines. One of them is shown.*

Figure 1g. *A pattern representing Plane Group ***P6mm*** has glide lines. Again, one of them is shown.*

Figure 1h. *A pattern representing Plane Group ***P6mm*** has glide lines. Again, one of them is shown.*

The

Figure 1i. *Total symmetry content of the Plane Group ***P6mm***.
All rotation axes are perpendicular to the plane of the drawing.
6-fold rotation axes are indicated by small blue solid hexagons.
3-fold rotation axes are indicated by small blue solid triangles.
2-fold rotation axes are indicated by small blue solid ellipses.
Glide lines are indicated by red dashed lines.
Mirror lines are indicated by solid lines (red and black).*
(

So we have found the (only) Plane Group (

Next we will consider the Class **6**.

Placing motifs with point symmetry **6**, i.e. motifs having only a 6-fold rotation axis, into a hexagonal net, such that the motifs are associated with the lattice nodes, yields a repeating pattern representing Plane Group **P6**. See Figure 2.

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

Figure 2a. *A unit mesh is chosen (yellow). This unit mesh has a point symmetry ***2***, and is primitive.*

If the pattern representing Plane Group

Figure 2b. *The translation-free residue of the Plane Group ***P6***.
The symmetry of this residue is *

The translation-free residue is, in the present case, also the motif s.l., which means that as such it tiles the 2-D plane completely, just like the translation-free residue of the pattern of Figure 1.

A pattern representing Plane Group **P6** has __no__ mirror lines and also no glide lines. The distribution of rotation axes is, however, exactly the same as in Plane Group **P6mm**. See Figure 2c for the **total symmetry content** of Plane Group **P6**.

Figure 2c. *Total symmetry content of Plane Group ***P6***.*

So we have found the (only) Plane Group (

Two distinct Plane Groups, **P3m1** and **P31m**, belong to Point Group **3m**. They have the same total symmetry content and shape, but the motifs differ in orientation with respect to the edges of the unit mesh. Let's start with Plane Group **P3m1**.

Placing motifs into a hexagonal net, but now motifs, having a point symmetry **3m** -- meaning that each motif has a 3-fold rotation axis going through its center (and perpendicular to the plane of the drawing), and three equivalent mirror lines, such that their point of intersection coincides with the center of such a motif -- and oriented in the net such that their mirror lines do not coincide with the connecting lines of the net, yields a pattern (of repeated motifs) representing Plane Group **P3m1**. See Figure 3.

Figure 3. *The insertion of motifs having ***3m*** symmetry and oriented as described above, into a hexagonal net, leads to a periodic pattern representing Plane Group ***P3m1***.
The symmetry of the motifs is indicated by their shape and by their coloration. *

Figure 3a. *A unit mesh is chosen (yellow). It is primitive, and its point symmetry is ***m*** , meaning that its only symmetry element is a mirror line.*

If the pattern representing Plane Group

Figure 3b. *The translation-free residue of the Plane Group ***P3m1***.
The symmetry of this residue is *

The translation-free residue is, in the present case, also the motif s.l., which means that as such it tiles the 2-D plane completely, just like the translation-free residue of the pattern of Figure 1 (See (click) Figure 1c).

The symmetries involved in a pattern representing Plane Group **P3m1** are 3-fold rotation axes, mirror lines and glide lines. See Figures 3c, 3d, 3e and 3f.

Figure 3c. *A pattern representing Plane Group ***P3m1*** has mirror lines. One of them is indicated.*

Figure 3d. *A pattern representing Plane Group ***P3m1*** has glide lines. One of them is indicated.*

Figure 3e. *A pattern representing Plane Group ***P3m1*** has 3-fold rotation axes perpendicular to the plane of the drawing. One of them is indicated.*

Figure 3f. *A pattern representing Plane Group ***P3m1*** has 3-fold rotation axes perpendicular to the plane of the drawing. Again one of them is indicated (small blue triangle).*

The

Figure 3g. *Total symmetry content of the Plane Group ***P3m1***.
3-fold axes are indicated by small solid blue triangles.
Glide lines are indicated by dashed red lines.
Mirror lines are indicated by solid *

For clarity we depict this same total symmetry content, but now referring only to one mesh of the net

Figure 3h. *Total symmetry content of Plane Group ***P3m1***, depicted for one mesh of the net. Only the red solid lines are mirror lines.*

The second, and last, Plane Group belonging to the Class

To continue, click HERE for Part Six.