Sequel to Group Theory

We'll start with reminding the reader about the

Sequel to Infinite two-dimensional periodic patterns

In the previous document we began the study of 2-D periodic patterns along a *synthetic* approach, which means that we investigate the symmetry of such patterns by *building them up*, using generators. This implies that we generate the *group* -- underlying the pattern -- by successively generating its elements. Every such group element is then represented by the location and orientation of a basic motif unit with respect to an *initial* such basic motif unit. But because the pattern contains infinitely many of such motif units, the number of group elements is infinite, so the process of generating the group will never be concluded. But from a certain point onwards it's becoming clear that no new generators will be needed and the pattern can be extended as far as we please. Such patterns are *periodic*, which means that we can detect a certain unit which is repeated in two directions while preserving its orientation. In crystallography (including also that of imaginary 2-dimensional crystals) such a unit is called a *unit-cell*.

The concept of *unit-cell*, in relation to certain point nets (two-dimensional point lattices), the symmetry of these point nets, and the the symmetry that results when motifs are placed into such point nets, is extensively treated in the *first part* of this website. The latter is accessible by the following link `back to homepage`, and the analytical approach with respect to studying the symmetry of two-dimensional periodic infinite patterns can be found in the Special Series, where the relevant documents (beginning after the documents on Crystal Systems) are named *Internal Structure of Crystals* (Part I -- XX).

In the previous document we generated a pattern according to the **P4mm** Plane Group. We did this by using four generators, acting on an asymmetric basic motif unit (a comma). This does not mean, however, that the *minimum* number of generators is necessarily four. By chosing other, appropriate, generators we can do the job with a minimum of *three* generators.

In order to understand two-dimensional periodic patterns (and through them, three-dimensional periodic patterns -- real crystals -- ) we must direct our attention to one-dimensional periodic patterns, which are patterns consisting of a unit that is repeated, while preserving its orientation, in one direction only (and within that direction, backwardly and forewardly repeated). Such patterns are called *frieze patterns* or *strip patterns*. As has been said, such patterns extend to infinity in one dimension. Because of the demand of periodicity there are only 7 types of such patterns, i.e. seven distinct patterns (for two dimensions there are 17 such patterns -- described by plane groups -- , and for three dimensions we have 230 such patterns -- described by space groups).

In the next table we show these **seven distinct one-dimensional patterns**, using again as motif unit the comma. The group elements they contain can be half-turns, mirror lines, translations and glide lines.

A half-turn is indicated by **s** ,

a mirror line is indicated by a red solid line,

a translation is evident from the pattern itself, and

a glide line (glide reflection) is indicated by a red dashed line.

In a second column we indicate some possible sets of generators, and in a third column we indicate the group theoretical structure of the pattern.

Pattern |
Generators |
Structure |

1 translationt |
||

1 glideg |
||

2 reflectionsa and b or reflection and translation (t = ba) |
||

2 half-turnsa and b or half-turn and translation (t = ba) |
||

1 reflection (a) and 1 half-turn (b) (g = ab) (t = (ab) ^{2} ) |
||

1 reflection (a) and 1 translation (t) or reflection and glide (g = at) |
||

3 reflections (a) (b) and (c).Other sets of generators may include t = cb g = acb, etc. |

With the frieze patterns, seven groups are involved as realizations of four

The groups C

The seven strip patterns will also be encountered as parts of two-dimensional periodic patterns.

We will now discuss the 2-D periodic patterns ('wall-paper patterns'). One of these, P4mm, we already spoke about earlier.

There are, as we have stated earlier, exactly seventeen distinct plane patterns for a given choice of motif. These plane patterns *must* contain translations in two different directions, and may also contain reflections, glide reflections and rotations. There are only five of the seventeen patterns in which the motifs are not 'turned over', i.e. which contain only translations and rotations. This is a consequence of a theorem, known as Barlow's theorem, which states that the only possible periods of rotation in such patterns are 2, 3, 4 and 6 (and of course the trivial rotation, the identity, which has period 1). These periods of rotation correspond to the possible presence of only 2-fold, 3-fold, 4-fold, 6-fold, or 1-fold rotation axes in such patterns, where with "such patterns" is meant infinite *periodic* patterns.

Let's take a closer look to these rotation axes.

A 2-fold rotation axis implies a rotation of 180^{0}, which is of period 2.

A 3-fold rotation axis implies rotations of 120^{0} and 240^{0}, both of period 3.

A 4-fold rotation axis implies rotations of 90^{0} and 270^{0}, both of period 4 (Moreover it implies a rotation of 180^{0} which is of period 2).

A 6-fold rotation axis implies rotations of 60^{0} and 300^{0}, both of period 6 (Moreover it implies a rotation of 120^{0} and of 240^{0}, both of period 3, and a rotation of 180^{0}, which is of period 2).

A 1-fold rotation axis is a rotation of 0^{0} or of 360^{0}, which are of period 1 (Recall that these rotations are not just any rotations, they are cover operations (cover transformations) resulting in the object that is being rotated to occupy the same space as it did before it was rotated. Only such transformations (including reflections and other cover operations) describe the *symmetry* of the given object).

A proof that this is all there is -- with respect to rotations -- in plane periodic patterns, as well as in three-dimensional periodic patterns, is given in the first part of our website accessible by `back to homepage`, and there within the Special Series in the document *Derivation of Crystal Classes* I.

We will now look at the **17 basic periodic plane patterns** from a *synthetic* viewpoint, which, as has been explained earlier, means that we consider the *building up* of them by generating the group elements from two or more possible generators (Recall that the consideration of them from an *analytic* viewpoint was already done in the first part of our website, in the documents on the *Internal Structure of Crystals* (Part I -- XX) in the Special Series of documents). The *groups* describing the symmetry of such patterns are called *plane groups*. Each such pattern consists of an imaginary point lattice (indicated by lines, the intersections of which are the points) in which *motifs* have been placed.

A letter **P** in the plane group symbol means that the underlying (imaginary) point lattice is *primitive*, which means that the unit cell (the unit that is periodically repeated) contains motifs only at its corners, while in a *centered* (2-D) point lattice there is also a motif in its center. Plane groups of patterns that are underlied by such a centered point lattice have the letter **C** in their symbols. In all the 2-D patterns that we will show, we can detetect a *unit cell*, which is outlined by the auxiliary lines indicating the point lattice (net), i.e. a single mesh of that net. It is a unit that indicates the periodicity of the pattern **:** The area of the unit cell is repeated throughout the pattern without (that area) being turned.

Figure 1. *Placing 2-D motifs with point symmetry ***1*** into an oblique 2-D point lattice yields a pattern that represents the Plane Group ***P1***. This is not the generation of the pattern, but a description. The synthetic approach considers the generation of such a pattern by (chosen) generators. Such generators are given group elements (symmetry transformations) that generate all the other group elements (geometrically represented by the position and orientation of motifs, in all cases of symmetry groups).*

By the generators

The pattern must be considered as (becoming to be) indefinitely extending in 2-D space.

( The angle between the auxiliary lines need not necessarily be the (particular) one that has been used in the Figure, it can be any angle, i.e. the angle between the two translations can be any angle.)

The

Figure 1a. *Total symmetry content of Plane Group ***P1***.
The only symmetry present is : simple translation.*

Figure 2. *Placing 2-D motifs with point symmetry ***2*** into an oblique 2-D lattice yields a pattern that represents the Plane Group ***P2***. The pattern must be viewed as extending indefinitely in 2-D space.
To *

Rotation by 180

A translation

In the next Figure we give the

Figure 2a. * The total symmetry content of the Plane Group ***P2*** : simple translation (two translation vectors), 2-fold rotation axes.*

The next Figure shows that the pattern as a whole will return as it was before (i.e. occupy the same (patch of) space as it did before) after a half-turn about the point

Figure 3. *The point ***R*** marks a two-fold rotation axis to be present in the P2 pattern. There are of course many such axes, but this particular one is chosen to be one of the generators of the pattern from an initial motif, (the latter) indicated by the numeral ***1***. The other generators needed are two translations.*

The next Figure shows how the P2

Figure 4. *The pattern -- and with it the group elements -- can be generated by two translations ***s*** and ***t*** , and the half-turn ***a**.

The pattern must be viewed as becoming to extend indefinitely in 2-D space.

Figure 5. *Arranging motifs with point symmetry ***3*** in a hexagonal 2-D lattice yields a pattern that represents Plane Group ***P3***. The pattern must be conceived as to be indefinitely extended in two-dimensional space.
A unit cell, outlined by the hexagonal net, is indicated (light blue).
Each motif consists of three motif units. One such unit is considered as the initial motif unit, and is indicated by the numeral *

As generators, for building up this pattern, we choose a (horizontal) translation

So from the initial motif unit

Wherever we have some (already generated) motif unit (representing a group element), we can generate a new motif unit (representing a new group element) by rotating it 120

The

Figure 5a. *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 and translations.*

Figure 6. *This Figure shows the tri-radiate nature of the P3 pattern with respect to the point ***R*** . ( That point is explicitly indicated in the previous Figure). The whole pattern returns as it was before (i.e. occupies the same space as it did before), when we rotate it 120^{0} about *R

The next Figures show the generation of the P3 pattern. A number of motif units are provided with the indication of

Figure 7. *From the element ***1*** the elements ***t ^{-1} , t, t^{2}**

From the element

From the element

From the element

From the element

Figure 8.

From the element **p*** the element ***tp*** is generated by applying ***t*** , and from ***tp*** the element ***ptp*** is generated by applying ***p*** , i.e. an anticlockwise rotation of 120 ^{0} about the point *

From the element

Figure 9.

From the element **p ^{2}**

Figure 10.

From the element **p ^{2}**

From

Figure 11.

From the element **1*** the element ***t*** is generated by applying the translation ***t*** , and from the element ***t*** the element ***pt*** is generated by applying the rotation ***p*** about the point *R* (see Figure 5), and from the element ***pt*** the element ***tpt*** is generated by applying the translation ***t*** . From the element ***tpt*** the element ***p ^{2}tpt**

Figure 12.

From the element **p ^{2}**

Figure 13.

From the element **p*** the element ***t ^{2}p**

The next Figure shows the overall result. In fact the generation of ever new group elements must be conceived to go on indefinitely.

Figure 14. *The ***P3*** pattern, generated by the transformations ***p*** and ***t** .

*( The pattern must be conceived as becoming to be extended indefinitely in 2-D space)*

Here we give a two-dimensional periodic pattern with motifs at the nodes of a square lattice. Each motif consists of four motif units -- commas, as used above -- such that the whole (composed) motif has a point symmetry **4** , i.e. a motif having a 4-fold rotation axis as its only symmetry element. It admits of four rotations **:** 0^{0}, 90^{0}, 180^{0} and 270^{0}.

Figure 15. *When we place four-fold motifs (in the present case motifs, each consisting of four partly overlapping commas) at the nodes of a square point lattice (square point net), we obtain a pattern according to the Plane Group ***P4*** . A unit cell is indicated (light blue). The point ***R*** is the location of a four-fold rotation axis (There are more such axes). The element ***p*** is obtained by an anticlockwise rotation of 90 ^{0} about the point *

The next Figure shows the

Figure 15a. *The total symmetry content of any pattern representing Plane Group ***P4***. There are neither mirror lines, nor glide lines present.*

Figure 16. *This Figure shows the four-fold symmetry at the point ***R*** , which means that the whole pattern returns as it was before (i.e. it will occupy the same space as it did before) after it has been anticlockwise rotated either by 90 ^{0}, 180^{0}, or 270^{0} about the point *

As the second generator we have chosen the translation

We will now show how the P4 pattern -- and with it the group P4 -- is generated by the generators

Figure 17. *Generation of five new motifs (each one is a composed motif, consisting of four motif units), from the initial motif (which also consists of four motif units : the initial motif unit, and the motif units *

Our

- The initial motif unit, denoted by
**1**.

- The motif unit
**p**, formed as an image of**1**under anticlockwise rotation of 90^{0}about the point**R**.

- The motif unit
**p**, formed as an image of^{2}**1**under anticlockwise rotation of 180^{0}about the point**R**.

- The motif unit
**p**, formed as an image of^{3}**1**under anticlockwise rotation of 270^{0}about the point**R**.

Our intitial composed motif consists of the four motif units **1 , p , p ^{2}** and

We're now going to produce the image of this new composed motif under an anticlockwise rotation of 90^{0} about the point **R** , i.e. we subject it to the transformation **p** . This means that

**t** becomes **pt**

**tp** becomes **ptp**

**tp ^{2}** becomes

In this way we have obtained yet another new composed motif consisting of **pt , ptp , ptp ^{2} , ptp^{3}** , which we can find as the second composed motif of the top row in Figure 17.

When we now subject this last found composed motif to successively the translations **t ^{-1} , t** , and

**pt** becomes **t ^{-1}pt**

An this yields yet another new composed motif, which we can see as the first one in the top row of Figure 17.

**pt** becomes **tpt**

**ptp** becomes **tptp** = (tp)^{2}

**ptp ^{2}** becomes

This yields the third composed motif in the top row of Figure 17.

**pt** becomes **t ^{2}pt**

And this yiels the fourth composed motif in the top row of Figure 17.

So we now have indeed generated five new composed motifs from the initial composed motif.

Next we shall generate still more composed motif of our P4 pattern **:**

The **third row of composed motifs** can be generated along the same lines as we did it for the first row. The difference is that we now use an (anticlockwise) rotation (about the point R (As it is denoted in Figure 15 )) by ** 270^{0}** (instead of 90

Figure 18. *To the composed motif to the right of the initial motif the transformation ***p ^{3}**

Next we generate the rest of the second row of composed motifs by applying respectively the translations

Figure 19. *Two more composed motifs of the second row are generated by the translations ***t ^{-1}**

To generate composed motifs of the fourth row, we start from the last one of the second row, subject it to the transformation

Figure 20. *The fourth row of composed motifs can be generated from the fourth composed motif of the second row, using the transformations ***p ^{3}**

Of course one should continue this process indefinitely, because the group is infinite. However, it is by now clear how to generate the pattern, and with it the group governing its symmetry.

In the

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