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  • Promorphology of Crystals Preparation IX
    parallel to but not coincident with the shorter diagonal of the rhomb shaped unit cell of the pattern of Figure 3 The next two Figures show that also the glide lines connecting the centers of two opposite edges of the unit cell are present Figure 8 Demonstration of the glide lines exemplified by one of them connecting the centers of two opposite edges of the rhomb shaped unit cell of the pattern of Figure 3 Figure 9 Demonstration of the glide lines exemplified by one of them connecting the centers of the two other opposite edges of the rhomb shaped unit cell of the pattern of Figure 3 We have now demonstrated that the pattern of Figure 3 belongs to the Plane Group P31m and not to the Plane Group P3m1 That the P3m1 pattern cannot be obtained by just rotating the point lattice describing the repetition in the P31m pattern is shown in the following Figure The net of Figure 6 was rotated anticlockwise by 30 0 and then superimposed onto te pattern of Figure 3 Figure 10 The net point lattice associated with the pattern depicted in Figure 3 and 6 was rotated anticlockwise by 30 0 and then superimposed onto the pattern of Figure 3 As was to be expected the rotated net is not compatible with the pattern showing that the Plane Group P3m1 cannot just like that be obtained by simply rotating the net describing a P31m pattern Also adapting the lengths of the line segments will not do because no true unit cell is outlined as the next Figure shows Figure 11 Adapting the dimensions of the superimposed net of Figure 10 does not yield a P3m1 pattern Two alternative cells are shown neither of which is a true unit cell The smaller one consists of non equivalent points while the larger one contains equivalent points in its interior resulting in a non unique point lattice type We can now return to our original problem how to e x t e n d the motifs s str as they are in Figure 1 such that they fill space completely We now know they cannot do so by just coming together and interlock among themselves involving a reshuffling of the motifs because then the space between them WHICH ALSO BELONGS TO THE PATTERN is deformed because of the reshuffling mentioned transforming the P3m1 pattern into a P31m pattern See Figure 4 The only thing we can do is to let the motifs s str o v e r l a p and so give in to the demand of not deforming space Figure 1a indicates how the overlap should take place The extra spaces must be filled up The next two Figures show the e x t e n s i o n of one motif s str Figure 11a The extension of one motif s str is indicated by the arrows One can see that overlap takes place The next Figure shows the result of the extension of the one motif as indicated in the previous Figure Figure 11b The extension of one motif s str as indicated in the previous Figure is carried out The black center in each original i e not yet extended motif s str is interpreted as belonging to that motif and it should be extended too resulting as we set it in the centra as we see them in the ensuing Figures Precisely the same as was done in the two previous Figures must be done with the remaining motifs When we do this we get the following pattern drawn at slighly smaller scale Figure 12 Space filling of the tri radiate motif according to the above considerations motif extension results in a periodic pattern according to the Hexagonal Point Lattice The total symmetry and its distribution configuration of the pattern is that of Plane Group P3m1 The next Figures indicate to what extend overlap and thus the possession of common parts has taken place The extended motifs overlap in all six hexagonal directions Figure 13 Pattern of Figure 12 Overlap of extended motifs in the oblique direction Green and yellow indicate common parts Figure 14 Pattern of Figure 12 Overlap of extended motifs in the horizontal direction Green and yellow indicate common parts All six hexagonal directions are equivalent so in all these direction the described overlap possession of common parts is present The next Figure highlights four motifs associated with the four corners of the rhomb shaped unit cell Yellow and green indicate common parts i e parts of the extended motifs commonly possessed Figure 15 Pattern of Figure 12 Repeat overlap of extended motifs in the two translation directions as given by the edges black of the rhomb shaped unit cell The four darkened centra of four motifs indicate the corners of the rhomb shaped unit cell The areas of overlap are given by a green and yellow color The next Figures demonstrate that the pattern obtained in Figure 12 has a total symmetry and distribution according to Plane Group P3m1 Figure 16 Pattern of Figure 12 A rhomb shaped unit cell is highlighted yellow The hexagonal net based on this unit cell is indicated dark blue The symmetry of the unit cell complies with Plane Group P3m1 Some of the many 3 fold rotation axes are indicated 3 They reside at the corners of each unit cell mesh of the net and in addition to that two of them are located in the interior of each unit cell First we give the symmetry content diagram of the Plane Group P3m1 in which all symmetry elements are indicated Figure 17 Symmetry content of the Plane Group P3m1 Glide lines are indicated by dashed lines Mirror lines are indicated by red solid lines and the 3 fold rotation axes by small triangles From Figure 16 it is obvious that the pattern contains 3 fold rotation axes at the corners of the rhomb shaped unit cell and

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  • Promorphology of Crystals Preparation X
    8 The motif of the left image of the above Figure is what we can call an e x t e n d e d m o t i f which means that the motif s str has fully extended over its background The interior of this motif is not interpreted as background The motif of the right image of the above Figure is a non extended motif which means that we have to do with a motif in the stricter sense a motif s str and its background The only specific feature of such a background is its shape At the end of Part Nine of the present series of documents we found out that if we extend the motif s str right image of above Figure properly it will tile the plane and still arranges according to Plane Group P3m1 See next Figure The scales of the underlying net of the pattern where the right image of Figure 7r came from and of the net underlying the pattern of the next Figure differ a little but this is immaterial for our discussion Figure 7t When the motifs of the P3m1 periodic pattern of Figure 7r are properly extended and invade the whole of their corresponding backgrounds they tile the plane neatly and still arrange according to Plane Group P3m1 See also Figure 23 of the previous document The extended motif of the above Figure 6s is at the same time the most symmetric unit cell and can serve directly as Complex Motif because elimination of the glide lines will not have any effect on the symmetry as was established in Part Eight The next Figure adds this extended motif as indicated in Figure 6s to the other two for comparison Figure 7u First image to the left Complex Motif of the P3m1 periodic pattern of Figure 6 Extended motif Second image Complex Motif as established in Figure 6s Extended motif Third image Complex Motif of the first P3m1 periodic pattern studied Non extended motif The slight differences in scale come from the different origins from where those Complex Motifs were derived As has been said they are immaterial for the present discussion After this diversion and comparison of the two P3m1 periodic patterns let us continue with the second of them the pattern of two sorts of discs red and blue that can stand for the structure of a two dimensional ionic crystal If we look to that structure as depicted in Figure 6 carefully we can detect several different d i r e c t i o n s that could become faces of the crystal possessing this structure i e the P3m1 structure These directions present several different atomic aspects to the nutrient environment of the growing crystal Faces displaying such different atomic aspects will in most cases grow at a different rate And this is decisive for the s h a p e that the crystal will finally adopt The slowest growing faces will eventually survive the faster growing faces except when they are perpendicular to each other The latter disappear after prolonged growth See for this phenomenon the next Figure and also the document on the Morphology of Crystals in the SPECIAL SERIES of the First Part of this website accessible by clicking on back to homepage Figure 7v A growing 2 dimensional crystal Because its y faces grow faster than its x faces the y faces will eventually grow themselves out of existence The next Figure depicts our structure and shows that it supports three types of faces each presenting a different atomic aspect to the nutrient environment of the growing crystal Figure 8 A periodic repetition of the motif of Figure 5 according to the Plane Group P3m1 The motif pattern depicted in Figure 6 supports three different types of crystal faces associated with different atomic aspects presented to the nutrient environment The point symmetry of 2 D crystals having this internal structure a structure according to the Plane Group P3m1 is 3m which is the translation free residue of all isogonal Plane Groups which here means of the Plane Groups P3m1 and P31m So a certain possible crystal face belonging to one of the three types found in Figure 8 will configure with its fellows such that the resulting face configuration has 3m symmetry If the faces of such a configuration are all of the same type say a faces then such a configuration is called a simple FORM When more than one type of faces is involved in such a configuration we call it a COMBINATION a combination of FORMS A simple FORM or a COMBINATION can only exist as a crystal For 2 D crystals this is of course an analogue of existing when it is closed i e when it encloses space completely For 2 D crystal 2 D space for 3 D crystals 3 D space There are six differently oriented faces capable of presenting the aspect a to the nutrient environment implying that hexagonal crystals can be formed bounded by these faces Although this FORM has a hexagonal shape it has a 3m symmetry not a 6mm symmetry which can be verified by inspecting the internal structure Because it is closed it can occur as a crystal The next Figure shows this hexagonal FORM Figure 9 A hexagonal two dimensional ionic crystal with an internal structure of which the symmetry is according to the Plane Group P3m1 The crystal has six equivalent faces a faces presenting the same atomic aspect to the environment A smaller hexagonal ionic crystal with the same internal structure P3m1 looks like this Figure 9a A hexagonal two dimensional ionic crystal with an internal structure of which the symmetry is according to the Plane Group P3m1 The crystal has six equivalent faces a faces presenting the same atomic aspect to the environment It is the same as in Figure 9 but smaller We just stated that the structure of the above Figure 9 and also Figure 9a depicted hexagonal 2 D crystal has a symmetry according to the Plane Group P3m1 which means that there are numerous glide lines mirror lines 3 fold rotation axes etc Of course this is only valid when we have a crystal of macroscopic dimensions consisting of microscopic atoms resulting in there being almost an infinity of atoms and unit cells So in the Figure the depicted crystal is much too small implying that there is only a small finite number of repeating units Indeed in a finitely and uniformly extended P3m1 structure as well as in a finitely extended P31m structure there can be only one 3 fold axis and only mirror planes that go through the center of the structure There can be no mirror lines parallel to each other and no rotation axes parallel to each other So a crystal and this is also valid for 3 dimensional crystals inherently has several different atomic aspects and each aspect can have copies that are regularly distributed across the structure in accordance with its point symmetry and resulting in regularly distributed crystal faces as we saw in Figure 9 and 9a with respect to the a faces Now we could be tempted to consider these regularly distributed aspects as representing the crystal s a n t i m e r s or counterparts suggesting that a crystal d i r e c t l y reveals its t e c t o l o g y which here means the number shape and configuration of its antimers from which the promorph or planimetric basic form could be directly read off And so there would be no need to derive the C o m p l e x M o t i f in order to determine the crystal s promorph However this is only seemingly so The following will demonstrate this If we consider the number of regularly distributed aspects of the same sort which have given rise to the six equivalent faces of the hexagonal crystal of Figure 9 and also of Figure 9a as the homotypic number of the crystal in other words if we consider the six regularly distributed copies of the aspect a giving rise to six a faces as representing the crystal s antimers then the crystal would have six antimers and would consequently possess an axis of six fold symmetry a 6 fold rotation axis But we know from the internal structure of the crystal that it does not have a 6 fold rotation axis but a 3 fold rotation axis Indeed if we follow the six aspects of the hexagonal crystal in Figure 9a we see that they are it is true all of the same aspect but their orientation is different in such a way that they are not repeated around some axis If we call a red disc A and a blue disc B and follow them around the crystal of Figure 9a then we get the following sequence AB AB ABA BA BAB AB ABA BA BAB AB ABA BA B So the aspect is not properly repeated around the axis going through the center of the crystal normal to the plane of the drawing Only alternate aspects are exactly repeated AB AB ABa ba bAB AB ABa ba bAB AB ABa ba b and also Ab ab aBA BA BAb ab aBA BA BAb ab aBA BA B So the crystal has a three fold rotation axis and thus has t h r e e antimers and not six And this in virtue of its i n t e r n a l structure If we want to indicate and recognize the three ANTIMERS in the 2 D hexagonal crystal of Figure 9a as a whole we must divide the crystal in three congruent parts that repeat around the 3 fold axis Figure 9b A hexagonal two dimensional ionic crystal with an internal structure of which the symmetry is according to the Plane Group P3m1 In accordance with its internal structure we could indicate three congruent a n t i m e r s in the crystal of Figure 9a The antimers of the crystal as depicted in the above Figure do not look very much like the antimers as we see them in organisms like polypes flowers echinoderms etc Those genuine antimers much more look like this Figure 9c A structure akin to that of Figure 9b with genuine a n t i m e r s These antimers are repeated around a three fold axis The structure also when it is imagined to be infinitely extended in 2 D space is NOT periodic while the crystal when imagined to be extended in 2 D space indefinitely IS The next Figure emphasizes this fact Figure 9d The crystal of Figure 9b furnished with the appropriate hexagonal net point lattice The possibility of letting the structure be underlied by a point lattice emphasizes the periodic nature of the crystal and consequently its non tectological structure The net thereby expresses that the antimers are not genuine antimers at least not antimers just like that In Organic Tectology as laid down on this website subordinated f o r m i n d i v i d u a l s of several types are being distinguished within an organismic body The antimers are such subordinated individuals to be precisely third order form individuals Well as can be seen in Figure 9b the antimers do not show any i n d i v i d u a l i t y whatsoever Even in organisms or parts thereof that have only two antimers unlike for instance star fishes which show up as simply to be their left and right body halves the boundary between those two antimers is often materialized for instance in the form of the spine in vertebrates or as the main vein in many plant leafs In this way the antimers stand out clearly Crystals on the other hand especially when they are considered from a macroscopical view are in fact homogeneous bodies And moreover in organisms also when they are inspected microscopically there is no sign of a p e r i o d i c structure as is typical for crystals The C o m p l e x M o t i f of crystals on the other hand shows a true tectological i e non periodic structure with genuine individualized antimers as we showed already for two dimensional crystals and will show for three dimensional crystals as well See the series of documents treating of 3 D crystals by clicking HERE to begin with Our discussions of two dimensional crystals is an instructive preamble to the structure of 3 D crystals and their Complex Motif and thus their Tectology and Promorphology All this points to the fact that if we want to consider CRYSTALS promorphologically like we did in the case of ORGANISMS we must look for a true tectological aspect in crystals an aspect that shows genuine antimers And this indeed is our C o m p l e x M o t i f The crystal of Figure 9a or Figure 9 for that matter has an external shape namely hexagonal that does as such not reflect the true symmetry of the crystal which is 3m This 3m symmetry was revealed by its internal structure But there are other crystals possessing our P3m1 structure or some conditions acting on the same crystal that can be imagined to show this lower symmetry already by their geometrical shape The faces corresponding to the b aspect the b faces an example of which is shown in Figure 8 and of which there are only three in different directions may grow slowly enough to appear at the growing crystal These faces differ from the a s in two important ways In the first place they present a quite different atomic aspect to the environment They will almost certainly grow at a different rate from the a faces In the second place there are only three faces capable of presenting that aspect in different directions They define a triangle not a hexagon From the three faces that would like to be like the b faces if the crystal had a 6 fold axis of symmetry one is indicated as c in Figure 8 Clearly the c faces and the b faces present different atomic aspects to the nutrient environment It would be very improbable to find these two sorts of faces growing at the same rate The environment would probably behave quite differently toward a face of blue ions and a face of red ions Even if you visualize coating the two sorts of faces with the same sort of ions you will reach the same conclusion There is a different disposition of the red ions directly beneath the blue ions in the coatings and the nutrient environment will respond to that difference also The nutrient environment sees through the coating as it were So we can imagine that in a second case in addition to the six a faces the three b faces also appear at the crystal while the c faces don t because their relative growth rate is too high We in this way obtain a second habit The first habit was the hexagonal shape of the crystal The crystal consists of six a faces while three of the six potential corners of the a hexagon are cut off by the b triangle as the next Figure shows Figure 10 A two dimensional ionic crystal with a symmetry of its internal structure according to the Plane Group P3m1 Two types of faces six a faces and three b faces have appeared They reveal the true symmetry of the crystal namely 3m three equivalent mirror lines and a 3 fold rotation axis A third habit can emerge when the b faces are the slowest growing faces while the others grow much faster In real crystals this could as in the case of the second habit be so either because the solution in which the crystal grows contains some ingredients that affect the relative growth rates or because we have to do with a crystal of a different chemical substance all together Because there are only three directions in which b faces can occur in our structure i e the structure depicted in Figure 6 the b faces form when they alone appear a triangle In our case an equilateral triangle in accordance with the 3m symmetry as the next Figure shows Figure 11 A two dimensional ionic crystal with a symmetry of its internal structure according to the Plane Group P3m1 Only one type of faces b faces have appeared They are three in number and equal and form a triangular 2 D crystal They reveal the true symmetry of the crystal namely 3m three equivalent mirror lines and a 3 fold rotation axis The size of the crystal s faces is 9 atoms discs long We call it a b 9 triangular crystal This crystal can be considered as a simple FORM of the 2 D Crystal Class 3m From the foregoing we see that the number of a n t i m e r s counterparts of a crystal is not in a constant way determined by the number of possible directions of a certain atomic aspect because that number can be different for different aspects In our case six for the a aspect three for the b aspect and three for the c aspect If on the other hand the c faces are the slowest growing faces while the a and b faces grow much faster then eventually only the c faces appear on the crystal three in number because also c faces have in our case only three possible directions So also in this case a triangular crystal will finally appear See next Figure Figure 12 A two

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  • Promorphology of Crystals Preparation XI
    P31m pattern a configuration based on the edges of the rhomb shaped unit cell coinciding with the mirror lines is the following Figure 10 Symmetry content of 2 D patterns according to the Plane Group P31m Glide lines are indicated by dashed lines Mirror lines are indicated by red and by black solid lines while the 3 fold rotation axes are indicated by small triangles Let s investigate whether the pattern of Figure 7 and 9 indeed does have or does not have the symmetry elements and their distribution as is indicated in the above symmetry diagram of the Plane Group P31m All this in fact means that we must verify that the u n i t c e l l as drawn in the Figures 7 and 9 possesses the symmetry distribution as given by the symmetry diagram of Plane Group P31m That the 3 fold rotation axes and mirror lines are indeed present in that unit cell is easy to verify in Figure 7 Figure 11 In the pattern of Figure 9 the unit cell indeed has mirror lines m as in the symmetry diagram Also the 3 fold rotation axes are present at their proper places One at each corner of the unit cell not indicated but evident and two in its interior also not indicated but evident namely coinciding with the motifs s str of that interior A 3 fold rotation axis goes through the center of each motif s str of the pattern But there are additional 3 fold axes indicated as green discs on the edges of the unit cell and two extra in its interior not present in the symmetry diagram If we interpret the pattern as a repetition of the total motifs as indicated in Figure 7 then we still find extra 3 fold axes on the edges of the unit cell not present in the symmetry diagram Figure 12 In the pattern of Figure 7 the unit cell indeed has mirror lines m as the symmetry diagram has it Also the 3 fold rotatation axes are present at their proper places One at each corner of the unit cell not indicated but evident and two in its interior also not indicated they coincide the two motifs s str of that interior A 3 fold rotation axis goes through the center of each motif s str of the pattern But the extra 3 fold rotation axes found above which are not present in the symmetry diagram for the Plane Group P31m Figure 10 are still present in the pattern indicated as green discs So the pattern is not according to the Plane Group P31m but still according to P3m1 The same result was obtained in the previous document with respect to the ionic P3m1 pattern There we chose a similar alternative unit cell but the pattern remained a pattern according to the Plane Group P3m1 See Figure 7h of the previous document Only when we interpret the motifs as indicated in Figure 7 as the total motifs of the pattern i e when we do NOT interpret the motif and all its equivalents indicated purple in the next Figure as total motif we finally get rid of the extra 3 fold axes and obtain a P31m pattern And this is consequently a d i f f e r e n t pattern different from the one we started with Figure 13 In order to obtain a P31m pattern from the pattern of Figure 7 we must NOT interpret the motif highligted purple as a genuine total motif which means that its center and those of all its equivalents must not be interpreted as a lattice point See next Figure Figure 14 Indication of the genuine total motifs in the pattern of Figure 7 Each AND ALL total motif s correspond s to a lattice point It is the lattice corresponding with the Plane Group P31m By correctly interpreting the total motifs the extra 3 fold axes have disappeared Figure 15 Indication of the genuine total motifs in the pattern of Figure 7 Each AND ALL total motif s correspond s to a lattice point It is the lattice corresponding with the Plane Group P31m As can be seen the points indicated black are not locations of 3 fold rotation axes anymore The locations of the 3 fold axes are now identical to those in the symmetry diagram of the Plane Group P31m Figure 10 Figure 16 Location of the 3 fold rotation axes in the unit cell of the pattern of Figure 14 It complies with the symmetry diagram for the Plane Group P31m So the pattern of Figure 9 must be adapted in order to be compatible with the inserted net of which the directions of the cell edges with respect to the motifs are such that the structure can be described by the Plane Group P31m See next Figure Figure 17 The array of motifs of Figure 9 is adapted to let the inserted net be such that it describes the pattern according to Plane Group P31m The rhomb shaped unit cell is indicated yellow Its angles are 120 0 and 60 0 as is also the case with the rhomb shaped unit cell of the pattern of Figure 4 in which the inserted net allows the description of the pattern according to Plane Group P3m1 The latter pattern is however a different pattern different from the one just obtained Because the non colored motif units in the previous Figure are all equivalent and have a higer symmetry than the colored ones i e they at the same time also comply with the three fold symmetry of the colored ones we can dispose of them all together See the next Figure Figure 18 In the array of the previous Figure the non colored motifs are erased The resulting pattern in conjuction with the inserted net now fully complies with the Plane Group P31m as the next Figures demonstrate Figure 19

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  • Promorphology of Crystals Preparation XII
    the pattern as given in Figure 2 as such belongs to the Plane Group P31m We will now derive the C o m p l e x M o t i f of the pattern of Figure 2 Recall that the Complex Motif is the translation free residue of a periodic motif pattern When this pattern is the internal structure of crystals the Complex Motif lies at the base of the crystal s tectological aspect The Complex Motif allows the crystal to be assessed promorphologically It determines the number and configuration of antimers essential for the crystal s promorph or basic form The Complex Motif should express all point symmetry that the given crystal has in store and it should be aperiodic in order to be tectological So to find the Complex Motif we must eliminate all translations inherent in the periodic pattern glide translations and simple translations Because all glide lines of our pattern have mirror lines parallel to each of them elimination of the glide translations will not increase the symmetry any further so that we can start right away with the content of a single unit cell by concentrating on a s i n g l e unit cell we automatically eliminate all s i m p l e translations This unit cell must however be the most symmetrical unit cell in order to exhibit all point symmetries of the given crystal The rhomb shaped unit cell of Figure 2 does not have all the symmetry which the crystal has in store which we know from the Crystal Class to which a crystal having a total symmetry of P31m belongs 3m so we must find another unit cell The most obvious one is a hexagonal unit cell as depicted in the next Figure Figure 9 A hexagonal unit cell choice is possible for the pattern of Figure 2 This unit cell which is not the smallest unit cell possible suggests a smaller but still hexagonal repeat unit Because the outline of this repeat unit is hexagonal the full 3m symmetry will be preserved There are two smaller hexagonal units possible but only one of them is a true repeat unit of the pattern Figure 10 A hexagonal unit cell choice for the pattern of Figure 2 suggests smaller hexagonal units green One of them is a true repeat unit of the structure and can represent the Complex Motif The hexagon green of the lower image of Figure 10 is not a true repeating unit as the next figure illustrates Figure 11 The hexagonal unit of the lower image of the previous Figure is not a true repeat unit and thus cannot represent the Complex Motif The hexagonal unit of the upper image of Figure 10 is a true repeating unit as the next Figure shows It can represent the Complex Motif of our P31m structure Figure 12 The hexagonal unit of the upper image of the Figure 10 is a true repeat unit and thus can represent

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  • Promorphology of Crystals Preparation XIII
    magnification of the ionic pattern of Figure 11 The pattern consists of two types of discs atoms ions here indicated by red and blue Figure 14 The periodic pattern of Figure 11 magnified The two ions negative and positive are indicated by colors here red and blue In Figure 12 we showed a net indicating the repeat of a unit This unit consists of two whole atoms and its shape is a rhombus This shape has not all the symmetry that a crystal having this stucture has in store namely 3m The next Figure suggests a larger and especially more symmetric repeat unit consisting of six whole atoms ions three of each of the two sorts Figure 15 The periodic pattern of ions also allows for a larger more symmetric repeat unit consisting of six atoms three of each sort The symmetry of this repeat unit fully complies with the point symmetry of any crystal having this structure The unit cell choice as in Figure 12 is also indicated here left in the image The hexagonal repeat units of the previous Figure repeat according to the net indicated dark blue in the Figure Compare this Figure with Figure 2 of the previous document This comparison shows that when the motifs s str extend till they meet we can still repeat these motifs according to the net indicated in that Figure 2 but some points which are in fact equivalent like the centers of the hexagons in the above Figure being equivalent to the meeting points of those hexagons are not considered to be equivalent anymore only the points at the corners of the rhombus are equivalent Nevertheless the hexagonal repeat unit is a good candidate for turning out to represent the Complex Motif of our present periodic pattern But first we must investigate whether our ionic pattern related to the proper net in which all points that initially were equivalent remain equivalent indeed belongs to the Plane Group P31m First we give again the symmetry diagram for the Plane Group P31m Figure 16 Symmetry content of 2 D patterns according to the Plane Group P31m Glide lines are indicated by dashed lines Mirror lines are indicated by red and by black solid lines while the 3 fold rotation axes are indicated by small triangles The next Figure shows that the 3 fold rotation axes as given and demanded by the symmetry diagram are indeed present in our ionic structure The positions of the 3 fold rotation axes however do not discriminate between the Plane Groups P3m1 and P31m because they are the same in both Groups Figure 17 The ionic pattern of discs as given in the Figures 11 and 14 indeed possesses the 3 fold rotation axes as demanded by the symmetry diagram But the mirror lines as given in the symmetry diagram are NOT present Figure 18 The ionic pattern of discs as given in the Figures 11 and 14 does NOT possess the mirror lines as demanded

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  • Promorphology of Crystals Preparation XIV
    Plane Group P31m The next Figure indicates a suitable repeat unit that can serve to represent the Complex Motif of the present pattern Figure 8 A hexagonal repeat unit of the present P31m pattern is indicated When we isolate this unit we automatically eliminate the simple translations and obtain the Complex Motif of the pattern See next Figure Figure 9 Complex Motif of the P31m pattern of Figure 1 and also of Figure Figure 27 of the previous document The hexagonal boundary line dark blue does not as such belong to the Complex Motif It belongs to the motif s background in fact half its thickness The next Figure clearly shows the point symmetry of this Complex Motif Three mirror lines intersecting at angles of 120 0 and a 3 fold rotation axis So the symmetry is 3m The motif has three congruent antimers Figure 10 3m point symmetry of the Complex Motif of the P31m pattern of Figure 1 and also of Figure Figure 27 of the previous document The hexagonal boundary line dark blue does not as such belong to the Complex Motif It belongs to the motif s background in fact half its thickness The Complex Motif has three congruent antimers The simplest planimetric figure expressing this symmetry is the Equilateral Triangle regular trigon which accordingly is the promorph or planimetric basic form of all crystals of the Plane Group P31m and with it of the Class 3m See next Figure Figure 11 The Equilateral Triangle regular trigon is the promorph or planimetric basic form of all crystals belonging to the Plane Group P31m and with it to the Class 3m In the next document we are going to c o m p a r e the Plane Groups P3m1 and P31m to detect their similarities and

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  • Promorphology of Crystals Preparation XV
    the rotated result or equivalently with the pattern of Figure 12 What we have done in fact in transforming the P3m1 pattern into that of a P31m pattern is the following In the homogeneous content of the triangles of the P3m1 pattern we inserted a tri radiate structure everywhere with the same orientation and also with the same orientation as those of the original motifs without destroying any existing mirror lines It is obvious that we in so doing change the pattern but it is also obvious that the total symmetry has not been changed See next Figure Figure 14 A part of the p3m1 pattern left of Figure 2 is compared with a part of the pattern right of Figure 12 The patterns show the same set of mirror lines yellow The orientations of the motifs is the same in both patterns In fact we have now demonstrated that the total symmetry content of the two Plane Groups P3m1 and P31m is the same while the corresponding patterns are qualitatively different The next Figure illustrates that the tri radiate figure inserted in every triangle has itself no mirror lines The way however the triangles are integrated in the whole pattern guarantees the preservation of pre existing mirror lines See also Figure 14 and note that the lines that constitute the net and also other auxiliary lines do not belong to the pattern Figure 15 Image taken from Figure 12 or equivalently from Figure 13 right image or equivalently from Figure 14 right image Although the inserted tri radiate figure in every triangle has no mirror lines it does not destroy any mirror line pre existent in the pattern In order to compare the pattern of Figures 12 13 right image and 14 right image with the P3m1 pattern of Figure 2 we undo the rotation as it was performed in Figure 12 Figure 16 Our present P3m1 pattern left compared with the P31m pattern derived from it by the above method Which consisted of the insertion of tri radiate figures into every triangle The subsequent rotation as we did it above for easy comparison is omitted If we eliminate all auxiliary lines we get the following Figure 17 Our present P3m1 pattern left compared with the P31m pattern right derived from it by the above method Which consisted of the insertion of tri radiate figures into every triangle All auxiliary lines erased The next Figure indicates te respective unit cells of the structures Figure 18 Our present P3m1 pattern left compared with the P31m pattern right derived from it by the above method Which consisted of the insertion of tri radiate figures into every triangle All auxiliary lines erased The unit cells of the patterns are indicated Both are rhomb shaped The structural difference between the P3m1 pattern and the P31m pattern is clearly exhibited Next we study the system of g l i d e l i n e s in both Plane Groups as they are visible

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  • Promorphology of Crystals Preparation XVI
    according to the ordering of the P6 pattern of Figure 1 They are true repeat units To derive the Complex Motif of our P6 pattern of extended motifs the only translations that must be undone are the simple translations because P6 patterns have no glide lines So in isolating the just established repeat unit we automatically eliminate the simple translations and wind up with the Complex Motif of our pattern That pattern is partially depicted in Figure 6c Indeed because there are no glide lines the motif is not partly telescoped outward i e it is not drawn out and thus needs not to be compressed i e telescoped inward again in order to obtain the translation free residue that can represent the Complex Motif of the pattern The latter is depicted in the next Figure Figure 6d The Complex motif of the pattern of Figure 6c which is a pattern consisting of a periodic repetition of extended motifs The point symmetry of the just derived Complex Motif of the P6 pattern of Figure 6c is 6 which means that its only symmetry element is a 6 fold rotation axis The simplest planimetric figure that expresses this symmetry geometrically is the Twisted Hexagon already depicted in Figure 6 The Plane Group P6mm A pattern according to Plane Group P6mm is given in the next Figure Figure 7 Motifs with point symmetry 6mm inserted in a hexagonal net in such a way that each lattice point is associated with such a motif yields a periodic motif pattern according to the Plane Group P6mm A mesh of the net can serve as unit cell The total symmetry content of Plane Group P6mm is depicted in Figure 8 Figure 8 Total symmetry content of the Plane Group P6mm All rotation axes are perpendicular to the plane of the drawing 6 fold rotation axes are indicated by small blue solid hexagons 3 fold rotation axes are indicated by small blue solid triangles 2 fold rotation axes are indicated by small blue solid ellipses Glide lines are indicated by red dashed lines Mirror lines are indicated by solid lines red and black One should not worry about the small discrepancies at some places in the drawing In order to derive the Complex Motif of the P6mm pattern of Figure 7 we can concentrate directly on a suitable repeat unit which should possess all the point symmetry that crystals belonging to this Plane Group have in store and that is 6mm i e one 6 fold rotation axis and two sets of mirror lines each set consisting of three equivalent mirror lines at 120 0 to each other We can do so because parallel to each existing glide line inherent in any P6mm pattern there exists a mirror line The next Figure gives the repeat unit that can represent the Complex Motif of the P6mm pattern of Figure 7 Figure 9 Hexagonal repeat units of the P6mm pattern of Figure 7 Such a unit consists of

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