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

The next motif we're going to insert in 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 Part. 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 Part Five).
(2). Orientation of the *

Placing two-dimensional motifs with

Figure 1a. *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*** .
The symmetry of the motifs is indicated by their shape and by their coloration.*

Figure 1b. *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 1c. *The translation-free residue of the pattern -- representing Plane Group ***P31m*** -- of Figure 1a. 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 1d. *A pattern representing Plane Group ***P31m*** has mirror lines. One of them is depicted here.*

Figure 1e. *A pattern representing Plane Group ***P31m*** has glide lines. One of them is depicted here.*

Figure 1f. *A pattern representing Plane Group ***P31m*** has 3-fold rotation axes. One of them is depicted here.*

The

Figure 1g. *Total symmetry content of Plane Group ***P31m***.
Mirror lines are indicated by solid lines (red and black).*

So we now have discussed the second and last Plane Group (

Finally we have arrived at the last 2-D Crystal Class, namely

If we place motifs, having a point symmetry **3** (i.e. the only symmetry element each one of them has is a 3-fold rotation axis), in a hexagonal net, then we obtain a periodic pattern (of motifs) representing the Plane Group **P3**. See Figure 2.

Figure 2. *Arranging motifs with point symmetry ***3*** in a hexagonal 2-D lattice yields a pattern that represents Plane Group ***P3***.
The symmetry of the motifs is indicated by their shape and by their coloration.*

Figure 2a. *A unit mesh is chosen (yellow), it is primitive and has point symmetry ***1***, i.e. it has no symmetry whatsoever.*

If we contract the pattern of Figure 2, representing Plane Group

Figure 2b. *Translation-free residue of the pattern of Figure 2. It represents the Point Group of the Plane Group ***P3***.*

The symmetry elements involved in a pattern representing Plane Group

Figure 2c. *A pattern representing Plane Group ***P3*** has 3-fold rotation axes. One of them is shown (small blue solid triangle).*

Figure 2d. *A pattern representing Plane Group ***P3*** has 3-fold rotation axes. One of them is indicated. It is situated at a node of the net.*

The

Figure 2e. *Total symmetry content of the Plane Group ***P3***.
There are no mirror lines and also no glide lines. The Plane Group only possesses 3-fold rotation axes.*

This concludes our discussion of the planar Point Groups and Plane Groups of the 2-D Hexagonal Crystal System, and with it we

The Point Groups represent the ten 2-D Crystal Classes. The Plane Groups describe the internal structure of a two-dimensional crystal insofar as (total) symmetry of the periodic pattern is concerned, and the orientation of the motifs in the 2-D lattice.

In the next Part we will

To continue, click HERE for Part Seven.