(Part Eleven)

We continue our investigation concerning the generation of group elements of the Plane Groups. Here we will consider the Group

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 1.

Figure 1. *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. So these motifs have a three-fold symmetry, not a six-fold symmetry.*

Figure 2. *A unit mesh is chosen (yellow), it is primitive and has point symmetry ***1***, i.e. it has no symmetry whatsoever. The sides of the unit mesh are equal in length, and include angles of 60 ^{0} and 120^{0}.*

If we contract the pattern of Figure 1, representing Plane Group

Figure 3. *Translation-free residue of the pattern of Figure 1. It represents the Point Group of the Plane Group ***P3***.*

The symmetry elements involved in a pattern representing Plane Group

Figure 4. *A pattern representing Plane Group ***P3*** has 3-fold rotation axes. One of them is shown (small blue solid triangle).*

Figure 5. *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 6. *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.*

Figure 7. *Arranging motifs with point symmetry ***3*** in a hexagonal 2-D lattice yields a pattern that represents Plane Group ***P3***. (We have here used somewhat different motifs than those used above, but with the same symmetry). The pattern must be conceived as to be indefinitely extended in two-dimensional space.
A unit cell (unit mesh), 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

Figure 8. *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 9. *From the element ***1*** the elements ***t ^{-1} , t, t^{2}**

From the element

From the element

From the element

From the element

Figure 10.

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 11.

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

Figure 12.

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

From

Figure 13.

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 7), 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 14.

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

Figure 15.

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 16. *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)*

Above we had the group elements represented by a motif unit of the motif

Now we shall determine the maximum background of such a motif unit, such that we obtain an area that tiles (not necessary periodically) the plane completely and at the same time represents a group element. In order to do so we first determine the appropriate

The next Figure shows how our P3 pattern can consist of periodically stacked

Figure 17. *Motifs ***s.l.*** (colored hexagons) of the ***P3*** pattern of Figure 7. They tile the plane in a periodic manner. The different colors do not signify qualitative differences. The pattern must be imagined to extend indefinitely over the plane.*

Figure 18. *Motif ***s.l.*** isolated.
Left image : Isolated motif s.l. with lattice lines and other auxiliary lines.
Right image : Isolated motif s.l. without those lines.*

The next Figure shows the partition of the motifs

Figure 19. *Partition of the motifs ***s.l.*** and with it a partition of the ***P3*** pattern of Figure 7. The resulting areas (each of them is one third of a hexagon) can represent group elements.
The different colors of the areas do not signify qualitative differences.*

The next Figure indicates the initial element

Figure 20. *The ***P3*** pattern of Figure 7 is partitioned into areas representing group elements (as was already the case in the previous Figure). Three group elements are explicitly indicated (i.e. specified), namely the initial element (initial motif unit s.l., identity element), and two generator elements.*

We're now going to generate the rest of the group elements of the displayed part of our

Figure 21. *Generation of the group elements ***p ^{2}**

The next Figure indicates how new elements can be generated by rotating existing elements 120

Figure 22. *Indication how new group elements can be generated by applying the rotations ***p*** and ***p ^{2}**

Figure 23. *Generation of new group elements by applying rotations about the point ***R** .

By means of the horizontal translation

Figure 24. *Generating the rest of the group elements of the first three rows of motifs s.l.*

The next Figure indicates the rotation to be performed in order to reach the fourth row of motifs s.l.

Figure 25. *Indication of the performance to be done to the group elements, i.e. to the motif units, of the last motif s.l. of the second row. This performance consists in applying the generator ***p*** two times to those motif units (s.l.), i.e. a rotation of 240 ^{0} anticlockwise about the point *

Figure 26. *Generation of some group elements of the fourth row of motifs s.l. of the pattern representing Plane Group *P3 .

When we now use translations again, we can complete the fourth row of motifs s.l.

Figure 27. *Completion of the fourth row of motifs s.l. by applying translations.*

We have now completed the generation of all group elements of the displayed part of our

The origin and growing of a crystal -- which is here exemplified by means of imaginary two-dimensional crystals -- is the gradual *explication* of a structure that is already present in its entirety within the Implicate Order (Here we 'relevate' only one aspect of such a crystal, namely its internal symmetry. The same goes for the other aspects pertinent to the origin and growing of such a crystal). This process of explication -- which in direct perception we experience as the coming into being and growing of a crystal in a solution, vapor or melt -- can apparently only proceed along certain more or less definite lines, making it possible to state certain laws governing such a process, in our case certain crystallization laws. Indeed the ultimate laws of the Holomovement, carrying all implicate orders, is apparently such as to force such a specific order of explication. So the crystallization law apparently at work when we see a crystal emerge and grow is just an aspect of a much larger Whole, and is as such not a primary law as soon as we give primacy to the Implicate Order. When we give this primacy then we must consider the presence of the whole crystal at once, while its components -- seen just as aspects of the whole crystal -- form a pattern of different degrees of implication that changes in time. But the role of time is not primary, because the whole crystal is already present, albeit that certain components are still in an implicate condition. This implies a different description of *structure*, different from what is usually done in this respect. In this new description the structure of the (whole) crystal concerns the pattern of different degrees of implication at a certain moment. This degree of implication varies over the different components (aspects) of the crystal. This means that even when the crystal is only at the point of emerging, or is just very small, i.e. still a microscopic fragment, its description is about the *whole* crystal, i.e. the fully formed crystal. In this way the description is holistic. So if we describe an emerging crystal in the latter way, we are considering it in a *broader context* than just focussing on its explicate state. In the explicate state we consider operations like reflections rotations, translations, dilations, etc, which are one-to-one correspondences of points. We can denote these operations as *transformations*, and, more specifically as Euclidean transformations. These transformations describe lengths, angles, congruence, similarity, etc. When, on the other hand, an aspect of, say, a crystal becomes implicate, i.e. when we compare the explicate state of that aspect with its corresponding implicate state, then we have to do with an altogether different operation, in which there is __no__ one-to-one correspondence between points, and we will call such an operation a *metamorphosis* (We can obtain an idea of it -- in the sense of an analogy -- when thinking of the changes as we see them in holometabolic insects, for example the metamorphosis of a caterpillar into a butterfly, in which everything alters in a thorough going manner while some subtle and highly implicit features remain invariant. But, as has been said, this is only an analogy, because when we observe the changes in these insects in the usual way, i.e. remaining in the Explicate Order, then we can in principle observe a one-to-one correspondence between points in the caterpillar to points in the butterfly).

So explicate objects can be described by the above mentioned Euclidean transformations **E _{1} , E_{2} , E_{3} . . . . **. When such objects are subjected to a metamorphosis

resulting in a similar (but different) operation (E') (Because operations can together form algebraic structures like groups, we can use the above relation which expresses the 'tranformation of a transformation') . What the just mentioned similarity means is that if any two operations, say, **E _{1}** and

The non-locality of the operations **E _{1}', E_{2}'** (as supposed by BOHM,

BOHM, 1988, (

In this way the emerging crystal must be seen as a totality including both explicate and implicate conditions, and in this way the whole crystal is present at any moment, which explains the cooperative phenomena observed in crystal growth.

Along **Neoplatonic lines**, the general *state of implication*, i.e. the state of something that is implicated (in a stronger or lesser degree), can be imagined as being an immaterial discursive thinking process taking place in Soul (i.e. the World-Soul), a process that we have imitated by our generation of group elements of the relevant symmetry Group. In Nous, the next higher metaphysical level with respect to Soul, this thinking is *intuitive*, i.e. without step by step derivation, and as such the relevant symmetry Group is present there all at once, and constitutes a genuine **Idea** in the Neoplatonic sense. While the several Ideas in Nous are distinguished from each other, in **The One** they are also present, it is true, but no longer as distinguished from each other, because **The One** is *totally one*. As has been said in previous documents, **The One** can be more or less equated with the Holomovement which carries all the implicate and generative orders. So an Idea is present in all the metaphysical levels (hypostases), but always in a way according to the conditions of the relevant hypostasis. And only the final explication, i.e. the emanation from Soul to Prime Matter (which is pure potentiality for receiving form), results in concrete separate structures as seen at that level, i.e. at the Explicate Order.

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