Sequel to Group Theory

We'll start with reminding the reader about the

Sequel to Infinite two-dimensional periodic patterns

Two distinct Plane Groups,

Placing motifs in 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 1.

Each motif in Figure 1 is subdivided into three units. Each individual unit is still symmetric, and is, consequently __not__ a **basic** motif unit. And only a __basic__ motif unit can represent a group element, in our case an element of the group P3m1. The units as depicted in the next Figure (i.e. augmented motif units) represent subgroup and cosets of the group P3m1.

Figure 1. *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***.
Each (composed) motif consists of three augmented motif units, such that the symmetry of the composed motif is *3m

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

The

Figure 2. *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 3. *Total symmetry content of Plane Group ***P3m1***, depicted for one mesh of the net. Only the red solid lines are mirror lines.*

As has been said, the motif units as depicted in Figure 1 are

or perhaps still better (i.e. a little more indicative) **:**

And to further highlight the **basic** motif units that legitimately represent group elements, we could use colors, *provided we do not interpret a difference of color as an asymmetry*. The two (differently colored) parts of an augmented motif unit represent different group elements, but are nevertheless symmetrically related to each other

So now we have **six basic motif units** making up one composed motif of the P3m1 pattern.

To generate the full group we need one such basic motif unit to represent the identity element **1** , another basic motif unit to represent a first (of the three needed) generator, namely an anticlockwise rotation **p** of 120^{0} about a certain fixed lattice point (which we will call the point R), yet another such unit to represent a second generator, namely a reflection in the line **m** making an angle of 30^{0} with the horizontal lattice connection lines, and passing through the point R, and, finally, yet another basic motif unit to represent the third generator, namely the translation **t** .

Let's indicate the identity element, the rotation (first generator) and the reflection (second generator) in an enlarged composed motif **:**

Figure 3a. *The composed motif at a chosen lattice point *R* . As such it consists of six basic motif units representing elements of the group P3m1. One unit is chosen to be the identity element ***1*** , the other two are the generators ***p*** and ***m** .

Let us now put all this in the context of the lattice. We then indicate the lattice points R and S, the identity element

Figure 3b. *Tri-radiate composed motifs are inserted into a hexagonal point lattice, such that the three mirror lines of those motifs do not coincide with the lattice lines (i.e. with the edges of the rhomb-shaped unit cell, as indicated in the Figure), resulting in a periodic pattern according to the Plane Group P3m1.
Each composed motif consists of three augmented motif units, while each augmented motif unit consists of two basic motif units (red and blue) representing elements of the group P3m1.
One such basic motif unit (of the composed motif at the lattice point R ) is chosen to represent the identity element *

One mirror line, the line

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

Figure 3c. *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 3d. *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 3e. *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 3b) can now be completed by applying the translations

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

Figure 3f. *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 3b) of composed motifs (to be generated), by applying

Figure 3g. *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

Next we will generate this same group P3m1 by means of

Such an augmented motif unit consists of two symmetrically related basic motif units. When using such augmented motif units a full composed motif will consist of three such motif units, and when we use the same notation as we did above, the composed motif at the lattice point R will look as follows

**or**

for that matter.

Because we now use *augmented* motif units, which themselves are symmetric, i.e. possess a mirror line, we do not need the generator **m** (i.e. an element resulting from a reflection of the initial element in a mirror line **m** ). The only generators we now need are **p** , which is an anticlockwise rotation of 120^{0} about the point R ( This generator is -- as augmented motif unit -- indicated in the above two images of the composed motif at the point R), and a (horizontal) translation **t** , represented by an augmented motif unit **t** . Also with respect to the augmented motif units, **p ^{3} = 1** holds. The augmented motif units are based on the subgroup

Now our newly conceived motif units, the augmented motif units (consisting of two symmetrically related

The next Figure shows the two chosen generators **:**

The 'element' **p** , represented by the augmented motif unit **p** , that results from the element (augmented motif unit) **1** (chosen to be the identity element, initial motif unit), by an anticlockwise rotation of 120^{0} about the point **R** .

The 'element' **t** , that results from the 'element' **1** by a translation **t** .

Figure 4. *Two chosen generators ***p*** and ***t*** that can generate the whole P3m1 pattern.*

Figure 5. *The whole P3m1 pattern can be produced by the generator elements ***p*** and ***t** .

This is the last pattern of the 17 fundamental two-dimensional periodic patterns.

The motif we're now 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** above. 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 6.

Figure 6.

(1). Orientation of the **3m*** motif, compatible with the Plane Group ***P3m1*** (discussed above).
(2). Orientation of the *

Both motifs (one for expressing the group P3m1, the other for expressing the group P31m) are placed in a same hexagonal net, i.e. a hexagonal net with rhomb-shaped meshes having two edges horizontally.

The

Thus placing two-dimensional motifs with

Figure 7. *When motifs, having point symmetry ***3m*** (i.e. having a 3-fold rotation axis and three equivalent mirror lines), are inserted in 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 pattern must be conceived as extending indefinitely in two-dimensional space.

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

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 8a. *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 8b. *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 8c. *Composed motif (consisting of six basic motif units) of the P31m pattern at the point R in Figure 8b.*

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

Figure 8d. *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 8e. *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 8f. *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 8b) can now be completed by applying the translations

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

Figure 8g. *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 8b) of composed motifs (to be generated), by applying

Figure 8h. *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 8i. *Composed motif (at the lattice point ***R*** (For its position, see Figure 8b)), 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 9. *The two chosen generators ***p*** and ***t*** for the P31m pattern.*

Figure 10. *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

This

A more extensive treatment of the generation of group elements of the 17 Plane Groups, and, moreover, embedded in a broader philosophical context, is given in the

After now having given a more or less general account of **Group Theory** (Part I -- XXIV), in the **next document** we continue this study, while (still) concentrating on the symmetry of **two-dimensional patterns**. First we will discuss **subpatterns** and **subgroups** of the 17 Plane Groups (all seventeen of which we have just treated globally above). Those subpatterns and subgroups will give us insight into the phenomena of **layers** as they occur in the growth of real (3-D) crystals, and in the phenomenon of symmetry-breaking or desymmetization (differentiation) occurring in natural processes and in (ornamental) art. But there are still more things that we should know about symmetry **:** The symmetry of the mentioned periodic patterns and also of their motifs, are all **isometric symmetries**, which means that they are transformations (in the present case transformations in the Euclidean Two-dimensional Plane E^{2} ) such that *distances* are preserved. When we generalize on that, we get the so-called **similarity symmetries**, in which distances are not preserved, while *angles and shapes* (still) are. A still further generalization leads to so-called **conformal symmetries** in which *circles* are preserved. In addition to isometric symmetries, the similarity and conformal symmetries -- which do not occur in crystals of whatever dimension, but do occur in organisms -- can provide a refinement and sophistication of quite a few *Promorphological Categories*, said differently, the study of the symmetries of two-dimensional patterns, including similarity and conformal symmetries, will deepen our insight in the shapes and structures of real crystals (isometries) and of organisms (isometries, similarity symmetries and conformal symmetries). With respect to **organisms** we will study their symmetries in the ensuing documents on **Promorphology** (starting after the Tectology Series).

When all this has been accomplished (i.e. the [group theoretic] study of the symmetries of two-dimensional patterns (periodic and non-periodic)), we are finally ready to embark on the **symmetry theory of three-dimensional crystals**, i.e. we study the pointgroups, lattices and space groups of three-dimensional (single) crystals by means of Group Theory.

When this is all done we conclude our study of three-dimensional crystals with discussions about chemical lattice types, twinning, and the thermodynamics of crystal growth. (And after this, the mentioned documents about **Organic Tectology** (general doctrine of structure in organisms) and ** (Organic) Promorphology** (general doctrine of symmetry in organisms **:** the stereometric basic forms of organisms) follow (and which are already finished, although Promorphology must still become group theoretically refined).

The documents about the subpatterns and subgroups of two-dimensional periodic patterns, about the similarity and conformal symmetries of two-dimensional (non-periodic) patterns, and about the symmetry theory of three-dimensional crystals, are to be found by clicking the next "*To continue click HERE *" link. But, of course one can skip all this, and directly proceed (by using the left frame of the present document) to the documents on lattice types, twins and crystal thermodynamics (which are however not yet finished), or directly to the Tectology and Promorphology documents (which *are* finished, except some group theoretic polishing up).

In order not to loose track of this admittedly complicated website, we will tabulate systematically the **context** of the previous documents on group theory, the present documents, and the documents that follow after having clicked the *To Continue* link below, and, finally, what follows after those documents (The complexification of LINKING is partially due to the limited available space on the webserver, that has caused us to place a whole intermediate series of documents (concerning the sequel of two-dimensional patterns and the symmetry theory of three-dimensional crystals) on *another* webserver, implying that we must link to that server and than, later, back again to the present server).

So the context and order of the relevant documents on **symmetry** is as follows **:**

- First Series of Documents
**:**(Part I -- XXIV) (present webserver) [The present document is Part XXIV of this Series].

General Group Theory - Second Series of Documents (Second webserver)
**:**(With the periodic ones we already made a beginning in the first series of documents [and also -- as an extensive introduction -- in the last Series of Documents of the First Part of the website, accessible by

Group theoretic exposition of the symmetry of two-dimensional patterns`back to homepage`] ).

Group theoretic Symmetry theory of three-dimensional crystals.

And from here to first webserver again, and staying there**:** - Third Series of documents (first webserver)
**:**(not yet finished)

Actual Structure and Growth of Three-dimensional Crystals. - Fourth Series of documents (first webserver)
**:**. Doctrine of general structure of Organisms, influencing the latter's symmetry.

Tectology Series - Fifth Series of documents (first webserver)
**:**. Symmetry of Organisms (Basic stereometric forms of organisms).

The Promorphological System - Sixth Series of documents (first webserver)
**:**Those orders were established in the Tectology Series.

Basic (stereometric) forms of the Six Individuality Orders in Organisms. - Seventh Series of documents (first webserver)
**:**. The symmetry of (3-D) crystals according to the Promorphological System of basic forms.

Basic (stereometric) forms of Crystals

This Series is concluded with a document concerning**Promorphological Theses and Tables**.

This is at the same time the**end**of the**Second Part of the website**.

The next *To continue --- * link brings us to the Second Series of documents (second webserver) that continues the group theoretic treatment of two-dimensional patterns and the symmetry theory of three-dimensional crystals (So also the link SEQUEL TO GROUP THEORY in the left frame).

**e-mail : **

To continue click HERE

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