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- implicate order 12

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 1 A second basic motif unit is chosen to be the generator p a third one is chosen to be the generator m and a fourth basic motif unit which is part of the composed motif at the point S is chosen to be the generator t One mirror line the line m is indicated It is the symmetry element of the generator m The identity of the remaining basic motif units of the composed motif at the lattice point R can now be determined p 3 1 Figure 13 The group elements of the composed motif at the point R Together they form the subgroup 1 p p 2 m mp mp 2 having the structure of D 3 The elements 1 p p 2 form a subgroup with the structure of C 3 The elements 1 and m form the subgroup C 2 We will now produce the composed motif at the point S Figure 12 We ll do this by subjecting all the elements basic motif units of the composed motif at the point R to a translation t i e we will form the left coset of the subgroup D 3 materialized as the composed motif at the point R by the element t See next Figure Figure 14 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 t The notations i e identities of the newly generated elements are placed at the perimeter of the image i e outside the image Next we determine the elements of the composed motif at the lattice point U See Figure 12 To generate those elements we must subject the elements of the composed motif at the point S to an anticlockwise rotation of 240 0 about the point R that is to say we must apply the transformation p 2 In this way we form the left coset of the subgroup D 3 by the element p 2 t See next Figure Figure 15 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 12 can now be completed by applying the translations t 2 t 1 t t 2 t 3 etc We will do so with respect to the composed motif at the

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of the elements around the lattice point next to R to its right The possible rotations are represented by the elements p and p 2 which are anticlockwise rotations of resp 120 0 and 240 0 about the point R See also next Figure for a better overview Figure 5 Smaller scale image of the presently used pattern representing the Group P3m1 also showing the possibility of generating new group elements by rotation about the point R The next Figure generates the group elements that result from the above mentioned rotations of the elements situated around the lattice point next right to the lattice point R Figure 6 The first and third rows of motifs s l have been reached by means of the above mentioned rotations about the point R The first and third rows can now be completed by repeatedly applying the generator t and its inverse t 1 Figure 7 Completion of the first and third rows of the motifs s l of the pattern according to the Plane Group P3m1 Next we will again use the rotation p 2 in order to obtain group elements of the fourth row of motifs s l This rotation is indicated in the next Figure Figure 8 We can reach the fourth row of motifs s l of our P3m1 pattern by rotating the elements of the last motif s l of the second row about the point R 240 0 anticlockwise which is equivalent to applying p 2 Figure 9 Reaching the fourth row of motifs s l of our P3m1 pattern by means of the rotation p 2 of the elements of the last motif s l of the second row This fourth row can now be completed by applying translations to the lastly obtained group elements Figure

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p representing an anticlockwise rotation of 120 0 about the lattice point R The basic motif unit m representing a reflection in the mirror line m The basic motif unit t representing a horizontal translation t Figure 10 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 R The initial basic motif unit 1 the generators p and m and the mirror line m are indicated Figure 11 Composed motif consisting of six basic motif units of the P31m pattern at the point R in Figure 10 The identity of the remaining basic motif units of the composed motif at the lattice point R can now be determined p 3 1 Figure 12 The group elements of the composed motif at the point R Together they form the subgroup 1 p p 2 m mp mp 2 having the structure of D 3 The elements 1 p p 2 form a subgroup with the structure of C 3 The elements 1 m form the subgroup C 2 We will now produce the composed motif at the point S Figure 10 We ll do this by subjecting all the elements basic motif units of the composed motif at the point R to a translation t i e we will form the left coset of the subgroup D 3 materialized as the composed motif at the point R by the element t See next Figure Figure 13 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 t The notations i e identities of the newly generated elements are placed at the perimeter of the image i e outside the image Next we determine the elements of the composed motif at the lattice point U Figure 10 To generate those elements we must subject the elements of the composed motif at the point S to an anticlockwise rotation of 240 0 about the point R that is to say we must apply the transformation p 2 In this way we form the left coset of the subgroup D 3 by the element p 2 t See next Figure Figure 14 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 10 can now be completed by applying the translations t 2 t 1 t t 2 t 3 etc We will do so with respect to the composed motif at the point X Figure 15 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 t 2 The notation of the

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Completion of the second row of motifs s l i e all the group elements of this row are now generated The next Figure indicates the rotations that can be performed on the newly generated elements i e anticlockwise rotations of 120 0 and 240 0 about the point R in order to generate new group elements Figure 4 Indication of the rotations about the point R for generating new elements in the first and third row of motifs s l Figure 5 The first and third row of motifs s l of our P31m pattern have been reached by rotations and new group elements are generated These first and third rows can now be filled in by means of translations Figure 6 Completion of the first and third rows of motifs s l The next Figure indicates how the fourth row of motifs s l can be reached by an anticlockwise rotation of 240 0 about the point R which is the transformation p 2 of the elements of the last motif s l of the second row Figure 7 Indication how the elements of the fourth row of motifs s l can be reached by a rotation Figure 8 The fourth row is reached by the rotation p 2 of the elements of the last motif s l of the second row about the point R The fourth row can now be completed by means of translations Figure 9 Completion of the fourth row of motifs s l of the P31m pattern by applying translations We now have generated all group elements of the displayed part of the P31m pattern In the next document we will generate the group elements of the Plane Group P6 e mail To continue click HERE for further study of the totally dynamic

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motifs See next Figure And from this lastly obtained composed motif we can produce the other composed motifs in that first row by applying translations Figure 4 Generation of several composed motifs from the initial composed motif From the third composed motif in the second row we can produce the fourth composed motif in that same row by applying the translation t and from this new composed motif in turn we can produce a composed motif in the fourth row the second one in that row by applying the transformation p 4 which is an anticlockwise rotation of 240 0 about the point R as indicated in Figure 1 This fourth row can then be completed by applying translations Directly from the third composed motif in the second row we can produce a composed motif in the third row by applying p 4 Also this third row can now be completed by applying translations The next Figure illustrates the production of the two new composed motifs obtained by applying the transformation p 4 Figure 5 Generation of two more composed motifs in addition to one obtained by a translation of the pattern according to the Plane Group P6 We will now generate the group elements of the group P6 again but now letting them be represented not only by a motif unit of the motif s str but by such a unit PLUS its corresponding background So the group elements will be represented by maximal areas of the pattern which are equally sized and shaped but not necessarily equally oriented and which tile the plane without any gaps or overlappings In order to determine such areas we first determine a suitable choice of motif s l Such motifs s l will then be partitioned into six areas representing group elements Figure 6 Motifs s l yellow hexagons of the P6 pattern of Figure 1 Each motif s l consists of one motif s str PLUS corresponding background Partition of such motifs s l will yield areas that can represent group elements The next Figure shows the partition of the motifs s l into equally sized and shaped areas that can represent group elements of our P6 pattern Figure 7 The motifs s l are each divided into six areas bi isosceles red and yellow triangles Each such area can represent a group element It consists of one basic unit of the motif s str or at least it represents it PLUS corresponding background The colors red and yellow in the partitioned motifs s l do not necessarily signify difference in symmetry So at the center of each motif s l there is a six fold rotation axis and not just a three fold rotation axis Figure 8 One partitioned motif s l isolated The areas A B C D E F represent group elements They are situated around a six fold rotation axis The next Figure indicates the areas that represent the initial element identity element and the two generators

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group elements represented by basic motif units indicated as they were produced by the translation t with respect to the composed motif at the point R Figure 8 The composed motif next to the right of the one at the point R The red motif units in Figure 8 together form the left coset of the subgroup 1 p p 2 p 3 p 4 p 5 with structure C 6 See Figure 6 by the element t All the twelve elements red and blue in Figure 8 together form the left coset of the D 6 subgroup See Figure 6 by the element t For the group theoretic concept c o s e t see our exposition of Group Theory in Second Part of Website In the present context it is not of direct importance Next we re going to generate the composed motif at the point Q in Figure 7 It can be obtained by rotating the previously generated composed motif the motif next to the one at point R 300 0 anticlockwise or which is the same 60 0 clockwise about the point R which means that we subject all elements of the previously generated composed motif to the action of p 5 The next Figure gives this new composed motif Inside the image we have left the notations for the elements as they were in the motif next to R while the identities of the newly generated group elements are given at the perimeter of the image Figure 9 The newly generated composed motif at lattice point Q as indicated in Figure 7 The names of the new elements are given at the perimeter of the image i e outside the image All the elements in Figure 9 red and blue together form the left coset of the D 6 subgroup See Figure 6 by the element p 5 t The third row Figure 7 of composed motifs can now be completed by means of applying to this lastly obtained composed motif the translations t 3 t 2 t 1 t t 2 t 3 And along the same lines we can complete row 2 of the pattern In order to reach the fourth row we first determine the composed motif at the point U by applying t 2 2 times applying the translation t to the elements of the composed motif at the point R and then rotate it 300 0 anticlockwise about the point R i e applying p 5 Figure 10 The composed motif at the point W Figure 7 is completed by applying p 5 to the composed motif at the point U The original element notations as they were at point U have been put inside the image while at its perimeter we have put the notations of the newly generated elements of the composed motif at the point W The six red motif units of the above Figure Figure 10 together form the left coset of the subgroup 1 p p 2 p 3 p 4 p 5 with structure C 6 by the element p 5 t 2 This fourth row can now be completed by means of translations It is clear that by using the rotations and translations we can generate in principle the whole group The generator m i e the reflection in the line m was needed to obtain a reflected motif unit which then was multiplied by the rotations and translations and these latter two transformations can never produce a reflected motif unit The same P6mm pattern can be generated by an augmented motif unit which could be for instance two symmetrically related basic motif units Figure 11 Composed motif of the P6mm pattern consisting of six symmetric motif units Two of them are provided with a notation Each such motif unit in fact consists of two symmetrically related basic motif units These non basic motif units cannot represent group elements The one denoted by 1 is in fact the subgroup 1 mp in the notation above See Figure 6 referring to basic motif units which has the structure of C 2 The element mp is of period 2 which can easily be verified in Figure 6 and a set consisting of the identity element and a period 2 element is a group of C 2 structure We will now generate the P6mm pattern by means of the non basic motif units established in Figure 11 We will use the same notation as we did with respect to the basic motif units Because we now use symmetrical motif units which as has been said consists of two symmetrically related basic motif units we do not need the generator m anymore We can do it with the generators p rotation and t translation So now to begin with we can generate the other elements of the composed motif representing the subgroup D 6 Figure 12 By the generator p the remaining elements of the subgroup D 6 are generated Given 1 initial element and p generator the element p 2 results after two times applying p to the initial element The element p 3 results from three times applying p etc while p 6 1 where p is an anticlockwise rotation of 60 0 about the point R The next Figure summarizes the two chosen generators as represented by non basic motif units augmented motif units that can generate the whole group P6mm The element augmented motif unit p resulting as just established from the element 1 which is chosen to represent the identity element by an anticlockwise rotation of 60 0 about the point R The element augmented motif unit t resulting from element 1 in virtue of the translation t We must realize that although speaking about elements they are not true elements of the group but subgroups or cosets If we were to insist that they are true group elements then we would not in fact have to do with the group P6mm

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m B5 t 1 ptp B6 t 1 ptm B7 t 1 ptp 2 B8 t 1 ptpm B9 t 1 ptp 3 B10 t 1 ptp 2 m B11 t 1 ptp 4 B12 t 1 ptp 3 m C1 ptp 5 C2 ptp 4 m C3 pt C4 ptp 5 m C5 ptp C6 ptm C7 ptp 2 C8 ptpm C9 ptp 3 C10 ptp 2 m C11 ptp 4 C12 ptp 3 m D1 tptp 5 D2 tptp 4 m D3 tpt D4 tptp 5 m D5 tptp D6 tptm D7 tptp 2 D8 tptpm D9 tptp 3 D10 tptp 2 m D11 tptp 4 D12 tptp 3 m E1 t 2 ptp 5 E2 t 2 ptp 4 m E3 t 2 pt E4 t 2 ptp 5 m E5 t 2 ptp E6 t 2 ptm E7 t 2 ptp 2 E8 t 2 ptpm E9 t 2 ptp 3 E10 t 2 ptp 2 m E11 t 2 ptp 4 E12 t 2 ptp 3 m F1 t 1 F2 t 1 p 5 m F3 t 1 p F4 t 1 m F5 t 1 p 2 F6 t 1 pm F7 t 1 p 3 F8 t 1 p 2 m F9 t 1 p 4 F10 t 1 p 3 m F11 t 1 p 5 F12 t 1 p 4 m 1 1 R2 p 5 m p p m m R5 p 2 R6 pm R7 p 3 R8 p 2 m R9 p 4 R10 p 3 m R11 p 5 R12 p 4 m G1 t G2 tp 5 m G3 tp G4 tm G5 tp 2 G6 tpm G7 tp 3 G8 tp 2 m G9 tp 4 G10 tp 3 m G11 tp 5 G12 tp 4 m H1 t 2 H2 t 2 p 5 m H3 t 2 p H4 t 2 m H5 t 2 p 2 H6 t 2 pm H7 t 2 p 3 H8 t 2 p 2 m H9 t 2 p 4 H10 t 2 p 3 m H11 t 2 p 5 H12 t 2 p 4 m I 1 t 2 p 5 tp I 2 t 2 p 5 tm I 3 t 2 p 5 tp 2 I 4 t 2 p 5 tpm I 5 t 2 p 5 tp 3 I 6 t 2 p 5 tp 2 m I 7 t 2 p 5 tp 4 I 8 t 2 p 5 tp 3 m I 9 t 2 p 5 tp 5 I 10 t 2 p 5 tp 4 m I 11 t 2 p 5 t I 12 t 2 p 5 tp 5 m J1 t 1 p 5 tp J2 t 1 p 5 tm J3 t 1 p 5 tp 2 J4 t 1 p 5 tpm J5 t 1 p 5 tp 3 J6 t 1 p 5 tp 2 m J7 t

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all the way down and which in itself is pure potentiality for receiving form is being informed resulting in material substances that are each for themselves in the strongest way possible separately existing i e separate from but interacting with other such material substances This material level is the Explicate Order As has been said the outflow from The One results in implicate orders This is however not a process of enfolding but a process of unfolding because The One is hyper implicate This unfolding continues until it has arrived at prime matter The latter is then informed resulting in maximally unfolded structures When on the other hand we look from material structures informed matter in the reverse direction of this emanation the return to The One then we see these material structures becoming enfolded again and so becoming part of the Implicate Order This enfolding intensifies in approaching The One until these structures are totally absorbed by the latter The crystallization process can be described within the Explicate Order with a high degree of approximation to the real process but in crystallization there are phenomena that seem to point to a certain degree of cooperation between the relevant chemical participants in a way that seems not to be explicable in terms of chemistry and physics as we see them in the Explicate Order These phenomena could therefore be indications that a deeper level is involved in determining aspects of the crystallization process This involvement is such that we experience it as cooperation In this crystallization process we have concentrated on one aspect only namely the internal symmetry of the emerging and growing crystal Together with the nature of the chemical participants symmetry determines much of the emerging structure of a crystal This internal symmetry can be described by exactly 230 Space Groups for real crystals while our two dimensional analogues can be described by exactly 17 Plane Groups In considering these analogues we have assumed that their symmetry is in an abstract and enfolded way already subsisting in some implicate order or hypostasis We can imagine the way it is so subsisting as being present as symmetry groups in the form of a thought process noetic process i e as continually self generating symmetry groups In the eighteen previous documents we have described a version of this noetic process for each Plane Group And this immaterial process corresponds to the mentioned deeper level which is responsible for the crystallization processes as we see them happening in the Explicate Order and which reveals itself by the phenomena of cooperation And because the 17 Plane Groups form a self contained logical whole we have treated them all i e we have shown the noetic generation process for all of them We will now summarize the 17 fundamental two dimensional patterns according to the 17 Plane Groups and the way those patterns can be partitioned in tesselating areas that represent group elements These are then the 17 periodic patterns partitioned according to

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