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- implicate order 4

by the horizontal dashed line The half turn r about the point R See also Figure 4 The vertical translation t First we generate some group elements by subjecting the initial element 1 repreatedly to the glide reflection g Figure 15 Generation of the group elements g 2 g 1 g 2 g 3 by applying the glide reflection the generator element g repeatedly to the initial group element 1 The above action of the glide reflection g is highligted in the next Figure Figure 16 The action of the glide reflection g The next Figure shows the generation of the group element rg 1 from the element g 1 by applying the half turn about the point R Figure 17 Generation of the group element rg 1 from the element g 1 obtained earlier We now generate new group elements from the element rg 1 by applying to it the translations t 1 and t 2 Figure 18 Generation of the group elements t 1 rg 1 and t 2 rg 1 from the element rg 1 by applying to it the translations t 1 and t 2 By means of the half turn r about the point R which is a generator we can generate yet a further group element Figure 19 Generation of the group element rg from the element g by appying to it the half turn about the point R Also from the element g we can generate new group elements tg and t 2 g by applying the translations t and t 2 Figure 20 Generation of the group elements tg and t 2 g By the half turn about the point R we can generate yet another group element Figure 21 Generation of the group element rtg Again by the half turn about the point R we can generate yet another group element Figure 21 Generation of the group element rt 1 rg 1 from the element t 1 rg 1 From the element g 1 we can generate the element t 2 g 1 Figure 22 Generation of the group element t 2 g 1 From the element r we can generate the elements tr and t 1 r Figure 23 Generation of the group elements tr and t 1 r From the group element g 2 we can generate two more group elements Figure 24 Generation of the group elements tg 2 and t 1 g 2 From the element rtg we can generate yet another element Figure 25 Generation of the group element t 1 rtg From the group element rt 1 rg 1 we can generate more elements by applying to it the glide reflections g 1 g g 4 which all are implied by the generator g Figure 26 Generation of the group elements g 1 rt 1 rg 1 grt 1 rg 1 g 4 rt 1 rg 1 by applying the glide reflections g 1 g g 4 to the element rt 1 rg 1 From

Original URL path: http://www.metafysica.nl/holism/implicate_order_4.html (2016-02-01)

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pattern representing Plane Group P2mg has mirror lines parallel to the y direction One of them is depicted Figure 8 The pattern representing Plane Group P2mg has glide lines parallel to the x direction One of them is depicted The total symmetry content of Plane Group P2mg is given in the next Figure Figure 9 Total symmetry content of the Plane Group P2mg Solid red lines indicate mirror lines Small solid red ellipses indicate 2 fold rotation axes perpendicular to the plane of the drawing The glide lines are all parallel to the x direction some going through the mid line of the rectangles others coinciding with their vertical sides They are indicated by red dashed lines Generation of the group elements of the Group P2mg The P2mg pattern as realized in Figure 2 consists of composed motifs Each such motif itself consists of two motif units partly overlapping commas which represent group elements and each group element is a symmetry transformation of the pattern For generating the P2mg pattern and with it the elements of the group P2mg we choose the following set of generators The element g which results from the chosen initial motif unit representing the identity element 1 as the effect of the glide reflection glide line g The element m which results from the reflection of the initial motif unit in the mirror line m The element t which results from the initial motif unit in virtue of a horizontal translation t See next Figure Figure 10 Three generators for the P2mg pattern are indicated The next Figure shows how all the group elements can be generated from the three generators g m and t Figure 11 Generation of the P2mg pattern from the three generators In accordance with Figure 4 we now tesselate the displayed part of our P2mg pattern with motifs s l Figure 12 Tiling of the plane with motifs s l red and blue rectangles of the P2mg pattern of Figure 2 The red and blue colors of the motifs s l do not indicate qualitative differences We can now divide the motifs s l into areas which are supposed to represent group elements The next Figure divides the motifs s l but closer inspection reveals that the way of division is not correct We show it anyway because it is instructive Figure 13 Division of the motifs s l in order to obtain areas that represent group elements and tile the plane completely As one can see however the division is not correct From Figure 11 we can see that each group element is associated with one motif unit i e with one half of a motif s str In the present Figure we see subareas resulting from the division that contain such a half motif s str However there are also areas supposed to be equivalent not containing such a half motif So the plane is not divided into tesselating areas such that each of them represents a group

Original URL path: http://www.metafysica.nl/holism/implicate_order_5.html (2016-02-01)

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line a The element b that results from the element 1 by a reflection in the line b The element s that results from the element 1 by a half turn about the point S These three generators together can produce all other group elements each such element may be represented by a motif unit Figure 9 The three chosen generators for the C2mm pattern Figure 10 The C2mm pattern can be generated by the elements a b and s The right part of the pattern can be reached as follows From the generated element absabsba lower right we can generate a new element by rotating it 180 0 about the point S This new element is situated to the left of the part of the pattern depicted i e it lies outside the depicted part of the pattern When we now reflect this new element represented by a motif unit in the reflection line b we will obtain yet another new element motif unit that lies far to the right again while reflection in the line a will yield a motif unit further down It is clear therefore that the whole pattern can be produced from these three generators As has been said earlier we can consider each mesh plus content as a motif s l As such it tesselates the plane completely See next Figure Figure 11 The motifs s l red and blue rectangles of the C2mm pattern as depicted in Figure 1 tesselate the whole plane The next Figure depicts one motif s l isolated Figure 12 A motif s l of the C2mm pattern of Figure 1 The following two Figures are about dividing the motifs s l in such a way that the resulting subareas correspond to group elements In the above Figures of our version i e chosen representative of the C2mm pattern we see that each motif s str consists of four motif units partly overlapping commas When we inspect Figure 10 we see that each such motif unit s str can represent a group element So if we want to represent the group elements by maximal areas of the pattern as we have done in all the previous cases each such area must contain one motif unit The next Figure shows a partition of the motifs s l that does not yet satisfy this demand Figure 13 Partition of the motifs s l in order to obtain areas that represent group elements Each area resulting from this partition red and blue rectangles contains two motif units s str and so does not represent a group element The areas must be divided again as is done in the next Figure Figure 14 Correct partition of the C2mm pattern of Figure 1 such that each resulting area contains one motif unit s str and so can as area contents represent a group element We will now explicitly indicate the initial group element and the three generator elements Figure 15 Areas of the C2mm

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to be the third generator The next Figure shows how two more group elements motif units are generated by repeatedly appling the quarter turn q to the initial element 1 Figure 7 Generation of the group elements motif units q 2 and q 3 Figure 8 Some more group elements motif units of our P4mm are generated Instead of concentrating further on the units of the motifs s str we will bring in the motifs s l as we have done in the previous documents in order to let every part of the pattern play a role in representing the group elements which means that the pattern its motifs complete background will be partitioned into areas in such a way that each such an area represents a group element For this purpose we first determine the motifs s l each one consisting of a motif s str PLUS corresponding and complete background See next Figure Figure 9 Motifs s l red and blue squares of the P4mm pattern of Figure 1 The red and blue colors do not signify qualitative differences The next Figure divides each motif s l into eight parts each of them representing a group element Figure 10 Division of the motifs s l results in red and blue triangular areas each representing a group element The colors red and blue do not signify qualitative differences In the above Figure we re now going to indicate the initial group element and the three generators as established in Figure 6 and insert the relevant symmetry elements mirror line and rotatation point associated with the generators Figure 11 The P4mm pattern of Figure 1 is partitioned as it was already in the previous Figure into triangular areas each representing a group element Four such elements are explicitly indicated The group element 1 representing the identity element The group element m representing the reflection in the line m It is chosen as a first generator The group element q representing an anticlockwise quarter turn about the point R It is chosen as a second generator The group element t representing a horizontal translation It is chosen as a third generator We will now start to generate more group elements by applying the generators to the initial group element and again to the results Figure 12 Some group elements of the P4mm pattern are generated by using the defined generators The next Figures depict our P4mm pattern enlarged in order to obtain enough space for the markings of the group elements Figure 13 To reach the first row of group elements we must use the generator q So we take the already generated element tq 2 and rotate it anticlockwise by 90 0 resulting in the element qtq 2 Figure 14 In the same way we take the already generated element tq 2 m and rotate it anticlockwise by 90 0 resulting in the element qtq 2 m Figure 15 In the same way we take the already generated element tq

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v Point k also is not equivalent to point s and also not to the points t u and v because the motif units at its left and right are oriented North West South East while they are oriented North East South West left and right from point s and the same goes for the points t u and v Consequently the unit mesh is not centered the mesh is primitive P We re now going to determine the total symmetry content of the pattern as depicted in Figure 1 representing the Plane Group P4gm To begin with the next Figure shows that there is a 4 fold rotation axis at each lattice node These axes are of course perpendicular to te plane of the drawing Figure 9 There is a 4 fold rotation axis at every lattice node One of them is shown The next Figure shows that there is also a 4 fold rotation axis at the center of each square defined by the lattice nodes Also these axes are perpendicular to the plane of the drawing Figure 10 The center of each square defined by the lattice nodes contains a 4 fold rotation axis One is shown From Figures 8 and 11 it is clear that point n lying exactly between points s and t contains a 2 fold rotation axis as well as any other point equivalent to n Figure 11 A 2 fold rotation axis goes through point n Figure 8 as well as through any other point that is equivalent to n As further symmetry elements we can detect mirror lines In the pattern representing Plane Group P4gm we can see one type of mirror line namely diagonal mirror lines Non diagonal mirror lines are not supported by the pattern See Figure 12 Figure 12 The pattern representing Plane Group P4gm has diagonal mirror lines m Four of them are depicted The pattern representing the Plane Group P4gm also has glide lines glide reflections diagonal and non diagonal glide lines See Figures 13 14 15 and 16 Figure 13 The pattern representing Plane Group P4gm has non diagonal glide lines g One of them is depicted dashed line Figure 14 The pattern representing Plane Group P4gm has further non diagonal glide lines g One of them perpendicular to the one depicted just above is shown dashed line Figure 15 The pattern representing Plane Group P4gm has also diagonal glide lines g One of them at 45 0 to the one depicted just above is shown dashed line Figure 16 The pattern representing Plane Group P4gm has further diagonal glide lines g One of them perpendicular to the one depicted just above is shown dashed line In the next Figure we depict the total symmetry content of patterns representing the Plane Group P4gm Figure 17 Total symmetry content of the Plane Group P4gm Glide lines are indicated by red dashed lines Mirror lines are indicated by red solid lines 4 fold rotation axes are indicated

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generator p which is an anticlockwise rotation of 90 0 about the point R to the initial group element motif unit s l Figure 2 Generation of the group elements p 2 and p 3 Figure 3 Rotating the element m anticlockwise by resp 90 0 180 0 and 270 0 we obtain the elements pm p 2 m and p 3 m Figure 4 New group elements are generated by applying the mirror reflection m to already existing elements Figure 5 More group elements are generated by repeatedly applying p anticlockwise rotation of 90 0 about the point R to already existing elements Figure 6 Still more group elements are generated by repeatedly applying p anticlockwise rotation of 90 0 about the point R to already existing elements Figure 7 One new element is generated by applying the reflection m Figure 8 Two more group elements are generated by repeatedly applying p anticlockwise rotation of 90 0 about the point R to already existing elements Figure 9 Some new elements are generated by applying the reflection m to already existing elements Figure 10 Two more group elements are generated by repeatedly applying p anticlockwise rotation of 90 0 about the point R to an already existing element Figure 11 Some new elements are generated by applying the reflection m to already existing elements Figure 12 Three more group elements are generated by repeatedly applying p anticlockwise rotation of 90 0 about the point R to an already existing element Figure 13 Again three more group elements are generated by repeatedly applying p to an already existing element Figure 14 Three new elements are generated by applying the reflection m to already existing elements Figure 15 Again two more group elements are generated by repeatedly applying p to an already

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which is the same as the previous Figure but now with the original colors of the lastly considered group elements restored Figure 16 All group elements of the displayed part of our P4gm pattern are generated from the initial element by the generators p and m which implies that all the group elements are expressed in these two generators While using ony two generators this was a long process to generate all the group elements of the displayed P4gm pattern If we had added a third generator a translation then the process would have been much easier One translation all by its own cannot generate the pattern from one ultimate asymmetric unit because a translation preserves orientation while the pattern contains differently oriented ultimate motif units in virtue of the rotations involved Similarly a translation and a rotation are not enough to generate the pattern from one asymmetric unit because they both are mechanical transformations that can physically be performed on an object But the pattern also includes reflected units and a reflection is not a mechanical transformation it cannot physically be performed on an object without using an extra dimension for instance a reflection in two dimensional space of a two dimensional object can only be performed by flipping that object through a third dimension As with the generation of the other Plane Groups done in the previous documents we see this generation as an objectively existing immaterial noetic process thinking process and as such representing the particular group as an Idea In Neoplatonic metaphysics this Idea in fact resides in Nous or Intelligence the second metaphysical level or hypostasis Nous directly intuits the Ideas from The One the First Hypostasis while Soul i e the World Soul or Third Hypostasis contemplates the Ideas from Nous by discursively thinking

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by applying the translation t Figure 15 From the element p the element t 2 p is generated by applying the translation t 2 i e applying the translation t two times and from the element t 2 p the element p 2 t 2 p is generated by the rotation p 2 And from the element p 2 t 2 p the elements t p 2 t 2 p t 2 p 2 t 2 p 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 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 motif s l and the latter must then be partitioned such that we get the areas having all the same shape and size but not necessarily the same orientation representing group elements The next Figure shows how our P3 pattern can consist of periodically stacked motifs s l 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 p is an anticlockwise rotation of 120 0 about the point R and t is a horizontal translation to the right The latter implies its inverse a horizontal translation to the left indicated by t 1 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 p 2 and by translation the elements tp and tp 2 The next Figure indicates how new elements can be generated by rotating existing elements 120 0 and 240 0 anticlockwise about the point R Figure 22 Indication how new group elements can be generated by applying the rotations p and p 2 to the elements t tp tp 2 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 240 0 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

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