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  • Promorphology of Crystals Preparation I
    in the array Already the rectangular outline of the pattern as drawn forbids the presence of such axes Also the two suggested mirror lines solid blue lines are not possible for such a pattern as it stands i e as it is drawn namely as a finite pattern The array must be considered as indefinitely extended in 2 D space and then the drawn symmetry elements are possible In the presently considered array we can distinguish and choose between several possibilities lattice points The lattice in this case turns out to be a Primitive Square Lattice Square net primitive here means that the unit cell has nodes only at its corners Figure 2 A two dimensional array of canadian maple leaves The pattern can be described and characterized by the Plane Group P4gm Nodes lattice points are indicated by small red discs We obtain the point lattice by connecting these points non diagonally See next Figure Figure 3 A two dimensional array of canadian maple leaves The pattern can be described and characterized by the Plane Group P4gm Nodes lattice points are indicated by small red discs These lattice points are now connected as indicated blue lines revealing a Primitive Square Lattice supporting the motif pattern In the present pattern of maple leaves we clearly see the presence of a 4 fold rotation axis at each node In the center of each unit cell we also see a 4 fold rotation axis But while the former are directed clockwise the latter are directed anti clockwise so the involved points are not equivalent which here means that the central point is not a lattice point so the lattice is not centered In our array a m o t i f consists of four maple leaves centered around each lattice node Each single maple leaf can be considered as a motif unit In the next Figure these units are given in different colors Figure 4 A two dimensional array of canadian maple leaves The pattern can be described and characterized by the Plane Group P4gm Nodes lattice points are indicated by small red discs Each node lattice point is associated with a motif consisting of four motif units One such motif is indicated The next Figure shows the periodic repetition of motifs according to the point lattice Figure 5 A two dimensional array of canadian maple leaves The pattern can be described and characterized by the Plane Group P4gm Nodes lattice points are indicated by small red discs Each node lattice point is associated with a motif consisting of four motif units The array consists of a periodic repetition of equivalent motifs as indicated What we have depicted in Figure 4 is the t o t a l m o t i f which here means the complete surroundings of the lattice point That s why this motif neatly adjoins to the next total motif belonging to the next lattice point And of course we can let the boundary of the total motif coincide with the perimeter of the square as the next Figure shows because all what we re looking for is a unit that is periodically repeated throughout the structure Figure 5a A second but equivalent way to conceive the total motif as the complete surroundings of a lattice point The boundary of such a total motif is indicated by red lines forming a square The next Figure shows the net and its nodes as in Figure 3 it shows the motifs and it outlines a possible u n i t c e l l Figure 5b A possible unit cell is outlined By repeating this unit cell periodically we obtain our structure pattern of maple leaves If we now compare the total motif as depicted in Figure 5a and a possible unit cell as depicted in Figure 5b then we see that they are exactly equal in size and equivalent with respect to content which means that THE TOTAL MOTIF IS EQUIVALENT TO A POSSIBLE UNIT CELL Such a total motif are the complete surroundings of a lattice point All this implies that when we telescope a crystal structure back i e eliminate all translations in order to find the c o m p l e x motif and on the basis of it the crystal s tectology we cannot push this so far as to even sqeeze a unit cell The process stops at the unit cell Recall that choosing a smaller unit cell than some given unit cell has nothing to do with an elimination of translations The smallest part of the lattice is the lattice point And as soon as we include content to a point lattice each lattice point is associated with its surroundings which have and represent content These surroundings are the total motif of the structure i e the given point lattice provided with content So the total motif is the smallest unity that we can get by eliminating translations and this total motif is as we just saw equivalent to a possible unit cell We can see that the pattern imagined to be extended indefinitely possesses m i r r o r l i n e s in the two diagonal directions Figure 6 A two dimensional array of canadian maple leaves The pattern can be described and characterized by the Plane Group P4gm Nodes lattice points are indicated by small red discs The pattern possesses mirror lines as indicated blue solid lines The pattern also possesses g l i d e l i n e s glide reflections They come in two varieties diagonal and non diagonal horizontal and vertical Let us first concentrate on the diagonal ones The next Figure depicts one such diagonal glide line Figure 7 A two dimensional array of canadian maple leaves The pattern can be described and characterized by the Plane Group P4gm Nodes lattice points are indicated by small red discs In addition to mirror lines the pattern possesses glide lines One such glide

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  • Promorphology of Crystals Preparation II
    of the whole pattern Concerning the other two motif units the superpositions should be carried out as indicated by the arrows in the next Figure Figure 29a The two superpositions indicated by arrows implied by the elimination of the translational component of the relevant horizontal glide line with respect to the the other two motif units from the surroundings of our lattice point Figure 29b Result drawn on the right of the two superpositions described above Figure 30 Result of the superpositions described above now in the context of the whole pattern We now must superimpose the result of elimination of the translational components of the horizontal glide lines onto the result of the elimination of the translational components of the vertical glide lines to obtain the total result of the elimination of the translation components of all non diagonal glide lines The next Figures show the result of this superposition Figure 30a Result of the superposition of the two results obtained above with respect to the non diagonal glide lines Superposition of the result in Figure 30 onto the result obtained in Figure 27 gives this same total result of the elimination of the translational components from the non diagonal glide lines now seen in the context of the whole pattern Figure 31 Result of the elimination of the translational components from the non diagonal glide lines We now must combine i e superimpose 1 the result concerning all the non diagonal glide lines and 2 the result concerning all diagonal glide lines We will do this in the next Figures Figure 31a Result of the elimination of the translational components from ALL the glide lines with respect to one node of the net This result is obtained by superimposing the results with respect to the non diagonal glide lines and the diagonal glide lines In the context of the repetition of this complex motif we will see that some overlappings occur so the real complex motif neatly fits into a square of the net and is as such periodically repeated It looks like this Figure 31b Complex motif as it is repeated according to the Square Net So we now have found how the surroundings of our node look like after elimination of the translational components of all glide lines By superposition of the result of Figure 31 concerning non diagonal glide lines onto the result of Figure 24 concerning diagonal glide lines we also will obtain those surroundings of our node after elimination of all translational components of the glide lines and now moreover in the context of the whole pattern See next Figure Figure 32 Result of the elimination of the translational components from ALL the glide lines with respect to one node This result is a complex motif associated with that node When this elimination is done with respect to every node of the lattice then every node is associated with such a complex motif These complex motifs are all identical See next Figure Figure 32a Result of the elimination of the translational components from ALL the glide lines with respect to EVERY node So in the above Figure the translational components of all the glide lines are eliminated Those glide lines are now transformed into mirror lines The s i m p l e t r a n s l a t i o n s however are still there and are responsible for the periodic repetition of the complex motif This repetition is according to the Square Net We can show this repetition in several ways as the next Figures illustrate Figure 32b Periodic repetition of the complex motif according to the Square Point Lattice Figure 32c Periodic repetition of the complex motif according to the Square Point Lattice The above Figure 32c finally precisely and correctly depicts the repetition of the complex motif When we now eliminate also all the simple translations we ll end up with ONE complex motif only This elimination of the simple translations boils down to isolating one of the complex motifs Figure 33 The isolation of one complex motif which is the same as eliminating all simple translations The result is the translation free residue of the periodic pattern of maple leaves Elimination of all simple translations means the superposition of all lattice points resulting in one lattice point only Because this point does not represent a lattice anymore we can remove it from the motif Figure 33a The complex motif If we study the complex motif of Figure 33a closely we see that its point symmetry is 4mm which is to be expected from the Plane Group of our maple leaf pattern P4gm Eliminating the translational component of all glide lines turns g into m Eliminating all simple translations removes the lattice i e turns it into one point and thus removes P from the symbol What is left is indeed 4mm Promorphology of single Crystals Now at last with obtaining this complex motif we have found the t e c t o l o g i c a l a s p e c t which underlies the possiblity and legitimacy to consider single c r y s t a l s promorphologically The complex motif has indeed a tectological structure in contradistinction to a periodic Our particular complex motif consists of four a n t i m e r s as the next Figure shows Figure 34 The complex motif as the translation free residue of the periodic pattern of maple leaves has a tectological structure It has four antimers When we look at a concrete single crystal i e when we consider the crystal macroscopically the translations involved in glide lines and in 3 D crystals the translations involved in glide planes and screw axes are not as such visible they are much too small This means that we see glide lines as mirror lines and in 3 D crystals glide planes as mirror planes and screw axes as rotational axes i e

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  • Promorphology of Crystals Preparation III
    crystal with Plane Group P1 The latter is the only Plane Group compatible with the 2 D Crystal Class 1 Figure 4 Geometric Figure representing the planimetric basic form of all crystals based on Plane Group P1 and thus of the Class 1 We have not developed a promorphological system of two dimensional objects but if we had done so the triangle of Figure 4 would certainly occupy a definite place in such a system The Plane Group P2 Patterns of motifs according to Plane Group P2 are also based on the Oblique Point Lattice Net But in this case the pattern has the full symmetry of the empty building block of the Oblique Net the parallelogram which has a 2 fold rotation axis See next Figures Figure 5 Pattern according to Plane Group P2 The pattern must be considered as to be extended indefinitely in 2 D space It has 2 fold rotation axes periodically repeated all over the pattern See next Figure Figure 6 Pattern according to Plane Group P2 It has 2 fold rotation axes normal to the plane of the drawing periodically repeated all over the pattern indicated by small black ellipses It is most convenient to locate the nodes of the net right on the symmetry elements So in the present case we locate the nodes of the underlying Oblique Net at 2 fold rotation axes in equivalent places within the pattern i e we only connect only those 2 fold axes which all lie in equivalent places which means that their surroundings are identical This is because the nodes of the net must indicate equivalent areas within the pattern And a net so constituted indicates the periodic nature of the pattern Figure 7 Pattern according to Plane Group P2 The underlying Oblique Net is inserted such that its nodes coincide with a part of the 2 fold rotation axes present in the pattern The net indicates the periodic nature of the pattern Scrutinizing the pattern of Figure 5 we can in addition to simple translations detect only 2 fold rotation axes No other symmetry elements are present So the lattice net indicating the pattern s periodic nature must be a version of the Oblique Net We have drawn it in Figure 7 There we see that an Oblique Net can accidentally have the shape of a square net Indeed the pattern shows the accidental nature of the net being square or almost square for that matter Isolating a unit cell means eliminating all simple translations And because there are no other translations involved the most symmetrical unit cell is identical to the Complex Motif See next Figure Figure 8 Isolated unit cell of the pattern of Figure 5 The content of this isolated unit cell as isolated possesses only one single symmetry element namely a 2 fold rotation axis at its center Of course if the unit cell is repeated indefinitely and in this way forming our pattern the 2 fold rotation axes present

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  • Promorphology of Crystals Preparation IV
    documents we want to integrate the i n o r g a n i c beings within our Promorphological System This system was originally geared to describe and systematize the stereometric basic forms of o r g a n i s m s Organisms were investigated as to their inherent and intrinsic symmetries that could be derived when all accidental and extrinsic form features were subtracted i e left out of consideration This was done by looking for and finding ideal body axes investigating the nature of their poles and determining the geometric nature of their constant and intrinsic body centra i e whether such a centrum was at all present and if so whether it was a point a line or a plane All this resulted in a stereometric characterization of the given organism This characterization was then given in the form of a geometric solid say a pyramid cylinder etc expressing the basic ideal symmetry content geometrically But inorganic real intrinsic beings that are mainly represented by individual c r y s t a l s also show definite shapes and symmetries and even clearer than most organisms do So the desire came up to integrate crystals into the promorphological system But the promorphology of organisms was based on an important feature of the structure of organisms namely their TECTOLOGICAL structure which means that their body is made up of repeating parts such that the orientation of these parts is not periodic which means that they normally group themselves around an axis resulting in their orientation to be different from each other These parts are called ANTIMERS Other parts build up the organismic body by repeating such that their orientation is it is true the same for al these parts but then this repetition is a repetition only in one direction These parts are called METAMERS Crystals to be precisely single non twinned crystals on the other hand are always periodically built A certain microscopic unit is repeated while maintaining exactly the same orientation and moreover this repetition always goes along three directions in such a way that space is completely filled So if we investigate crystals we do not find antimers and metamers And especially the antimers played a crucial role in determining the stereometric basic form or promorph of the organism namely the number symmetry and configuration of antimers As has been said in single non twinned crystals we do not find such antimers which fact could indicate that crystals do not yield to any promorphological assessment at most doing so as a vague analogy So we are only theoretically legitimized to include crystals in our promorphological investigation if they in spite of their periodic nature possess some t e c t o l o g i c a l a s p e c t that determines in some way their shapes and symmetries in such a way that those symmetries can be promorphologically interpreted One indication of their tectological aspect is the fact that they are finite bodies with definite intrinsic symmetries on the basis of which they indeed are attributed to Crystal Classes of which there are thirty two But this is not enough We must explicitly demonstrate that they possess a genuine and intrinsic tectological aspect Assuming they have this aspect we can assess their promoph on the basis of their intrinsic point symmetry reflected in their belonging to a certain Crystal Class In accordance with our way of work within organic Promorphology we then look for and find the s t e r e o m e t r i c s o l i d that geometrically expresses this symmetry Not all crystals are such that their geometric shape even when they have grown undisturbedly in a uniform medium displays this symmetry Sometimes we must physically investigate their faces to detect symmetries or the lack of symmetries for instance by means of etch figures In these cases we must come up with a geometric solid having a shape that does not as such occur in the crystals involved in those cases In the present documents we are concerning ourselves with the demonstration of the tectological aspect in single non twinned crystals Thereby we make also use of the geometic properties of 2 D crystals in order to help the understanding of 3 D crystals We know that crystals are constituted by chemical atoms ions atomic or ionic groups And these chemical entities do have a tectological nature They are not themselves crystals i e they do not show a full fledged p e r i o d i c nature So we know the direction in which to find the tectological aspect in crystals the c h e m i c a l m o t i f s thay reside in them Well we now have to find such a non periodic motif in every crystal species How do we do this We can do this by eliminating everything that is responsible for the crystal s periodic non tectological nature And it are the TRANSLATIONS that are responsible for that periodic nature First we have the so called simple translations They are responsible for the periodic repetition of the mentioned microscopic entity the unit cell The unit cell is a building block of the given crystal furnished with chemical motifs Then we have the translational components of of certain symmetries glide planes in 2 D crystals glide lines and screw axes Because screw axes themselves involve three dimensional space they do not occur in 2 D crystals Screw axes and glide planes lines draw as it were the chemical motifs apart so that parts of them become displaced and separated from each other This lowers the symmetry of crystals insofar as they are considered microscopially which here means insofar as they are p e r i o d i c structures But in order to find the tectological aspect in crystals we must do away with these translations in

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  • Promorphology of Crystals Preparation V
    have to do with a centered rectangular net When we isolate the unit cell of Figure 3 which is fully equivalent to a total motif we see that it contains of 1 2 1 2 1 2 motifs s str See next Figure Figure 7 An isolated unit cell or total motif from the pattern of Figure 3 which is the same pattern as that of Figure 1 The repeating unit depicted in Figure 7 consists of 1 2 1 2 1 2 motifs s str while the repeating unit that we obtained in Figure 2h derived from the same pattern but with the net placed in the original position Figure 1 contains 4x1 4 1 2 motifs s str But while the symmetry of the latter Figure 2h is m i e it has one mirror plane there is no symmetry at all in the former Figure 7 So as unit cells these two units are equivalent but as Complex Motif not The unit of Figure 7 is not a Complex Motif because it has not yet the symmetry of the Class which is m It therefore turns out that in order to find the Complex Motif we must take that net point lattice position which has the highest symmetric unit cell And indeed we see that the unit cell of the alternative position of the net see Figure 3 is less symmetric than the one of the initially chosen net position Figure 1 The Complex Motif obtained from the latter and obtained without eliminating the glide translations can be considered as the p e n u l t i m a t e Complex Motif while the Complex Motif obtained by eliminating the glide translations can be considered as the u l t i m a t e Complex Motif Figure 2f and finally Figure 2g The penultimate Complex motif already possesses the symmetry of the Crystal Class associated with the Plane Group Cm namely m so the elimination of the glide translation was not strictly necessary The next Figure recapitulates the penultimate Complex Motif and the ultimate Complex Motif Figure 8 The penultimate Complex Motif and the ultimate Complex Motif associated with Plane Group Cm The symmetry of the Complex Motif consists of one mirror line as its only symmetry element The planimetric Figure that geometrically represents this symmetry is the Isosceles Triangle The isosceles triangle is accordingly the promorph or planimetric basic form of all 2 D crystals belonging to Plane Group Cm and with it to the Class m Figure 9 The Isosceles Triangle as the promorph or planimetric basic form of all 2 D crystals with a total symmetry described by Plane Group Cm and with it belonging to the Class m The Plane Group P2gg Periodic patterns of motifs according to Plane Group P2gg are based on a representative of the Primitive Rectangular Point lattice See next Figures Figure 10 Periodic two dimensional pattern according to the Plane Group P2gg The pattern must be considered as indefinitely extended in 2 D space A unit mesh choice is given yellow This unit mesh is primitive and has a point symmetry of 2 Point n is not equivalent to point a because the orientations of the motifs left and right to them are different So indeed the unit mesh is primitive The P2gg pattern contains two sets of glide lines indicated in the next Figure Figure 11 Periodic two dimensional pattern according to the Plane Group P2gg The pattern must be considered as indefinitely extended in 2 D space The pattern possesses two sets of g l i d e l i n e s indicated by red dashed lines Because there are no mirror lines parallel to the glide lines we know that the Complex Motif is drawn apart by these glide lines and effecting a lower symmetry as can be seen by the fact that the symmetry of the unit cell is 2 and thus not possessing the point symmetry of the crystal 2mm as can be deduced from the Plane Group symbol So to obtain the Complex Motif we must undo the glide translations We first handle the vertical translation involved in the vertical glide lines We take a copy of the pattern shift it vertically by half a vertical unit cell edge in order to undo the vertical translation and superimpose the shifted copy of the pattern onto the original Then we make a copy of the just obtained result shift that copy horizontally by half a horizontal edge of the unit cell and superimpose this shifted copy onto the first result We then obtain a pattern of which the smallest repeated unit is the Complex Motif i e the condition of being drawn apart of the motif is now undone In all this it is sufficient to just take a p a r t of the pattern of unit cell size or a little bigger because the pattern is periodic The next Figures illustrate this whole process Figure 12 A copy of the pattern P2gg is shifted upward indicated by the blue arrow by half the vertical length of the unit cell Figure 13 The shifted copy of the pattern P2gg is superimposed upon the original An area of the size of the unit cell is indicated Figure 14 We make a copy of the result of the previous Figure and shift it to the right indicated by the arrow by half the length of the horizontal unit cell edge Figure 15 We superimpose the shifted copy onto the the first result i e onto the original with respect to that copy From the just obtained result we can easily recognize the Complex Motif It is depicted in the next Figure Figure 16 The Complex Motif of the P2gg pattern as given above Its symmetry is 2mm which means that it possesses a 2 fold rotation axis in its center normal to the plane of the drawing and two mirror lines intersecting in the motif s center The light blue surroundings also belong to the Complex Motif We stated that the symmetry of the Complex Motif is 2mm However when we look carefully to its drawing we see a slight discrepancy with respect to its symmetry But this is resulting from slight errors inaccuracies that can easily creep in while making digital drawings From all the foregoing Figures it is clear that the symmetry of the Complex Motif is 2mm The simplest planimetric figures that geometrically express this 2mm symmetry are the Rectangle and the Rhombus Although the Rectangle and the Rhombus both possess the same symmetry namely 2mm Promorphology discriminates between them Let us explain this The shape of the Rectangle as well as of the Rhombus allows for two homopolar axes the d i r e c t i o n a l a x e s to be distinguished They are perpendicular to each other and coincide with the two mirror lines Motif units insofar as they are the Complex Motif s a n t i m e r s counterparts can have different relations to these two directional axes When the Complex Motif has f o u r antimers we can state the following Whether the promorph must be represented by a Rectangle or by a Rhombus depends on the fact whether the directional axes of the Complex Motif are i n t e r r a d i a l or r a d i a l which means that they either go between the antimers of the Complex Motif or they go right through them See also Figure 27 When the Complex motif with symmetry 2mm has only t w o congruent antimers one directional axis is radial it passes through the two antimers while the other directional axis is interradial it passes between the two antimers and the promorph must be expressed by a Rhombus See Figure 28 And so the R h o m b u s accordingly represents the promorph or planimetric basic form of all the 2 D crystals possessing a 2mm Complex Motif having either four antimers provided that its two directional axes are radial or having two antimers And thus a promorph based on a Rhombus just like that consists of two species one with four and one with two antimers depending on the tectology of the Complex Motif When none of the two directional axes are radial the promorph must be represented by a R e c t a n g l e and in that case there are always four antimers present See next Figures Figure 17 Rhombi representing a promorph with two and four antimers In the case of two antimers left image one directional axis is radial the other interradial In the case of four antimers right image both directional axes are radial Figure 17a A Rectangle representing a promorph with four antimers and with i n t e r r a d i a l directional axes Compare with Figure 27 right image Let s see if we can find it out for our Complex Motif In order to do this we must inspect its morphology The next Figure gives a magnification of the Complex Motif Although this magnification is slightly different from the original all the tectological features here the number and configuration of the antimers are preserved Figure 18 The Complex Motif of the P2gg pattern as given above and its magnification The two d i r e c t i o n a l a x e s red are indicated in the magnification To judge from the Complex Motif s magnification it consists of two oppositely positioned antimers analogous to the 3 D Orthostaura diphragma Heteropola heterostaura autopola or amphitect pyramids with only two antimers The latter feature is dictated by the given organism or inorganism See next Figures Figure 19 We can inscribe a Rhombus in the Complex Motif of the P2gg pattern to express the radiality of at least one directional axis Figure 20 Indication of the t w o antimers of the Complex Motif The Complex Motif derived from the P2gg motif pattern of Figure 10 has two antimers Figure 20 so its promorph or planimetric basic form must be represented by a Rhombus in which the TWO antimers are indicated Figure 21 The Rhombus as the promorph or planimetric basic form of all 2 D crystals of Plane Group P2gg and of the Class 2mm with two antimers The Plane Group P2mm Two dimensional periodic motif patterns according to the Plane Group P2mm are based on the Primitive Rectangular Point lattice The next Figure displays such a pattern Figure 22 Motif pattern according to the Plane Group P2mm A unit cell mesh is indicated yellow The pattern must be imagined as indefinitely extended in 2 D space The P2mm pattern does not possess glide lines So the Complex Motif is identical to the content of a unit cell which always means the most symmetric unit cell if there are several possible choices of unit cell A unit cell as depicted above contains 4x1 4 1 motif s str If we want to avoid fragments of motifs s str we can take a total motif i e a complete motif associated with one of the lattice nodes Such a total motif is fully equivalent to the unit cell See next Figure Figure 23 A total motif associated with the upper right lattice point of the unit cell Recall that all lattice points are equivalent and also all lattice points their corresponding complete surroundings Such a complete environment surroundings consists of the motif s str or parts thereof its corresponding background The total motif as established in the previous Figure is equivalent to the C o m p l e x M o t i f of the pattern of Figure 22 Indeed when we isolate it effecting an elimination of the simple translations that are responsible for the repetition of the unit cell or total motif we get the Complex Motif See next Figure Figure 24 The Complex Motif of the periodic pattern of Figure 22 The symmetry of the motif s str dictates the symmetry of the whole motif The Complex Motif as established in Figure 24 has a symmetry of 2mm which means that it has two non equivalent mirror lines and a two fold rotation axis going through the intersection of these mirror lines and normal to the plane of the drawing To judge from the morphology of the motif s str the Complex Motif has two antimers so the promorph or planimetric basic form must be expressed by a Rhombus in which the presence of the two antimers is indicated See next Figures Figure 25 The Complex Motif of the periodic pattern of Figure 22 and a magnification of it to show tectological features In the magnification the two directional axes are indicated red lines They coincide with the two mirror lines Figure 26 Complex Motif of the P2mm pattern of Figure 22 Left image In the Complex Motif a Rhombus can be inscribed indicating the radiality of at least one directional axis Right image The motif s str indicates the presence of two antimers Therefore one directional axis is radial the other interradial It is perhaps instructive to enumerate the other two possible promorphs for the 2mm symmetry See next Figure Figure 27 Possible 2 D motifs with 2mm symmetry two mirror lines perpendicular to each other and one 2 fold rotation axis that contains the intersection of the mirror lines and is perpendicular to the plane of the drawing Both motifs consist of four antimers The directional axes are indicated by red lines They coincide with the two mirror lines In the left image the directional axes are r a d i a l which means that they pass through the antimers The promorph must accordingly be expressed by a Rhombus In the right image the directional axes are i n t e r r a d i a l which means that they pass between the antimers The promorph must accordingly be expressed by a Rectangle See also Figure 17a A motif clearly displaying its composition out of only t w o antimers is shown in the next Figure Here one directional axis is radial while the other is interradial Figure 28 A third possibility for the morphology of 2mm motifs The motif consists of only two congruent antimers Directional axes are indicated by red lines One directional axis is radial it passes through the antimers while the other is interradial it passes beteen the antimers The promorph must be expressed by a Rhombus Our derived Complex Motif derived from the P2mm pattern of Figure 22 and depicted in the Figures 24 25 and 26 belongs to this type be it less clearly In the next Figure we depict the planimetric figure which expresses the 2mm symmetry of the Complex Motif Figures 24 25 and 26 of the P2mm pattern of Figure 22 and which shows i e expresses the fact that the Complex Motif consists of two antimers Figure 29 The Rhombus provided with the indication of the presence in the Complex Motif of two antimers is the planimetric basic form of all 2 D crystals of the Plane Group P2mm and with it of the Crystal Class 2mm of which the Complex Motif consists of two antimers The Plane Group P2mg The symmetry of two dimensional periodic patterns according to the Plane Group P2mg is based on the Primitive Rectangular Point Lattice The next Figure depicts such a pattern Figure 30 A motif pattern according to the Plane Group P2mg A unit cell is indicated yellow The pattern must be imagined as indefinitely extended in 2 D space The point n is not equivalent to the point a which is evident in their different surroundings This means that if the point a is a lattice point then the point n is not a lattice point so the net is not centered This pattern has vertical glide lines Figure 31 A motif pattern according to the Plane Group P2mg A unit cell is indicated yellow The pattern must be imagined as indefinitely extended in 2 D space In addition to horizontal mirror lines green lines there are vertical glide lines g indicated by red dashed lines The pattern also has 2 fold rotation axes but these are not shown here In order to find the Complex Motif we must undo the translations involved in the glide lines We accomplish this by our Copy Shift Superimpose method And in the present case the shift must be a vertical one The length of the shift is equal to the length of one half of the vertical side of the unit cell See next Figures Figure 32 A copy red is made of the P2mg pattern of Figure 30 The copy is shifted vertically upward and then copied onto the original The fact that small areas top and bottom are not involved in the superposition is a consequence of the fact that we use only a finite part of the pattern This is legitimate because the pattern is periodic The next Figure properly gives the final result Figure 33 Final result of the Copy Shift Superimpose operation applied to the motif pattern of Figure 30 One repeating unit is highlighted light blue One can see that the vertical glide lines that were initially present are now transformed into mirror planes as an effect of the Copy Shift Superimpose operation See next Figure Figure 34 The initial vertical glide lines are now turned into vertical mirror lines indicated by red solid lines The black lines indicate the mirror lines that were originally present See Figure 31 The repeating unit as indicated in Figure 33 is our sought for Complex Motif If we isolate this motif then at the same time all simple

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  • Promorphology of Crystals Preparation VI
    of the pattern of Figure 11 maple leaf pattern according to P4gm by sucessively undoing the diagonal vertical and horizontal glide translations and then superimposing the results while concentrating on the immediate surroundings of one lattice node only We see that the two results are identical not only with respect to the symmetry 4mm but also with respect to the shape of the resulting motif As is evident from Figure 18 we already have reached the 4mm symmetry Motifs with 4mm symmetry are repeated When we isolate such a motif plus its corresponding background i e if we eliminate all simple translations we end up with the Complex Motif Above we executed the copy horizontal shift superimpose operation onto the result of the copy vertical shift superimpose operation In Part Two Figure 30 there of the present series of documents on the other hand we eliminated the horizontal translations while concentrating on the immediate surroundings of one lattice node only not from the result of the elimination of the vertical glide translation but from the initial pattern like was done with respect to the vertical glide translations It is instructive to execute the copy horizontal shift superimpose operation not on the result of the corresponding vertical operation but on the initial pattern Figure 11 Figure 18b A copy is made from the pattern P4gm of Figure 11 The copy is shifted horizontally by half the length of a side of the unit cell Figure 18c The shifted copy of the previous Figure is superimposed onto the original An area within the region where the superposition actually has taken place is highlighted blue and green for comparison with earlier results Although we do not need to consider the d i a g o n a l glide translations it is nevertheless instructive to investigate them We begin with the undoing of the diagonal translations from our just obtained result This means that we copy the pattern of Figure 18 shift the copy by half a unit cell diagonal and superimpose the shifted copy onto the original The next Figure gives the result of all this Figure 19 Copy shift superimpose operation in the diagonal direction executed on the result obtained in Figure 18 As was to be expected the pattern doesn t change because the begin and end points that determine the shift vector are equivalent The next Figure shows the result of the copy shift superimpose operation in the diagonal direction in which the amount of the shift is equal to half a unit cell diagonal executed on the initial pattern i e on the pattern of Figure 11 Figure 20 Result of the diagonal copy shift superimpose operation in which the amount of shift is equal to half a unit cell diagonal In the next figure we isolate the area that was actually involved in the superposition in fact of course always the whole infinite pattern is involved in the superposition Figure 21 The area actually involved in the

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  • Promorphology of Crystals Preparation VII
    The only symmetry elements are simple translations responsible for the repetition of the unit cell and 3 fold rotation axes First we could think of the Compex Motif being represented by the content of the unit cell as chosen in Figure 11 of even more conveniently the total motif i e the complete surroundings of a lattice point Figure 12 Periodic two dimensional pattern according to the Plane Group P3 The pattern must be considered as indefinitely extended in 2 D space In addition of a possible unit cell yellow a corresponding total motif is indicated green It is the complete surrounding of the upper right lattice point of the lattice chosen When we isolate this total motif thereby eliminating all simple translations we obtain the following Figure 13 Isolated total motif It consists of one whole motif s str plus its corresponding background This total motif however cannot represent the Complex Motif Recall that in addition to the motif s str the tri radiate figure its corresponding background green including the shape of that background belongs to the total motif associated with the given lattice point From the Plane Group symbol P3 we can deduce that 3 i e the possession of one 3 fold rotation axis as the only symmetry element is the translation free residue of the Plane Group P3 So the symmetry of the Complex Motif must be 3 Well although the symmetry of the motif s str within the total motif Figure 13 is indeed 3 the symmetry of the whole total motif is lower In fact the total motif has no symmetry at all All this implies that we have to look for another unit cell Earlier we had established that in order to find the Complex Motif we must start with the most symmetric unit cell choice Indeed the rhombic unit cell as chosen in Figure 11 is certainly not the most symmetric unit cell The pattern clearly suggests a h e x a g o n a l u n i t c e l l And such a unit cell has definitely more symmetry than the rhomb shaped unit cell has See next Figure Figure 14 An alternatif unit cell choice The regular hexagon We obtain it after we have triangulated the net i e we have inserted NW SE connection lines One such hexagon is indicated yellow The other colors serve to show that the hexagon is indeed a repeating unit Recall that the net must be imagined as to be extended infinitely in 2 D space If we isolate this hexagonal unit cell we get the following Figure 15 An isolated hexagonal unit cell of the periodic motif pattern of Figure 11 It can serve as the Complex Motif The above isolated unit cell consists of 6x1 3 1 3 motifs s str tri radiate figures So the Complex Motif consists of three motifs s str We can try to isolate just one whole motif s str one tri

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  • Promorphology of Crystals Preparation VIII
    the rhomb shaped unit cell Although the total motif as depicted in Figure 6a is a genuine repeating unit it is not a p r e f e r a b l e unit cell choice because of the following reason to which we referred already when treating of the Plane Group P3 The corners of the hexagon as can be seen in Figure 6a are not equivalent to the central point because in that point something is present which is absent in the corner points This implies that the hexagonal net point lattice based on such a unit cell would necessarily be primitive i e the central point which is furnished with a motif s str is not a lattice point All this further implies that the motifs s str do not coincide with the lattice points Although such a coincidence is theoretically not mandatory it is very convenient and instructive But in spite of the fact that the total motif is less suited to figure as unit cell and as such defining a lattice i e a net it can figure and preferable so as C o m p l e x M o t i f It is smaller than that of Figure 6 Instead of 3 it consists of just one whole motif s str So it does not contain half or third motifs str If we isolate such a total motif we indeed have a preferable version of the penultimate Complex Motif See next Figure Figure 6b Preferable because smaller and because it does not contain halfs or thirds of motifs s str penultimate Complex Motif equivalent to a total motif of the rhomb shaped unit cell Figure 1 or of the larger hexagonal repeat unit Figure 5 It consists of one whole motif s str plus corresponding background Note that the shape of the background is as is the shape of the motif s str hexagonal but differently orientated by 30 0 Although we now have the Complex Motif it will nevertheless be instructive to eliminate all glide translations and see what happens We will do this with our method of Copy Shift Superimpose Let s start with the first out of three set of glide lines Figure 2 Figure 7 A copy is made of the pattern of Figure 1 and shifted by the height of the rhomb shaped unit cell as depicted in Figure 1 the shift is indicated by the blue arrow and then superimposed onto the original Compare with Figure 2 In Figure 7 we shifted the copy upward Exactly the same result will be obtained when we shift it downward This is easy to grasp If we in Figure 7 consider the copy to be the original and the original to be the copy then the copy has been shifted downward This corresponds to the translation direction of any such glide line We can read this symmetry operation as REFLECT AND TRANSLATE UPWARD See Figure 2 but we can

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