We continue our investigation concerning the generation of group elements of the Plane Groups. Here we will consider the Group P3 .
The Plane Group P3
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 600 and 1200.
If we contract the pattern of Figure 1, representing Plane Group P3, such that all translations are eliminated, then we end up with a figure having a point symmetry 3. This figure then represents the Point Group (Crystal Class) 3 to which that Plane Group belongs. The figure is at the same time the motif s.l. of the pattern. It tiles the 2-D plane completely. See Figure 3.
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 P3 are 3-fold rotation axes only. Through every node of the net there is such an axis, and a pair of them is situated in each mesh. See Figures 4 and 5.
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 total symmetry content of the Plane Group P3 is given in the next Figure.
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.
Generation of the group elements of the Plane Group P3
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 1.
As generators, for building up this pattern, we choose a (horizontal) translation t and an anticlockwise rotation of 1200, denoted by p about the point R .
So from the initial motif unit 1 (representing the identity element of the group) the motif unit p is generated (which represents the group element p ). And when -- on the initial motif unit -- the rotation p is twice applied, we get the motif unit p2, and with it the corresponding group element p2. Three times applying p yields the identity element, so p3 = 1.
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 1200 about the point R . Another new motif unit can be produced by shifting it according to the translation t. (See also Figure 8).
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 1200 about R . The point R is not the only point with this property : There are many such points. But the rotation about the specific point R by 1200 anticlockwise, is chosen as a generator of the group. Any already existing motif unit (representing a group element) of the pattern yields a new motif unit (representing a new group element), when it is subjected to rotation of 1200 or of 2400 about the point R . In addition to this particular generator a second generator is needed, which is chosen to be a horizontal translation to the right (and implying also such a translation to the left). By this translation the motif units (representing group elements) t-1, t , t2, etc. are generated from the initial motif.
The next Figures show the generation of the P3 pattern. A number of motif units are provided with the indication of how they can be generated by means of the two generators p (rotation of 1200 about the point R (indicated in Figure 7)), and t (translation horizontally to the right).
From the element 1 the elements t-1 , t, t2 , etc. are generated by the element t .
From the element 1 is generated the element p by applying the transformation p , which is an anticlockwise rotation by 1200 about the point R (indicated in Figure 7).
From the element p is generated the element p2 by (again) applying p .
From the element p are generated the elements t-1p , tp , t2p , etc. by applying t .
From the element p2 are generated the elements t-1p2 , tp2 , t2p2 , etc. by the element t .
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 1200 about the point R (as indicated in Figure 7).
From the element ptp the elements tptp , t2ptp , etc. are generated by applying t .
From the element p2 the element tp2 is generated by applying t , and from tp2 the element ptp2 is generated by applying p (rotation of 1200 anticlockwise about R ), and from ptp2 the elements tptp2 , t2ptp2 , etc. are generated by t .
From the element p2 the element tp2 is generated by applying t , and from the element tp2 the element p2tp2 is generated by applying p2 (A rotation of 2400 anticlockwise about the point R (As given in Figure 7).
From p2tp2 are generated the elements t-1p2tp2 , tp2tp2 , etc. by applying t .
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 p2tpt is generated by applying the rotation p2 (about the point R ).
From the element p2 the element t2p2 is generated by applying the translation t2 , and from the element t2p2 the element p2t2p2 is generated by the rotation p2 . And from the element p2t2p2 the elements t-1p2t2p2 , t p2t2p2 , etc. are generated by applying the translation t .
From the element p the element t2p is generated by applying the translation t2 (i.e. applying the translation t two times), and from the element t2p the element p2t2p is generated by the rotation p2 . And from the element p2t2p the elements t p2t2p , t2 p2t2p , etc. are generated by applying the translation t .
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 s.str.
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 s.l. of the above Figure. Each resulting area is one third of a hexagon and contains one motif unit of the motif s.str. As such those areas can represent group elements.
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 1 and the two generator elements p and t .
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 P3 pattern.
Figure 21. Generation of the group elements p2 and, by translation, the elements tp and tp2 .
The next Figure indicates how new elements can be generated by rotating existing elements 1200 and 2400 anticlockwise about the point R .
Figure 22. Indication how new group elements can be generated by applying the rotations p and p2 to the elements t, tp, tp2 .
Figure 23. Generation of new group elements by applying rotations about the point R .
By means of the horizontal translation t , its inverse, and their repetitions, we can now finish the first three rows of motifs s.l., i.e. fill in the symbols of the group elements involved in these motifs :
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 2400 anticlockwise about the point R .
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 P3 pattern. Of course the process must be continued indefinitely.
Implicate and Explicate Order
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 E1 , E2 , E3 . . . . . When such objects are subjected to a metamorphosis M (which, as has been said, is a special operation) they become, in virtue of that, implicate. The resulting implicate structure, as such residing in, and according to, the Implicate Order, can in turn be described by operations that are appropriate to the (conditions of the) Implicate Order. The metamorphosis M effects the transform (in the group theoretical sense) E' of an Euclidean operator E according to the relation
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, E1 and E2 , are related in a certain way in the description of a specified structure, then there is a set of operations E1' and E2' , describing non-local 'enfolded' operations that are related in a similar way :
The non-locality of the operations E1', E2' (as supposed by BOHM, Wholeness and the Implicate Order, 1988, Ark edition, p. 165) means that the relations between the 'parts' of the implicated object cannot be described by Euclidean operations, in which distances and the like play a role. And this way of implicatenes (i.e. its non-locality) fits well with the conjecture that implicate structures are immaterial objective thought structures (See below).
BOHM, 1988, (Op. cit.) suggests that the explicate and implicate conditions of something are equivalent to each other in the sense that they are implicate relative to each other (or explicate relative to each other), and so extending the principle of relativity with respect to coordinate systems corresponding to different velocities to implicate and explicate conditions. We cannot, however, agree with him on this point : The implicate condition represents a different ontological status, different from the explicate condition. In the Implicate Order everything is enfolded into everything else, while this is not so in the Explicate order. The idea of (ordinary) space is a space constituted of a set of unique and well-defined points, related topologically by a set of neighborhoods and metrically by a definition of distance. This idea is (also according to BOHM) no longer adequate as an absolute and universal description. Indeed, each set of Euclidean operations E defines such a set of points, neigborhoods, measures, etc., which are explicate in an absolute sense (and not in a relative sense as BOHM proposes), while those defined by another (corresponding) set E' (related by E' = MEM-1) are implicate in an absolute sense. So the notion of space as a set of points with a topology and a metric is thus merely (also according to BOHM) an aspect of a broader totality which includes explicate and implicate order.
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.
In the next document we will generate the elements of the Plane Group P3m1 .
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