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  • Promorphology of Crystals I
    characterized by two equivalent horizontal crystallographic axes perpendicular to each other and one other non equivalent vertical axis perpendicular to the other two axes For the concept of crystallographic axes see below The Class 4mm has a mirror plane through one of the horizontal crystallographic axes and another mirror plane through the other axis So these mirror planes are equivalent belonging to one and the same type They will be represented by only one m in the Hermann Mauguin symbol But the Class has two more mirror planes namely the ones bisecting the angles between the horizontal crystallographic axes These two mirror planes are equivalent to each other but non equivalent to the mirror planes that go through the horizontal crystallographic axes So they are also represented by only one m in the Hermann Mauguin symbol And because the Class possesses only one 4 fold rotation axis and so also just one type only one 4 will appear in the Hermann Mauguin symbol So the full symbol for this Class is 4mm The Hermann Mauguin symbol of a Crystal Class consists of one two or three parts that relate to the following directions For the Crystal Systems and the crystallographic axes i e a axis b axis etc mentioned in the table see the OVERVIEW OF CRYSTAL SYSTEMS AND CLASSES bvelow Crystal System First Part Second Part Third Part Monoclinic Ortho axis Orthorhombic a axis b axis c axis Tetragonal c axis a and b axis bisectors between a and b axes Hexagonal c axis a b and d axis bisectors betw a b and d axes Isometric a b and c axis body diagonals plane diagonals Regarding the Isometric System Body diagonals are lines perpendicular to the octahedral faces while plane diagonals are lines that are parallel to one of the cube faces Regarding the Monoclinic System The Ortho axis is the crystallographic axis of the Monoclinic System that is perpendicular to the two other monoclinic crystallographic axes Regarding the Triclinic System It consists of only two Classes having respectively 1 and 1 as their Hermann Mauguin symbols Let us treat another example The symbol 4 m 2 m 2 m refers to a tetragonal crystal i e a crystal belonging to the Tetragonal Crystal System 4 also occurs in the isometric Classes but the symbols for such crystals are directly recognized by the 3 at the second part of the symbol This Class has a 4 fold rotation axis along the c axis and a mirror plane perpendicular to it so 4 m comes as the first part of the symbol It has moreover 2 fold rotation axes along the a and b axes each of them having a mirror plane perpendicular to it so 2 m will figure as the second part of the symbol there are two such rotation axes coupled with a mirror plane but because they are equivalent they are represented in the symbol by only one 2 m Further the Class has two more 2 fold rotation axes along the bisectors of the a and b axes and a mirror plane perpendicular to each one So they are also represented by 2 m which however will figure as the third part of the Hermann Maugin symbol of the Class Sometimes the Hermann Maugin symbols are used in an abbreviated form namely in those cases where some symmetry elements are implied by others The implied ones are then omitted if there is no possibility of confusion On this website we will however always use the complete symbols Overview of the 32 Symmetry Classes of Crystals In this Overview we use the following abbreviations and terminology cs center of symmetry p polar axis present absent u v nbsp u v fold rotation axes present for example three 4 three 4 fold rotation axes present 4 means a 4 fold roto inversion axis for example one 4 one 4 fold roto inversion axis present An expression like Hemimorphic of Holohedric means that the hemimorphic Form is directly derived from the holohedric Form The same applies to expressions like hemimorphic of pyramidal hemihedric which means that the hemimorphic Form is derived from a in this case pyramidal hemihedric Form and consequently not from a holohedric Form one one one means three items present for example three mirror planes but belongong to three different types The Hermann Maugin symbols are written down below each Class name The next six tables are going to list all the 32 Crystal Classes one table for each Crystal System The Isometric System Division Class mirror planes axes cs Holohedric Hexakisoctahedric 4 m 3 2 m 3 6 three 4 four 3 six 2 Tetrahedric Hemihedric Hexakistetrahedric 4 3 m 6 three 2 four 3 p Pentagonal Hemihedric Duakisdodecahedric 2 m 3 3 three 2 four 3 Plagihedric Hemihedric Pentagonikositetrahedric 4 3 2 three 4 four 3 six 2 Tetartohedric Tetrahedric pentagondodecahedric 2 3 three 2 four 3 p The Tetragonal System Division Class mirror planes axes cs Holohedric Ditetragonal bipyramidal 4 m 2 m 2 m 1 2 2 one 4 two two 2 Hemimorphy of Holohedric Ditetragonal pyramidal 4 m m 2 2 one 4 p Pyramidal Hemihedric Tetragonal bipyramidal 4 m 1 one 4 Hemimorphy of Pyramidal Hemihedric Tetragonal pyramidal 4 one 4 p Trapezohedric Hemihedric Tetragonal trapezohedric 4 2 2 one 4 two two 2 Sphenoidic Hemihedric Tetragonal scalenohedric 4 2 m 2 one two 2 Sphenoidic Tetartohedric Tetragonal bisphenoidic 4 one 4 The Hexagonal System Division Class mirror planes axes cs Holohedric Dihexagonal bipyramidal 6 m 2 m 2 m 1 3 3 one 6 three three 2 Hemimorphy of Holohedric Dihexagonal pyramidal 6 m m 3 3 one 6 p Pyramidal Hemihedric Hexagonal bipyramidal 6 m 1 one 6 Hemimorphy of Pyramidal Hemihedric Hexagonal pyramidal 6 one 6 p Trapezohedric Hemihedric Hexagonal trapezohedric 6 2 2 one 6 three three 2 Rhombohedric Hemihedric Ditrigonal scalenohedric 3 2 m 3 one 3 three 2 Rhombohedric Tetartohedric Trigonal rhombohedric 3 one 3 Trigonal Hemihedric Ditrigonal bipyramidal 6 m 2 1 3 one 3 three 2 p Hemimorphy of Trigonal Hemihedric Ditrigonal pyramidal 3 m 3 one 3 p Trigonal Tetartohedric Trigonal bipyramidal 6 1 one 3 Hemimorphy of Trigonal Tetartohedric Ogdohedric Trigonal pyramidal 3 one 3 p Trapezohedric Tetartohedric Trigonal trapezohedric 3 2 one 3 three 2 p The Orthorhombic System Division Class mirror planes axes cs Holohedric Rhombic bipyramidal 2 m 2 m 2 m 1 1 1 one one one 2 Hemimorphy Rhombic pyramidal m m 2 1 1 one 2 p Hemihedric Rhombic bisphenoidic 2 2 2 one one one 2 The Monoclinic System Division Class mirror planes axes cs Holohedric Prismatic 2 m 1 one 2 Hemimorphy Sphenoidic 2 one 2 p Hemihedric Domatic m 1 The Triclinic System Division Class mirror planes axes cs Holohedric Pinacoidal 1 Hemihedric Asymmetric 1 Now we are ready to continue our p r o m o r p h o l o g y of crystals Recall that we assess the promorph of a single crystal by establishing the simplest geometric body that geometrically exhibits the symmetry of the particular Class to which that crystal belongs This simplest geometric body should moreover be such that it cannot be further divided without losing its specific symmetry We d promised to explain this further and in order to do so we will give an example Within the Class 4 m 2 m 2 m the Ditetragonal bipyramidal Class of the Tetragonal Crystal System the simplest geometric body that represents this symmetry content geometrically is a tetragonal prism Figure 1 A tetragonal prism It consists of six faces The vertical faces are rectangles the top and bottom faces are squares The tetragonal prism has here as a geometric solid not as a crystallographic Form six faces while the tetragonal bipyramid also expressing 4 m 2 m 2 m has eight faces Figure 2 A tetragonal bipyramid It consists of eight faces The equatorial plane is a square But the prism can be divided by any plane perpendicular to the c axis along which runs the 4 fold rotation axis and the resulting parts still possess the full 4 m 2 m 2 m symmetry This is not possible with the tetragonal bipyramid one but the simplest body i e comes next to the tetragonal prism So we will choose the t e t r a g o n a l b i p y r a m i d as the b a s i c f o r m promorph of the Class 4 m 2 m 2 m and as such it is a member of the I s o s t a u r a o c t o p l e u r a which is one of the many categories of our Promorphological System Antimers Organic promorphs are assessed mainly on the basis of the h o m o t y p i c n u m b e r number of antimers or counterparts and further on the basis of the form relationships of those antimers and with it of the relationships of the axes which they imply and finally on the basis of absence or presence of a main axis and the nature of its poles Most important are the antimers In crystals we would as it initially seems have to consider the tectology of the c h e m i c a l m o t i f that remains after elimination of all translations But because the crystal as a whole i e as a macroscopical object does not possess a tectological structure and so does not possess true antimers we cannot determine the promorph on the basis of prevailing antimers The symmetry of the given Crystal Class leads us to a certain simple geometric body representing this symmetry geometrically The number of antimers must now be determined from this geometric body implying that sometimes several numbers of antimers are equally possible say six or eight two or four etc In these cases we will give both numbers So in the case of Crystals we determine the homotypic basic number of and from the basic form itself instead of determining it from that real object of which we consider the basic form promorph In principle however the number of antimers the homotypic basic number can be determined from the real object itself namely from that chemical motif that remains after eliminating all translations But in most cases such a determination assessment will not be easy to carry out successfully and without errors so we will stick to the assessment of the number of antimers from the basic form itself New Promorphological Categories In some cases we will have to introduce some new promorphological categories in order to accommodate for some crystal classes Those new categories that as far as I can see are not materialized in any organism and are moreover not expected to be materialized ever will be presented in separate INFRAMES in order to prevent the Promorphological System to become more or less messy and unwieldy and in order to keep the system organic Promorphology is a system of organic forms When on the other hand a new category is as to my knowledge not materialized in any organism but could reasonably be expected to be so materialized at some time or place it will be incorporated and fully integrated into the Promorphological System but not extensively be dealt with When all this is done below we will have established a connection between the crystallographic system of crystals and the promorphological system of organic forms Our motivation for this is to emphasize the u n i t y of the material world The inorganic world of genuine beings as represented first of all by single crystals is at a fundamental level not different from the organic world We have established this on the basis of their c o m m o n o n t o l o g y in all the philosophical essays documents of the First Part of this website accessible via back to homepage Twins Many crystals occur as t w i n s i e not as single crystals but as regular and lawful aggregations of several single crystals Many such twinned crystals have accordingly acquired a t e c t o l o g i c a l structure consisting of genuine antimers Such twinned crystals can be directly promorphologically assessed as to one or another promorphological category In the following we will however concentrate on single crystals and assess their promorphology Each determined p r o m o r p h stereometric basic form can be accessed by clicking on its name When one does so one will end up IN the Promorphological System so that one can see the proper place of the determined promorph within the system Isometric System REMARK In the Hermann Mauguin symbol expressing the symmetry we use for typographical reasons an asterisk for example 3 A symmetry element denoted by 3 is a 3 fold rotoinversion axis rotation of 120 0 immediately followed by inversion in a point on that axis In the crystallographic literature and also in our Figures this asterisk is replaced by a horizontal score above the relevant numeral Hexakisoctahedric Class 4 m 3 2 m The geometric solid taken according to the above established criteria to represent the stereometric basic form promorph of all the crystals of this Class is the Regular Octahedron Figure 3 The promorph is accordingly that of the Octaedra regularia Polyaxonia rhythmica Figure 3 Regular Octahedron Stereometric Basic Form of all the single crystals of the Hexakisoctahedric Class of the Isometric Crystal System Adapted from HURLBUT C KLEIN C 1977 Manual of Mineralogy Hexakistetrahedric Class 4 3 m The geometric solid taken according to the above established criteria to represent the stereometric basic form promorph of all the crystals of this Class is the Regular Tetrahedron Figure 4 The promorph is accordingly that of the Tetraedra regularia Polyaxonia rhythmica Figure 4 Regular Tetrahedron Stereometric Basic Form of all the single crystals of the Hexakistetrahedric Class of the Isometric Crystal System Symmetry content is indicated Small squares with solid oval 4 fold rotoinversion axis Small triangles 3 fold rotation axes The right image displays the prevailing mirror planes After HURLBUT C KLEIN C 1977 Manual of Mineralogy Pentagonicositetrahedric Class 4 3 2 The geometric solid taken according to the above established criteria to represent the stereometric basic form promorph of all the crystals of this Class is the Gyroid or Pentagonal Icositetrahedron Figure 5 The promorph is accordingly that of the Subendosphaerica gyroidea Polyaxonia subendosphaerica Figure 5 Gyroid Stereometric Basic Form of all the single crystals of the Pentagonicositetrahedric Class of the Isometric Crystal System After HURLBUT C KLEIN C 1977 Manual of Mineralogy Duakisdodecahedric Class 2 m 3 The geometric solid taken according to the above established criteria to represent the stereometric basic form promorph of all the crystals of this Class is the Pyritohedron or Pentagonal Dodecahedron Figure 6 The promorph is accordingly that of the Subendosphaerica pyritoidea Polyaxonia subendosphaerica Figure 6 Pyritohedron Stereometric Basic Form of all the single crystals of the Duakisdodecahedric Class of the Isometric Crystal System Symmetry content is indicated Small triangles with holes 3 fold rotoinversion axes Small black ovals 2 fold rotation axes In the right image the prevailing mirror planes are indicated The next Figure again depicts the Pyritohedron but without all the indications After HURLBUT C KLEIN C 1977 Manual of Mineralogy Figure 6a Pyritohedron or Pentagonal Dodecahedron Stereometric Basic Form of all the single crystals of the Duakisdodecahedric Class of the Isometric Crystal System Adapted from HURLBUT C KLEIN C 1977 Manual of Mineralogy Tetrahedric pentagondodecahedric Class 2 3 The geometric solid taken according to the above established criteria to represent the stereometric basic form promorph of all the crystals of this Class is the Tetartoid or Tetrahedral Pentagonal Dodecahedron Figure 7 The promorph is accordingly that of the Subendosphaerica tetartoidea Polyaxonia subendospherica Figure 7 Tetartoid Stereometric Basic Form of all the single crystals of the Tetrahedric pentagondodecahedric Class of the Isometric Crystal System After HURLBUT C KLEIN C 1977 Manual of Mineralogy Tetragonal System Ditetragonal bipyramidal Class 4 m 2 m 2 m The geometric solid taken according to the above established criteria to represent the stereometric basic form promorph of all the crystals of this Class is the Tetragonal Bipyramid Figure 8 The promorph is accordingly that of the Isostaura octopleura Stauraxonia homopola Figure 8 Tetragonal Bipyramid Stereometric Basic Form of all the single crystals of the Ditetragonal bipyramidal Class of the Tetragonal Crystal System Ditetragonal pyramidal Class 4 m m The geometric solid taken according to the above established criteria to represent the stereometric basic form promorph of all the crystals of this Class is the Tetragonal Pyramid Figure 9 The promorph is accordingly that of the Isopola tetractinota Stauraxonia heteropola homostaura Figure 9 Tetragonal Pyramid Stereometric Basic Form of all the single crystals of the Ditetragonal pyramidal Class of the Tetragonal Crystal System Tetragonal bipyramidal Class 4 m The geometric solid taken according to the above established criteria to represent the stereometric basic form promorph of all the crystals of this Class is the Tetragonal Gyroid Bipyramid Figure 10 Versions of the gyroid bipyramid in which the receding angles are replaced by their corresponding anti receding angles can occur in crystals The promorph is accordingly that of the Isosigmostaura quadrimera Stauraxonia homopola Figure 10 Oblique top view of the Tetragonal 4 fold Gyroid Bipyramid Stereometric Basic Form of all the single crystals of the Tetragonal bipyramidal Class of the Tetragonal Crystal System Because of the view direction the lower pyramid is not visible Tetragonal pyramidal Class 4 The geometric solid taken according to the above established criteria to represent the stereometric basic form promorph of all the crystals of this Class is the Tetragonal Gyroid Pyramid

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  • Promorphology of Crystals II
    of this Class is the Trigonal 3 fold Gyroid Twisted Apical Bipyramid Figure 11 The promorph is accordingly Isosigmostaura pseudorhomboedra Stauraxonia homopola sigmostaura In the next Figures we will construct the Trigonal Gyroid Twisted Apical Bipyramid Figure 8 Construction of one of the equal bases of the Trigonal Gyroid Twisted Apical Bipyramid from the base of a regular trigonal pyramid The 3 fold rotation axis is in the center of the depicted form as indicated Figure 9 Slightly oblique top view of a trigonal 3 fold gyroid pyramid The final geometric solid to be constructed the Trigonal Gyroid Twisted Apical Bipyramid consists of two such pyramids connected to each other by their tips and rotated 60 0 with respect to each other By means of superimposing two bases of 3 fold gyroid pyramids combined with a rotation by 60 0 of one such base with respect to the other we illustrate the construction of the Trigonal Gyroid Twisted Apical Bipyramid Figure 10 Superposition and rotation 60 0 of two bases of trigonal pyramids in order to construct the Trigonal Gyroid Twisted Apical Bipyramid The next Figure finally gives the Trigonal Gyroid Twisted Apical Bipyramid It is the Stereometric Basic Form of all the single crystals of the Trigonal rhombohedric Class of the Hexagonal Crystal System I e it is the geometric solid body that geometrically depicts the symmetry of that Class which here means the possession of a 3 fold rotoinversion axis as its only symmetry element Figure 11 The Trigonal Gyroid Twisted Apical Bipyramid the Stereometric Basic Form of all the single crystals of the Trigonal rhombohedric Class of the Hexagonal Crystal System The symmetry of the just depicted gyroid bipyramid can be further illustrated by the following diagram representing the 3 fold rotoinversion symmetry Figure 11a The Symmetry of the Trigonal Gyroid Twisted Apical Bipyramid Figure 11 diagrammed Ditrigonal bipyramidal Class 6 m 2 The geometric solid taken according to the above established criteria to represent the stereometric basic form promorph of all the crystals of this Class is the Trigonal Bipyramid Figure 12 The Promorph is accordingly that of the Isostaura polypleura hexapleura Stauraxonia homopola Figure 12 Trigonal Bipyramid Stereometric Basic Form of the single crystals of the Ditrigonal bipyramidal Class of the Hexagonal Crystal System Ditrigonal pyramidal Class 3 m The geometric solid taken according to the above established criteria to represent the stereometric basic form promorph of all the crystals of this Class is the Trigonal Pyramid Figure 13 The Promorph is accordingly that of the Anisopola triactinota Stauraxonia heteropola homostaura Figure 13 Trigonal Pyramid Stereometric Basic Form of the single crystals of the Ditrigonal pyramidal Class of the Hexagonal Crystal System Trigonal bipyramidal Class 6 The geometric solid taken according to the above established criteria to represent the stereometric basic form promorph of all the crystals of this Class is the Trigonal 3 fold Gyroid Bipyramid Figure 14 Versions of the gyroid bipyramid in which the receding angles are replaced by their corresponding anti

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  • Promorphology of Crystals III
    numeral The geometric solid taken according to the above established criteria to represent the stereometric basic form promorph of all the crystals of this Class is a Triclinic Bipyramid or Oblique Rhombic Bipyramid Figure 10 and 11 Such a bipyramid has no symmetry axes but it does have a body center namely its center of symmetry Of course it has a 1 fold rotation axis but such an axis is never unique Every geometric solid is mapped onto itself by any such axis whatsoever because every geometric body maps onto itself by a rotation of 360 0 about any axis such an axis is a 1 fold rotation axis In the pyramid we can nevertheless draw three axes They are the crystallographic axes i e the three triclinic axes They intersect in the center of symmetry but do not involve angles of 90 0 The crystallographic axial system is wholly oblique Promorphologically the poles of these axes do not represent specific body parts i e specific anatomical parts of the organic individual that realizes this basic form We can draw as many axes through the center of symmetry as we wish to None of these axes stands out neither does a subset of those axes and this is equivalent to there being no genuine axes at all The situation looks a little like we see in the Homaxonia spheres but here all axes are identical The promorph is accordingly that of the Anaxonia centrostigma i e bodies having no promorphological axes but which do have a body center Figure 10 Triclinic Bipyramid Stereometric Basic Form of the Pinacoidal Class of the Triclinic Crystal System The angle between the equatorial plane and the main axis is different from 90 0 The next Figure depicts this same bipyramid but now with the triclinic axial system inserted Figure 11 Triclinic Bipyramid Stereometric Basic Form of the Pinacoidal Class of the Triclinic Crystal System The angle between the equatorial plane and the main axis is different from 90 0 Triclinic axial system inserted The three axes do not involve angles of 90 0 The next Figure depicts the equatorial plane of the Triclinic Bipyramid It is not a square nor a rectangle and especially not a rhombus Figure 12 Equatorial plane of a Triclinic Bipyramid as seen from a direction perpendicular to that plane and thus not seen along the direction of the bipyramid s main axis The angle between the equatorial plane and the main axis is different from 90 0 Two triclinic axes inserted red they meet in the bipyramid s center of symmetry i In order to make matters more clear the next Figures depict a parallelopipedum which is a geometric body also possessing only a center of symmetry But because we can divide this body such that the resulting parts are still parallelopipeda each still possessing a center of symmetry in other words because this body is idem specie divisible it cannot serve as the geometric body that represents the

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  • Promorphological Theses
    In them we determine the relative intersection distances of a crystal face with repect to the crystallographic axial system and the axial center i e intersection point of the crystallographic axes In this way the relative orientations of crystal faces are determined But as has been said with these orientations not all symmetries are yet determined some properties of the faces need to be investigated in order to obtain the true symmetry point symmetry of the given crystal Because the mentioned Complex Motif from which the crystal s point symmetry is immediately evident is just a theoretical construct one determines the crystal s symmetry in the just described way 13 On this firm promorphological basis a mathematical knowledge of the organic forms is possible as it is in crystals III Theses of the constitution of the individual basic forms 1 The promorph or stereometric basic form which lies at the basis of every organic form in which either a center is present or in addition to it a set of axes or at least one axis is directly implied with mathematical necessity by the number and size positioning and connection the equality or inequality differentiation of the constituent form elements 2 In simple organisms i e those that are just a single individual of first order a single Cell the promorph is consequently implied by the number and size the positioning and connection the equality or inequality differentiation of the constituent molecules especially the macromolecules and their complexes which are composed of all atoms within the organic body The promorph of a cell that is a constituent of a tissue is also partly determined by its immediate surroundings 3 In composed organisms on the other hand i e those which are represented by an intrinsic pattern of two or more first order form individuals and this means in all organic individuals of second or higher order the promorph is directly implied i e formally caused by the number positioning and connection of the constituent form individuals of the next lower individuality order If a certain form individual is not a biont i e not an independently living unit but a constituent of a higher order form individual then its promrph is co determined by its immediate surroundings Here we see how the promorph is wholly determined by the tectological built up of the given organic form unit without it being necessary to consider the Tectology as developed on this website in an a b s o l u t e way i e without necessarily attributing an absolute status to the different order form individuals cells organs antimers metamers persons and colonies So the Promorphology is independent of the more or less problematic i e speculative assumption of the absoluteness of the different order form individuals 4 The stereometric basic form of Organs or second order form individuals is consequently implied i e formally caused by the number positioning and differentiation of the constituent cells first order form individuals especially by the number and positioning of those cell groups which as p a r a m e r s lie around a common center If the organ is not a biont its promorph is co determined by its surroundings surroundings still within the organism i e belonging to the organism s morphological make up 5 The stereometric basic form of Antimers or third order form individuals is in the same way formally caused by the number positioning and differentiation of the constituting organs especially the paramers i e now paramers with respect to organs or 2nd order form individuals insofar as those paramers are co determining an antimer co determining determining the antimer together with other organ paramers If the antimer is not a biont which is almost always the case then the promorph of the antimer is co determined by its surroundings The latter still within the organism 6 The stereometric basic form of Metamers or fourth order form individuals is implied by the number positioning and differentiation of the constituting antimers If the metamer is not a biont its promorph is co determined by its surroundings The latter still within the organism 7 The stereometric basic form of Persons or fifth order form individuals is implied by the number positioning and differentiation of the constituting metamers and because of that at the same time by the constituting antimers If the Person is not a biont this occurs in the majority of plants its promorph is co determined by its surroundings The latter still within the organism 8 The stereometric basic form of Colonies or sixth order form individuals is implied by the number positioning and differentiation of the constituting persons off shoots IV Theses of the geometric center differences of the basic forms 1 All stereometric assessible basic forms of organic individuals as well as inorganic ones divide with respect to their intrinsic body center in three main groups that we call the Centrostigma Centraxonia and the Centrepipeda Centrostigma occur in addition to some Axonia also in some Anaxonia namely those that have a center of symmetry that is then their body center Anaxonia centrostigma 2 In the Centrostigma the stereometric forms with an intrinsic center the natural center of the form i e the planimetric body part with respect to which all other parts of the organic or inorganic body have a definite positional distance and direction relation is a p o i n t This is the case in the mentioned Anaxonia centrostigma in which that center is their center of symmetry the Homaxonia spheres and in the Polyaxonia endospheric polyhedra 3 In the Centraxonia the stereometric forms with a mid line axis the natural center of the form is a l i n e main axis or longitudinal axis This is the case in all Monaxonia Spheroid Bicone Ellipsoid Cylinder Egg Cone Hemispheroid truncated Cone further in the Bipyramids Regular Pyramids and the Amphitect Pyramids i e in all Stauraxonia except the Heterostaura allopola It also is the case in those Spiraxonia spiral forms of which the transverse section is a circle an ellipse a regular polygon or an amphitect polygon 4 In the Centrepipeda the stereometric forms with a mid plane the natural center of the form is a p l a n e median plane or sagittal plane This is the case in the Heterostaura allopola or Zeugita of which the general basic form is half an amphitect pyramid 5 The Centrostigma are the lowest and least perfect forms while the Centrepipeda are the highest and most perfect The Centraxonia occupy a middle position in this respect 6 All the different basic forms that appear as subordinated form species of these main groups allow to be ordered according to their stepwise increasing differentiation of their axes and poles This sequence of increasing differentiation reflects at the same time the increasing perfection of the form 7 So there exists a degree of promorphological perfection of every organism that is implied just by the degree of differentiation of its promorph and which is first of all independent of its degree of tectological perfection i e it is independent whether the biont is a first second third fourth fifth or sixth order form individual For inorganic beings crystals a degree of promorphological perfection is not relevant because crystals are not f u n c t i o n a l entities V Theses of the Lipostauric basic forms 1 With respect to the general presence condition and positioning of the axes all basic forms divide in two large groups Forms with cross axes S t a u r o t a and forms that lack cross axes L i p o s t a u r a The S t a u r o t a include all Stauraxonia i e all pyramidal forms bipyramids Stauraxonia homopola regular pyramids Heteropola homostaura amphitect pyramids Heterostaura autopola and half amphitect pyramids Heterostaura allopola They i e the Staurota also include those Spiraxonia spiral forms that have a cross section that allows for axes to be distinguished The L i p o s t a u r a on the other hand i e forms lacking cross axes include all Anaxonia i e forms having no axes at all further the Homaxonia spheres then all Polyaxonia endospheric polyhedra and finally all Monaxonia i e all forms having only one axis 2 The Lipostaura or basic forms without cross axes are generally much lower on the differentiation ladder than the Staurota or basic forms with cross axes The former are mainly present in lower organic form individuals while the latter are predominantly found in higher organic form individuals 3 The Lipostauric basic forms either possess no axes at all Anaxonia or all equal axes Homaxonia or a definite and finite number of constant axes that are however all equal Polyaxonia or finally possessing only one constant axis Monaxonia With respect to all these forms neither the expression radiate nor the expression bilateral or symmetric is applicable 4 All Lipostauric forms are characterized by a lack of a definite number of meridian planes that intersect in a single main axis and by which the body is divided into a definite number of equal or similar parts 5 All Lipostaura consequently lack definite a n t i m e r s or paramers if one understands by antimers and paramers only those body parts that lie around the main axis antimers or paramers or around some other axis paramers In the Polyaxonia we can in a way speak of antimers They are the pyramidal parts that come together with their tips in the center of the body i e they meet in a point instead of in a line axis 6 In the Polyaxonia we could interpret the above mentioned pyramidal parts that meet in the center as metamers or epimers sequential parts instead of antimers or paramers counter parts along every axis of such a polyaxonic form then lie two metamers or epimers But because this is only the case where the axes of every two oppositely positioned pyramidal parts are each other s extension we do not favor this interpretation and keep interpreting those pyramidal pars of the Polyaxonia as antimers or one could call them by the neutral term perimers VI Theses of the Staurotic basic forms 1 All Stauraxonia or basic forms with cross axes are higher and more perfect basic forms than all Lipostaurotic basic forms basic forms without cross axes because in virtue of the presence of certain cross axes that intersect in the main axis a greater diversity and possibility of differentiation is given than in any one lipostaurotic basic form 2 The general stereometric basic form of all Stauraxonia is the pyramid namely either the bipyramid Stauraxonia homopola or the single pyramid Stauraxonia heteropola 3 The expression regular or radiate forms if one wants to retain these expressions should be limited to the form categories of the Homopola isostaura regular bipyramids and Heteropola homostaura regular pyramids 4 The expression symmetric or bilateral forms should if one wants to retain those espressions be limited to the Zeugita or Centrepipeda Heterostaura allopola 5 All Stauraxonia are characterized and as such essentially distinguished from the Lipostaura by the possession of a definite number of meridian planes which intersect in a single main axis and by which the body is divided into a definite number of identical or similar parts 6 The corresponding constituent parts of the stauraxonic body which by their number positioning and differentiation equality or unequality further determine the basic form of the staurotic individual either are p a r a m e r s in the first to third order form individuals For paramers see HERE and also HERE in Part II of Tectology or a n t i m e r s in metamers and in linear persons or m e t a m e r s in planar persons The metamers of planar persons are not sequentially ordered but in a branched way or p e r s o n s in colonies Promorphologically the most significant are generally the antimers then the paramers Their basic form is always pyramidal 7 All Stauraxonia divide into two main groups according to whether the body center is one of the meridian planes Zeugita i e Heterostaura allopola or the main axis Stauraxonia centraxonia i e all Stauraxonia except the Heterostaura allopola 8 The centraxonic stauraxonians in which the body center is a line are either I the regular bipyramids Homopola isostaura or II the regular pyramids Heteropola homostaura or III the amphitect bipyramids Homopola allostaura or IV amphitect pyramids Heteropola heterostaura autopola In all these forms the two poles of every cross axis or at least of both directional axes which are perpendicular to each other are equal which means the right body side never is different from the left body side and also the back side dorsal half is never different from the belly side ventral half Dorsal half and ventral half are congruent as well as right half and left half 9 The centrepipedal stauraxonians or zeugites on the other hand in which the body center is a plane the median plane either are I half amphitect pyramids homopleural zeugites or II irregular pyramids heteropleural zeugites The zeugites are identical to the Heterostaura allopola Here at least one cross axis is unequipolar So in all cases the dorsal side is different from the ventral side and in the heteropleural zeugites also the right side from the left They never are congruent Where the left side equals the right side as in the homopleural zeugites they are symmetric not congruent i e there is no mechanical procedure say a rotation or translation that maps them onto each other VII Theses of the Zeugitic basic forms 1 The form group of the Zeugites or Centrepipeda Heterostaura allopola forms as half an amphitect pyramid the highest and strongest differentiated basic form of organisms While HAECKEL 1866 reckoned the spiral forms to the Dysdipleura which belong to the Zeugites we on the other hand have erected a special promorphological category for them i e for the spiral forms namely the Spiraxonia forms with a spirally curved main axis Although it is not clear where precisely the Spiraxonia should be placed on the ladder of promorphological perfection they occupy a high position on that ladder 2 The Zeugites or Centrepipeda are distinguished from all remaining organic forms by the possession of three unequal ideal axes directional axes euthyni of which either two are heteropolar and the third one homopolar or all three heteropolar Also the Spiraxonia can sometimes show these differentiations of the axes and their poles The spirally curved main axis is always heteropolar and when the cross section of the spirally curved tube is not a circle ellipse or rhombus it will possess only one homopolar cross axis or none at all just like we see it in the Zeugites 3 The three directional axes of the Zeugites are perpendicular to each other and correspond to the three dimensions i e independent directions of space They can accordingly be indicated as 1 longitudinal axis Axis longitudinalis 2 axis of thickness Axis sagittalis and 3 axis of width Axis lateralis 4 The two poles of the longitudinal axis or main axis are generally in Promorphology indicated as mouth pole Polus oralis or peristomial pole and as counter mouth pole Polus aboralis or antistomial pole regardless whether they lie up or down in front or at the rear 5 The two poles of the axis of thickness sagittal axis or dorso ventral axis are generally in Promorphology indicated as dorsal pole Polus dorsalis and as ventral pole Polus ventralis regardless whether they lie up or down in front or at the rear 6 The two poles of the axis of width or lateral axis generally in Promorphology are indicated as right pole Polus dexter and left pole Polus sinister regardless whether they are equal to each other or not 7 By the three ideal axes which are perpendicular to each other and which correspond to the three space dimensions three planes that are normal to each other the directional planes Plana euthyphora are determined which are of the greatest promorphological significance 8 The first directional plane is the median plane or main plane Planum medianum sagittal plane bisection plane which divides the whole body of the Zeugites or Centrepipeda in two symmetric parts left and right part pars sinistra pars dextra It is determined by the longitudinal axis and the dorso ventral axis 9 The second directional plane is the lateral plane Planum laterale which divides the whole zeugite body into two unequal parts the dorsal and ventral half pars dorsalis pars ventralis It is determined by the longitudinal axis and the lateral axis 10 The third directional plane is the equatorial plane Planum equatoriale which divide the whole zeugite body into two unequal parts oral half and aboral half pars oralis pars aboralis It is determined by the lateral axis and the dorso ventral axis In most of our drawings figuring in the Promorphological System we have drawn the two latter axes in the base of the pyramid instead of in its equatorial plane but this difference is immaterial 11 The p h y s i o l o g i c a l terms taken from the locomotion of the mobile Zeugites or from the attachment of the sessile Zeugites and their relative direction with respect to the earth s axis and the horizon namely frontal and rear side upper and lower side horizontal and vertical axis should be banished and replaced by the above determined purely m o r p h o l o g i c a l terms The complete elimination of the topographic physiological terms front and rear up and down horizontal and vertical out of the whole of Morphology is

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  • implicate order
    then it is even more essential for the understanding of life In the explicate and sequential orders life appears to arise as a fortuitous chance combination of molecules which leads in a more or less mechanical determined way to further developments which produce ever higher and more complex forms While this approach can be admitted as significant for study it is now seen as an abstraction and an approximation in the light of the generative order Its deeper meaning is to be understood by exploring how it reveals the inward generative order of the whole stream that is constantly present Here we see that the implicate order in general is dynamic but all the dynamics is as a whole already present so that dynamics is not according to a time succession In this connection some scientists notably Erwin Schrödinger have suggested that the quantum theory with its new feature of wholeness could explain the basic qualities of life However current work in biology hardly takes the quantum theory into account even though it is necessary for understanding the very existence of molecules The current approach is however justified by pointing out that the relatively high temperatures at which life becomes possible make long range quantum connections as we see it for example in superconductivity taking place at very low temperatures not particular important although it may turn out that in certain macromolecular process such as protein folding long range quantum correlations may indeed be relevant In conclusion therefore it does not appear likely that the essence of life is to be understood in terms of the details of conventional quantum theory Rather it is necessary for the understanding of life to go beyond the quantum theory and the superimplicate order into an infinity of generative and implicate orders from which present biological theory has been as it were abstracted In doing so however it is not intended to seek the ultimate origin of life in a reductionist way by going for example to an even more fundamental microscopic theory than the quantum theory Rather it is being proposed that a deeper generative order is common to all life and to inanimate matter as well It is not therefore an attempt to explain life in terms of matter but rather to see how both emerge out of a common overall generative order Within this order there is room for new kinds of pools of information from which life could be generated The wholeness of the living being and even more of the conscious being can then be understood in a natural way rather as the wholeness of the molecule and the superconductive system is understood although it must not be forgotten that life is much subtler and more complex than molecules and superconducting systems Life is no longer seen as the result of somewhat fortuitous factors which perhaps happened only on an isolated planet such as Earth Rather it is seen to be enfolded universally deep within the generative order BOHM and PEAT 1987 pp 200 It must be emphasized that the domain of implicate and generative orders is not a transcendent domain like for instance Plato s domain of Ideas i e it is not separately existing separate from the things that we can observe with our senses and instruments but is immanent in those things In other words the implicate orders are immanent in the explicate order Those orders explicate and implicate form one indivisible dynamic whole It is perhaps instructive to bring in some elements of the closely related theory of morphogenetic fields of SHELDRAKE into the discussion of implicate orders which is not carried as far by BOHM and BOHM PEAT as I would wished they had Although matter in general has evolved from the Big Bang onwards crystals and the like do not evolve further But organisms do They have experienced an evolution having lead to ever more complex forms all the way to conscious beings How do we relate this to the above mentioned generative orders or formative fields Well existing organisms are connected we could say tuned in to their respective formative fields implicate generative orders that sustain them But a change in the environment of a population of an organismic species can trigger a tuning in to new ranges of the generative order resulting in a new sequence of genetic changes which are thus not the result of random mutations but of directed and ordered mutations although under certain circumstances random mutations which are often disadvantageous or lethal can and will occur These genetic changes will give rise to a new sequence of successive organisms in the explicate order So analogously as is the case in quantum mechanical experiments the observed randomness is caused by an underlying enfolded order an order one is only dimly aware of All the cases of apparent cooperation between a host of changes that together must result in for example the transition from Reptiles to Birds must be attributed to the enfolded generative orders or formative fields The tuning in to such a field by an explicate entity which is always partly enfolded into the generative order is in fact further enfoldment injection into the generative order while the resulting action of the formative field is an unfoldment from the implicate to the explicate order At the same time we must assume that the formative field is itself changed by it being tuned in This language of tuning in is very relevant It stems from the radio technique that a part of the radio is able to resonate with a particular frequency of an incoming radio wave And because it is reasonable to assume that everything whatsoever is in fact one or another vibration which can form interference patterns we can imagine that such a tuning in can take place For the theory that everything is in fact a certain vibrational phenomenon see ZWART Het Wezen van het Zijn The Essence of Being in Dutch The mentioned phenomena of cooperation of parts in creating a whole is widely encountered in biological phenomena But even in inanimate things we sometimes find such a cooperation There are many solids crystals whose atoms can adopt more than one kind of orderliness In such solids one orderliness is usually more stable above some definite temperature another below it As the solid in the form of a crystal is heated through that temperature the crystal structure may change abruptly from one form to the other HOLDEN MORRISON p 219 this source is more fully cited below So some crystal species can undergo a spontaneous transformation while they remain in a solid state A beautiful example of this can be observed in mercuric iodide It is especially spectacular because the change of crystal structure brings with it a change in color from red to yellow The red tetragonal form stable at room temperature changes to a yellow form belonging to the orthorhombic crystal system as the temperature rises through 126 0 centigrade On cooling the yellow form transforms more slowly back to the red HOLDEN MORRISON 1982 Crystals and Crystal Growing p 221 remark the follwing about this crystal transformation When we visualize what the atoms must do to accomplish a structural transformation of a solid it seems even clearer than in visualizing melting that co operative action of the atoms might be necessary In the structural transformation the atoms must not only move they must move to the new right places Since the locations of its neighbors determine what is the right place for an atom the atoms must feel one another out quite extensively in the course of the transformation Sometimes the atoms will not shift over to the other arrangement even when they would find that arrangement more stable The element carbon comes in at least two crystalline varieties Graphite its crystals belonging to the Hexagonal Crystal System and Diamond its crystals belonging to the Cubic Crystal System Graphite is more stable than Diamond at all temperatures at ordinary pressure But diamonds do not change spontaneously into graphite crystals In a way not yet 1982 understood some of the carbon in nature crystallized as diamonds and now those carbon atoms are frozen in that arrangement So a state of minimum energy garanteeing stability is not in all cases realized which means that states corresponding to higher energy are sometimes able to persist in time This observation may be important when we ask ourselves how it is possible that in the large majority of crystal species their minimum energy states which is translated in a minimum energy configuration of their atoms is found very quickly by them and can be exactly repeated in all individual cases i e for example calcium carbonate under certain thermodynamic conditions always crystallizes in the form of the mineral Calcite in the Hexagonal Crystal System while under other conditions it always crystallizes in the form of the mineral Aragonite in the Orthorhombic Crystal System How exactly are these atomic minimum energy configurations realized by the atoms involved So also in such inanimate cases one could suspect the action of some formative field Now if we go down the complexity ladder within the explicate order we get the sequence CONSCIOUS BEINGS LIVING BEINGS MOLECULES and CRYSTALS ATOMS and finally ELEMENTARY PARTICLES These entities in order for them to be generated and sustained reach back to their corresponding levels of and in the implicate order s as the next diagram illustrates Figure 2 Each type of explicate structure depends for its generation and sustenance on its own layer of implicate generative order These layers are not transcendent but immanent with respect to the explicate structure This means that the explicate structure is tuned in to the a c t i v e i n f o r m a t i o n of the relevant layer in the implicate order The above diagram shows that if we really causally go down from say the category of conscious beings to their genuine causes causes that make this type of structure namely conscious beings possible in the first place we directly end up in the implicate order So all the structural types that are to be found in the explicate order atoms molecules living beings etc have a common origin and this origin is the implicate order This means further that the ultimate ground of all explicate things is more complex and subtle than those things themselves The inward laws of the holoflux which carries the implicate generative orders are such that a d e r i v a t i o n as done by science of conscious beings from lower living beings and the derivation of living beings from molecules and the derivation of molecules from atoms and the latter from elementary particles turns out to be a good approximation to what has really happened although here and there this approximation is not so good The ink in glycerine model of the relationship between the implicate and explicate orders We will now outline the promised mechanical analogy given by BOHM 1980 that visualizes the relationship between the explicate and implicate orders Suppose we have two glass cylinders a smaller one placed concentrically inside the other We fill the space between the two cylinders with a viscous fluid like glycerine Then we take insoluble ink for example ink composed of carbon particles We then place a droplet of that ink onto the surface of the fluid So what we then see is a dark more or less circular droplet floating on a colorless medium If we now rotate the outer cylinder slowly then any fluid element i e any small region of the fluid will be drawn out because the parts of that fluid element that are closer to the outer cylinder are moving faster than those that lie more inward And with this fluid element the ink droplet will also be drawn out After prolonged rotation the ink droplet will be stretched into a long thread and as the process proceeds the droplet will finally disappear and the thread will be so thin that it becomes invisible to the naked eye For the latter the ink has been totally mixed up with the glycerine Figure 3 The ink in glycerine model for explicate and implicate order Two concentric cylinders top view of unequal size The space between the cylinders is filled with glycerine and a droplet of insoluble ink is placed onto the fluid When the outer cylinder is rotated arrow the ink droplet will be drawn out into a long thread So now the droplet which stands for a particle in the Explicate Order is enfolded into the Implicate Order Although in fact in this mechanical analogue the thread is still present as such in the fluid i e still as an isolated object we consider it to have become infinitely thin and thus be present in any fluid volume however small that may be But we do not suppose it to have been irreversibly diffused through the fluid The information with respect to its origin that was its condition as a droplet has not been lost Indeed when we now reverse the rotation of the outer cylinder the carbon particles will approach each other more and more thereby the thread becoming thicker and thicker until finally the ink droplet appears again in about the same state as it was in before the initial rotation started It is now explicate again With this mechanism we can simulate several phenomena thereby sometimes using more than one ink droplet or colored droplets etc First we will describe the more or less prolonged explicate local existence of an object say a particle represented by an ink droplet and its non local existence in the Implicate Order In the next four Figures the implicate entities are indicated by threads These threads must not be taken literally They are supposed to symbolize the enfolded condition of entities in the Implicate Order i e the entities are homogeneously diffused all over the Implicate Order The different colors of the thread and the explicate object does not symbolize a qualitative difference but only a sequential difference reflected in the order of appearance in the Explicate Order In the present case the degree of enfoldment of a droplet is weak when it is in the vicinity of location A and stronger for enfolded droplets further away from that location The four Figures represent successive situations of enfoldment injection and unfoldment projection resulting in this particular case in a relative constant local presence of the same object in the Explicate Order and at the same time a non local presence of the enfolded state of the same object in the Implicate Order The latter is carried by the Holoflux like the modulation of a radio wave is being carried by the carrier wave Figure 4 The object indicated by a black ball is explicate d in virtue of the unfolding of a black thread at location A Just a moment before it had been made explicate by the unfolding of a blue thread which is now being enfolded back again into the Implicate Order In the next moment the object will become represented by the unfolding of the red thread leaving the black thread implicate again See next Figure We imagine here that in this particular process of unfolding and enfolding no new objects are being created The replacements represent the same object and this same object is non locally and in an implicate way present in the Implicate Order and locally present in the Explicate Order at location A The next images relate to what happens further during prolonged rotation of the outer cylinder Figure 5 Now the red thread is contracted resulting in its becoming explicate as a droplet at location A while the previous droplet black has become implicate again and in the model present as a black thread in the glycerine But already the next thread purple is in the process of contracting Figure 6 The purple thread has now become fully contracted and appears as droplet in the Explicate Order at location A while the red droplet is again drawn out into a thread and is implicate again But the green thread is already in the process of contracting and will soon become explicate Figure 7 The green thread is now fully contracted and has become explicate as a droplet at location A while at the same time the purple droplet is drawn out again into a thread and so is implicate again But the blue thread is already contracting As has been said the above four Figures symbolize the more or less sustained existence of the same object in the Explicate Order But and now considering another possible state of affairs the four Figures can also be interpreted as replacements of that object by another object by the successive unfoldment and enfoldment of qualitatively different threads this qualitative difference now being indicated by the different colors In this interpretation we see in the Explicate Order an object that is gradually changing on the spot i e at location A In both cases however the object in the Explicate Order is just an abstraction from the Implicate Order and ultimately from the Holoflux As such it is just a relatively and temporarily stable and isolated object which in fact is not isolated but connected with the whole which is everywhere in the Implicate Order It is like a semi stable vortex in a steam enjoying a transient explicate existence while being integrated with and dependent on the implicate whole which sooner or later will absorb it again within itself The condition of unfoldment depends on the pattern of degrees of implication of the threads inside the Implicate Order As has been said the threads must not be taken literally as threads but as some sort of diffused entities wholly taken up by the Implicate Order but in contrast with the case of real diffusion they can be recovered again and become explicate The pattern of degrees of implication is an expression and consequence of the laws that reign in the Implicate Order These laws are approximated by the laws of Quantum Mechanics Maybe not all known natural laws governing observable processes can be reduced to the laws of present day quantum mechanics because even without considering the possible existence of the implicate and generative orders the quantum theory cannot be the final theory about all of Reality in virtue of the fact that it as it stands today cannot be harmonized with the theory of relativity Anyway we can imagine the real laws to be present in the implicate and generative orders in the form of mathematical entities to which structures in the Explicate Order can tune in as it were in order to be guided in their processes With respect to the individual development of crystals for example we could say that the locally existing chemical motifs atoms or sets of bonded atoms are guided by group theory present in the implicate generative order as a set of principles of symmetry symmetry groups in order to find their minimum energy configuration demanded by quantum laws by actually taking up one or another particular symmetry corresponding to that minimum energy configuration With the mentioned real laws we could mean all the laws that we earlier have called Dynamical Laws namely in the documents of the First Part of our website Such a dynamical law represents the so called Essence of that intrinsic thing that is generated by the dynamical system that is governed by that law In this vision of deeper levels of enfolded generative orders that organize by their active information the entities in the Explicate Order it is demanded that those entities particles atoms molecules etc must be of sufficient complexity in order to be able to respond to this active information The order of matter has in this vision but also already in quantum mechanics become far removed from that of the billiard balls and falling apples of the Newtonian world Rather than nature being reduced to the material the whole notion of the material has been extended into regions of indefinite intangibility While in material processes like say crystallization the actual driving force is still the quantum mechanical desire to minimize energy by moving toward the configuration of atoms that corresponds to the lowest energy valley the actual process is given form by information in the enfolded generative orders or formative fields themselves So in in principle quantum mechanics can say something about how a lowest energy configuration of atoms must look like but to actually find this configuration i e to actually take up this configuration by the relevant atoms themselves which must move to certain places is accomplished by the active information from the enfolded order Thereby the atoms chemical motifs move in virtue of so to say their own energy but this energy is directed by the much weaker energy of the formative field The still deeper and most subtle enfolded orders or formative fields are it seems close to some form of objective intelligence that wells up from an underlying creative source So we see that in this vision the whole philosophy as presented in the First Part of this website is in a way turned upside down Matter in a way stems from some mind like substance instead of mind ultimately stemming from matter The discrepancy between the two visions in one and the same website should not be taken negatively but positively It means that it makes possible a fertile dialogue between say self critical reductionists and self critical holists After having had this intermezzo we continue with the ink in glycerine model of the explicate and implicate orders And we will once and for all set that in all ensuing images of explicate and implicate ink droplets unless stated otherwise each droplet must without drawing all that again be imagined as consisting of a whole ensemble of implicate threads all belonging to the same explicate object This implies that whatever droplet we draw as explicate it is at the same time also implicate in virtue of its fellow threads which means that every object visible in the Explicate Order is at the same time present everywhere in the Implicate Order And because of this it is not an isolated particle but represents the Whole The transformation of becoming implicate of something explicate and vice versa must be considered to be a radical one Therefore we will call it a metamorphosis rather than just a transformation Such just transformations are for example linear displacements translations rotations reflections dilations etc while a metamorphosis can be best compared with what we see in certain insects for instance the change from caterpillar to butterfly Metamorphosis often results in different orders To see what this means let us first illustrate a case where this is not so In these illustrations we investigate the process of enfolding and the orders that can result therefrom and for clarity in these cases only we assume that when something is explicate it is not implicate and vice versa The simplest notion of order is that of a sequence or succession The essence of such an order is in the series of relationships among distict elements A B B C C D For example if A represents one segment of a line and B the succeeding one etc the sequentiality of segments of the line follows from the above set of relationships Returning to our ink in glycerine analogy suppose that we have inserted into the fluid a large number of droplets set close to each other and arranged in a line These we label as A B C D E See next Figure Figure 8 A row of explicate droplets After they all are thus placed at the surface of the glycerine rotation of the outer cylinder will begin We then turn the outer cylinder many times so that each of the droplets gives rise to an ensemble of ink particles enfolded as infinitely thin threads in so large a region of space of glycerine that particles from all the droplets intermingle We label the successive ensembles A B C D E See next Figure Figure 9 In virtue of the rotation of the outer cylinder all droplets have simultaneously become implicate Although the ensembles of ink particles infinitely thin threads purple are intermingled with each other they can be recovered back into the Explicate Order by reversing the rotation of the outer cylinder The earlier explicate state of the droplets is indicated by soft colors It is clear that in some sense an entire linear order has been enfolded into the fluid This order may be expressed through the relationships A B B C C D This order is not present to the senses Yet its reality may be demonstrated by reversing the motion of the fluid so that the ensembles A B C D will unfold to give rise to the original linearly arranged series of droplets A B C D In the above we have taken a pre existent explicate order consisting of ensembles of ink particles arranged along a line and transformed i e metamorphosed it into an order of enfolded ensembles which is in some key way similar We shall next consider a more subtle kind of order not derivable from such a transformation metamorphosis Suppose now that we insert a droplet A and rotate the outer cylinder n times We then insert a second ink droplet B at the same place and again rotate the cylinder n times We keep up this procedure with further droplets C D E F The next Figure schematically shows what happens When later droplets are inserted and the cylinder is rotated the degree of implication increases for earlier droplets One should read the Figure from top to bottom Figure 10 Intrinsically implicate order From top to bottom We insert a droplet A onto the fluid at location X We rotate the outer cylinder n times The droplet A becomes implicate It becomes the ensemble thread a Then we insert a droplet B onto the fluid again at location X We rotate the outer cylinder n times The droplet B becomes implicate It becomes the ensemble thread b The degree of implication of A increases Then we insert a droplet C onto the fluid again at location X We rotate the outer cylinder n times The droplet C becomes implicate It becomes the ensemble thread c The degree of implication of A increases still further and that of B increases Then we insert a droplet D onto the fluid again at location X We rotate the outer cylinder n times The droplet D becomes implicate It becomes the ensemble thread d The degree of implication of A and B increases still further and that of C increases Then we insert a droplet E onto the fluid again at location X We rotate the outer cylinder n times The droplet E becomes implicate It becomes the ensemble thread e The degree of implication of A B and C increases still further and that of D increases Then we insert a droplet F onto the fluid again at location X We rotate the outer cylinder n times The droplet F becomes implicate It becomes the ensemble thread f The degree of implication of A B C and D increases still further and that of E increases We can if we want continue this procedure The resulting ensemble of ink particles a b c d e f will now differ in a new way for when the motion of the fluid is reversed the ensembles will successively come together to form droplets again in an order opposite to the one in which they were put in For example at a certain stage the particles i e the carbon particles of the ink of ensemble d will come together i e the thread forming that ensemble contracts and becomes an explicate droplet again after which it will be drawn out into a thread again This will happen for the particles of c then to b etc It is clear from this that ensemble d is related to c as c is to b and so on So these ensembles form a certain sequential order However this is in no sense a transformation of a linear order in space as was that of the sequence A B C D that we considered earlier for in general only one of these ensembles will unfold at a time When any one is unfolded the rest are still enfolded In short we have an order which cannot all be made explicate at once and which is nevertheless real as may be revealed when successive droplets become visible as the cylinder is turned Bohm Wholeness and the Implicate Order calls this an intrinsically implicate order to distinguish it from an order that may be enfolded but which can unfold all at once into a single explicate order The just temporal i e time successive order in the Explicate Order in the present case corresponds to a simultaneous order in the Implicate Order So we have here an example of how an explicate order is a particular case of a more general set of implicate orders Let us now go on to combine both of the above described types of order We first insert a droplet A in a certain position and turn the cylinder n times We then insert a droplet B in a slightly different position and turn the cylinder n more times so that A has been enfolded by 2n turns We then insert C further along the line AB and turn n more times so that A has been enfolded by 3n turns B by 2n turns and C by n turns We proceed in this way to enfold a large number of droplets We then move the cylinder fairly rapidly in the reverse direction If the rate of emergence of droplets is faster than the minimum time of resolution of the human eye what we will see is apparently a particle moving continuously and crossing the space See next Figure which only depicts the enfoldment of the droplets Figure 11 Movement in the Explicate Order This Figure reading it from top to bottom illustrates the successive enfoldment of droplets at sequential locations that lie very close to each other on a straight line When such an enfoldment of a large number of droplets has thus been accomplished the cylinder is rapidly turned in the opposite direction The droplets will then become explicate again one after the other while at the same time they will also one after the other be enfolded back again into the glycerine What we then actually see i e see what happens in the Explicate Order is a continuous movement of a droplet along a straight line The sequential locations X 0 X 1 X 2 X 3 X 4 X 5 etc are supposed to be very close to each other while lying on a straight line From top to bottom We insert a droplet A onto the fluid at location X 0 We rotate the outer cylinder n times The droplet A becomes implicate It becomes the ensemble thread a Then we insert a droplet B onto the fluid at location X 1 We rotate the outer cylinder n times The droplet B becomes implicate It becomes the ensemble thread b The degree of implication of A increases Then we insert a droplet C onto the fluid at location X 2 We rotate the outer cylinder n times The droplet C becomes implicate It becomes the ensemble thread c The degree of implication of A increases still further and that of B increases Then we insert a droplet D onto the fluid at location X 3 We rotate the outer cylinder n times The droplet D becomes implicate It becomes the ensemble thread d The degree of implication of A and B increases still further and that of C increases Then we insert a droplet E onto the fluid at location X 4 We rotate the outer cylinder n times The droplet E becomes implicate It becomes the ensemble thread e The degree of implication of A B and C increases still further and that of D increases Then we insert a droplet F onto the fluid at location X 5 We rotate the outer cylinder n times The droplet F becomes implicate It becomes the ensemble thread f The degree of implication of A B C and D increases still further and that of E increases We continue this for a large number of droplets As has been said when the cylinder is now rapidly rotated in the reverse direction we see a droplet moving continuously along a straight line Such enfoldment and unfoldment in the Implicate Order may evidently provide a new model of for example an electron which is quite different from that provided by the current mechanistic notion of a particle that exists at each moment only in a small region of space and that changes its position continuously with time We must not be confused by the fact that the described model is itself wholly mechanical The model is supposed to be an analogy The status of the ink ensembles must be interpreted as totally implicate in the fluid just like the image of an illuminated object is genuinely implicate enfolded in the interference pattern of a hologram discussed earlier What is essential to this new model is that the electron is instead to be understood through a total set of enfolded ensembles which are generally not localized in space At any given moment one of these may be unfolded and therefore localized but in the next moment this one enfolds to be replaced by the one that follows The notion of continuity of existence is approximated by that of very rapid recurrence of similar forms changing in a simple and regular way rather as a rapidly spinning bicycle wheel gives the impression of a solid disc rather than of a sequence of rotating spokes Of course more fundamentally the particle is only an abstraction that is manifest to our senses What is is always a totality of ensembles all present together in an orderly series of stages of enfoldment and unfoldment which intermingle and inter penetrate each other in principle throughout the whole of space It is further evident that we could have enfolded any number of such electrons in different ways so that we could then mimic a set of interacting particles Since classical physics traditionally aims to explain everything in terms of interacting systems of particles it is clear that in principle one could equally well treat the entire domain that is correctly covered by such classical concepts in terms of this new model of ordered sequences of enfolding and unfolding ensembles What this theory of the Implicate Order proposes here is that in the quantum domain this model is a great deal better than is the classical notion of an interacting set of particles Thus although successive localized manifestations of an electron for example may be very close to each other so that they approximate a continuous track this need not always be so In principle discontinuities may be allowed in the manifest tracks and these may of course provide the basis of an explanation of how an electron can go from one state to another without passing through states in between This is possible of course because the particle is only an abstraction of a much greater totality of structure This abstraction is what is manifest to our senses or instuments but evidently there is no reason

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  • implicate order 2
    in the given group In order to grasp what follows the reader should consult the documents on two dimensional crystals in the First Part of this website and the documents on Group Theory in the Second Part of the website See the following links First Part of Website Second Part of Website A point symmetry i e the point symmetry of a given object is a set of symmetry transformations which are such that at least one point is not moved by any such transformation The Group that is formed by a set of such symmetry transformations is called a point group If on the other hand we have a group with group elements such that among these elements symmetry transformations there is at least one that moves all points of the object to another place then we have a Plane Group for two dimensional patterns or a Space Group for three dimensional patterns One may wander how a rigid transformation that moves all points of an object can be such that the object still occupies the same patch of space after it has been subjected to such a transformation Well this is possible only for homogeneous infinite objects or when such an object is not fully homogeneous for an infinite periodic object Although c r y s t a l s are not infinite objects they can nevertheless be considered as infinite because from a microscopic perspective they contain an almost unlimited number of microscopically small translations distance of lattice points so that a shift according to one such translation leaves the crystal virtually in the same patch of space as it was before being shifted over such microscopic distance These translational symmetries simple translations in contrast to glide reflections and screw axes of a crystal define a lattice A crystal lattice is an imaginary set of equivalent points equivalent here means that every point has the same kind of surroundings The lattice looks exactly the same from one such point or from any other such point These points we call nodes Each node can now be associated with a certain motif generally consisting of motif units and representing in our case chemical units This associated motif must be the same everywhere in order to preserve the equivalence of the nodes When indeed a lattice is thus provided with motifs motifs having a symmetry that is compatible with that of the lattice we have a Plane Group two dimensions or a Space Group three dimensions because we can then indicate the motif units as relating to each other by rigid transformations of the whole pattern and all these transformations leave that whole pattern occupying the same patch of space as it did before so these transformations are symmetry transformations and together form a Group in which they figure as group elements Two dimensional periodic patterns seen as two dimensional crystals Chemical motifs and the formative role of Group Theory In order to speculate on the formative field insofar as the individual developement of c r y s t a l s is concerned let us first of all confine ourselves to imaginary two dimensional crystals because their symmetries are easier to visualize Let us begin with those two dimensional crystals that have a total symmetry content according to the Plane Group P4 We can visualize the internal structure of the crystal as an infinite set of motifs representing chemical units motifs that are repeated according to a square point lattice net This Square Net can accommodate for a motif having a point symmetry 4 and representing the Class 4 of the 2 D Tetragonal Crystal System See nextFigure Figure 2 A periodic pattern of motifs based on a 2 D square lattice net This particular pattern represents the Plane Group P4 It must be imagined to extend indefinitely over the 2 D plane As can be seen from Figure 2 the symmetry elements of the pattern representing Plane Group P4 are simple translations 2 fold rotation axes and 4 fold rotation axes The next Figure shows the total symmetry content of that Plane Group Figure 3 The total symmetry content of any pattern representing Plane Group P4 There are neither mirror lines nor glide lines present Small solid squares indicate 4 fold rotation axes Small solid ellipses indicate 2 fold rotation axes The black line segments connecting the nodes of the net indicate horizontal and vertical simple translations The diagram of Figure 3 is a pattern of symmetry elements the vertical and horizintal directions of the simple translations the 4 fold rotation axes and the two fold rotation axes As such this pattern of symmetry elements completely determines the Group because now all the symmetry transformations are implied We can place only motifs having determined symmetries into the net and these motifs must moreover have orientations compatible with the pattern of symmetry elements In Figure 2 the motifs are depicted as black four fold structures but in fact also the proper surroundings belong to such a motif Together motif in the strict sense proper surroundings we will call it a motif s l i e a motif in the broad sense The next Figure depicts one such motif s l Figure 4 A complete motif motif s l of the pattern of Figure 2 The auxiliary lines drawn within the motif do not belong to it These motifs s l tile as they should the whole plane when periodically repeated as indicated in the next Figure Figure 5 The complete motifs motif s l of the pattern of Figure 2 fill up the whole plane as they are periodically repeated Four such motifs are highlighted The auxiliary lines drawn within the motifs do not belong to them In the next Figure the tiling of the plane is extended Figure 6 The plane covered with complete motifs motifs s l which repeat periodically and as such representing the Plane Group P4 The red and blue motifs s l are considered to be totally equivalent Having said something of the possible motifs making up a periodic pattern according to the Plane Group P4 we re now going to treat all this group theoretically First we only consider the motifs s str i e just the black 4 fold figures each of them consisting of four motif units commas that have come together forming while partly overlapping a four fold figure which is called composed motif below that has as its only symmetry element a 4 fold rotation axis going through its center perpendicular to the plane of the drawing i e its point symmetry is 4 and in group theoretic notation C 4 the cyclic group of order 4 From an initial motif unit which represents the identity element of the Group not only of the Point Group of the motif but also of the Plane Group we generate the whole pattern and with it the whole Group by using two group elements symmetry transformations as generators of the Group Figure 7 When we place four fold motifs in the present case motifs each consisting of four partly overlapping commas at the nodes of a square point lattice square point net we obtain a pattern according to the Plane Group P4 A unit cell is indicated light blue The point R is the location of a four fold rotation axis There are more such axes The group element p is defined by an anticlockwise rotation of 90 0 about the point R and is chosen as a generator of the group A second generator is needed We choose it to be the horizontal translation t which is a horizontal shift of the whole pattern to the right over a distance that is equal to that between two horizontally adjacent nodes of the lattice The pattern must be imagined to extend indefinitely over 2 D space The inverse of this translation which is consequently an equal shift to the left is denoted by t 1 The group elements and with it the whole pattern can then be generated by applying the symmetry transformations represented by the generators over and over again and starting with the initial motif unit 1 Note that Whenever we apply two transformations a and b after each other and we write for this combined transformation ab then this means that we first have applied the transformation b and then on the result the transformation a And when we write say p 2 this means that we have applied the transformation p two times immediately after each other And in the same way p 3 means that transformation p was applied three times Figure 8 This Figure shows the four fold symmetry at the point R which means that the whole pattern returns as it was before i e it will occupy the same space as it did before after it has been anticlockwise rotated either by 90 0 180 0 or 270 0 about the point R indicated in the previous Figure The group element representing a 90 0 rotation about this point R maps the initial element 1 See previous Figure onto the location that is denoted by p See previous Figure and letting p represent the mentioned rotation it is chosen as a generator of the group in addition to this generator a second one is needed A 180 0 rotation about R of the initial element generates the group element p 2 represented by the motif unit denoted as p 2 See previous Figure while a 270 0 rotation about R generates the element p 3 represented by the motif unit denoted as p 3 See previous Figure Finally a 360 0 rotation about R is equivalent to 0 0 rotation which means that we can say p 4 1 As the second generator we have chosen the translation t See subscript of previous Figure We will now show how the P4 pattern and with it the Group P4 is generated by the generators p and t This means that given the identity element and its representation as an initial motif unit and the elements p and t represented by the position and orientation of corresponding motif units with respect to those of the initial motif unit we can in principle generate all other elements of the group and with it all motif units commas in our case of the pattern and thus generate the pattern itself by combining the elements p and t We shall show this in several stages We follow the transformations of each of the four individual motif units commas of a composed motif and connect the latter with the resulting composed motif also consisting of four motif units by a i e one green line So such a line represents the transformation of a whole composed motif into another whole composed motif Figure 9 Generation of five new motifs each one is a composed motif consisting of four motif units from the initial motif which also consists of four motif units the initial motif unit and the motif units p p 2 p 3 Explanation is given directly below Our initial motif consists as has been said of four motif units The initial motif unit denoted by 1 The motif unit p formed as an image of 1 under anticlockwise rotation of 90 0 about the point R The motif unit p 2 formed as an image of 1 under anticlockwise rotation of 180 0 about the point R The motif unit p 3 formed as an image of 1 under anticlockwise rotation of 270 0 about the point R We will now follow in detail the generation of the above mentioned five new composed motifs by considering what happened to each of the four components of our initial image See Figure 9 above Our intitial composed motif consists of the four motif units 1 p p 2 and p 3 If we produce the coresponding images of these under the translation t which is a shift to the right along a certain fixed distance then we get respectively t tp tp 2 tp 3 which form a new composed motif to the right of the initial composed motif See Figure 9 We re now going to produce the image of this new composed motif under an anticlockwise rotation of 90 0 about the point R i e we subject it to the transformation p This means that t becomes pt tp becomes ptp tp 2 becomes ptp 2 tp 3 becomes ptp 3 In this way we have obtained yet another new composed motif consisting of pt ptp ptp 2 ptp 3 which we can find as the second composed motif of the top row in Figure 9 When we now subject this last found composed motif to successively the translations t 1 t and t 2 we get the following Application of t 1 pt becomes t 1 pt ptp becomes t 1 ptp ptp 2 becomes t 1 ptp 2 ptp 3 becomes t 1 ptp 3 An this yields yet another new composed motif which we can see as the first one in the top row of Figure 9 Application of t pt becomes tpt ptp becomes tptp tp 2 ptp 2 becomes tptp 2 ptp 3 becomes tptp 3 This yields the third composed motif in the top row of Figure 9 Application of t 2 pt becomes t 2 pt ptp becomes t 2 ptp ptp 2 becomes t 2 ptp 2 ptp 3 becomes t 2 ptp 3 And this yiels the fourth composed motif in the top row of Figure 9 So we now have indeed generated five new composed motifs from the initial composed motif Next we shall generate still more composed motifs of our P4 pattern The third row of composed motifs can be generated along the same lines as we did it for the first row The difference is that we now use an anticlockwise rotation about the point R As it is denoted in Figure 7 by 270 0 instead of 90 0 of the earlier obtained composed motif the motif at the right of the initial composed motif See next Figure Figure 10 To the composed motif to the right of the initial motif the transformation p 3 is applied resulting in a new composed motif second of the third row From this new composed motif three other composed motifs are generated repectively by the transformations t 1 t t 2 Next we generate the rest of the second row of composed motifs by applying respectively the translations t 1 and t 2 to the initial composed motif Figure 11 Two more composed motifs of the second row are generated by the translations t 1 and t 2 To generate composed motifs of the fourth row we start from the last one of the second row and subject it to the transformation p 3 which is an anticlockwise rotation of 270 0 about the point R as indicated in Figure 7 This results in a new composed motif the second one in the fourth row From this new composed motif we can then generate the rest of the composed motifs of that row by applying translations Figure 12 The fourth row of composed motifs can be generated from the fourth composed motif of the second row using the transformations p 3 t 1 t and t 2 Of course one should continue this process indefinitely because the Group is infinite However it is by now clear how to generate the pattern and with it the Group governing its symmetry We will now relate these generated group elements which are represented by the respective motif units of the composed motifs i e the whole motifs s str to the motifs s l as we depicted them in Figure 6 Figure 13 Relationship of group elements with motifs s l in a periodic pattern representing Plane Group P4 The red squares and the blue squares both representing motifs s l are totally equivalent They are the motifs s l that are repeated according to the point lattice which is a square net While having in the above Figure emphasized the relationship between the motifs s l and the group elements in the next Figure we explicitly indicate the group elements Figure 14 The red and blue squares represent the group elements of the pattern according to the Plane Group P4 The square marked by 1 is the bottom right quadrant of the corresponding motif s l That quadrant is associated with the initial motif unit comma and represents the identity element of the Group By subjecting it to the generators p and t and then continually applying those generators to the results will produce the Group However because this Group is infinite the process of generating new group elements will never end In the subscript of Figure 14 we stated that the quadrant marked 1 is associated with the initial motif unit comma and represents the identity element of the Group The next Figure indicates this association What was meant that initially we have let that comma represent the identity element of the group By means of the generators all the other commas were then generated and as such representing the group elements But of course a more abstract association of the group element 1 with the motif s str can be considered and the same for all other group elements This concludes our discussion of the generation of the Group P4 i e the generation of its group elements from the initial element by means of two generator elements The guiding process in terms of the pattern and distribution of s y m m e t r y e l e m e n t s defining the given Group Remark The rest of this document will adress the guiding process that directs chemical units to their proper places This guiding process takes place just in virtue of the fact that all interactions take place via the Whole forcing us as observer in some cases to speak of cooperation between parts i e to speak in a holistic fashion Along the way we will speculate about the metaphysical form and status of these guiding agents i e we will slowly built up some proposals maybe we d better say trials as to putting those guiding agents into a broader philosophical context This is not an easy matter because it turned out that we have to characterize these agents in many diverse ways which is by the way typical for genuine philosophical concepts for example as quantum potential implicate order formative field active information generative order objective mathematical structure objective thought structure objective thinking dynamics the latter of NOUS as it subject the Second Hypostasis of the Neoplatonic metaphysics etc Because all this could become highly confusing we have repeated i e said the same but shifted the context a little some expositions about the guiding agents active in crystal formation in order to expose all this from several more or less different angles We further must stress that the guiding activity as such is not very evident i e does not reveal itself clearly to us in the explicate order of crystals i e it is not evident because this guiding activity is apparently such as to allow a classical chemical description of crystal formation i e with moving parts to be a good approximation except in cases of observed cooperation The hints to parallels with Neoplatonic metaphysics will in a later document be more fully worked out The meaning of what follows is to train ourselves to talk about such common a process like crystallization fully in terms of the theory of the Implicate Order In order to apprehend the following discussion about the crystallization process we begin with stating succinctly the general conclusion about how to visualize the crystallization process in terms of the Theory of the Implicate Order Because of supersaturation of a solution or because of a decrease in temperature in a melt the system of chemical units becomes unstable Quantum mechanics prescribes how a stable situation stable arrangement of chemical units must look like The potential field of the ordinary forces can now going to become actual in virtue of the instability The ordinary explicate relatively strong forces coming from and acting on the chemical units are now being modulated by the much weaker though more subtle forces of the activated group theoretic formative field resulting in the actual attainment of the stable end destination of the chemical units This stable destination is the periodic arrangement of the chemical units according to the appropriate lattice such that the symmetries associated with the activated Group as field aspect present in the Implicate Order are satisfied The situation of point 4 is such that it demands to be described in a number of alternative ways in order to fully express its subtlety and complexity In crystal formation the action of the formative field is often such that it allows for a conventional chemical description to be a good approximation except in observed cases of cooperation Group Theory at least insofar as it deals with the symmetry groups concerning the symmetries of three dimensional crystals and also the basic stereometric forms of organisms can be seen as an aspect of the metaphysical generative order of those crystals and organisms It is at the same time an aspect of Nous Objective Intelligence the Second Hypostasis of the Neoplatonism of the Greek philosopher Plotinus third century And Nous can be identified with an aspect of or the whole of the Implicate Order s The latter is carried by the Holomovement as its modulation and this undefinable and unmeasurable Holomovement can be identified with The One being the First Hypostasis of the Neoplatonism of Plotinus The One is the ultimate creative source of all there is We are now ready to discuss how the formation of the lattice supplied with motifs of an initial seed of a crystal belonging to this Plane Group P4 will be guided by an aspect of the Implicate Generative Order This aspect is supposed to be some small relevant patch of group theoretic structure enfolded within the Implicate Order To elaborate on this further we first reproduce Figure 3 As has been said the diagram of this Figure is a definition of the Group P4 in terms of symmetry elements Recall that symmetry elements are those geometric objects points lines and planes with respect to which the symmetry transformations such as rotations reflections etc are being performed Figure 15 The total symmetry content of any pattern representing Plane Group P4 There are neither mirror lines nor glide lines present Small solid squares indicate 4 fold rotation axes Small solid ellipses indicate 2 fold rotation axes The black line segments connecting the nodes of the net indicate horizontal and vertical simple translations The next Figure reproduces again Figure 2 of our pattern according to the Plane Group P4 Figure 16 A periodic pattern of motifs based on a 2 D square lattice net This particular pattern represents the Plane Group P4 It must be imagined to extend indefinitely over the 2 D plane As can be seen from Figure 16 and from the next Figure the symmetry elements of the pattern representing Plane Group P4 are simple translations 2 fold rotation axes and 4 fold rotation axes Figure 17 The pattern of Figure 16 with the distribution of its symmetry elements indicated according to Figure 15 The pattern of the above Figures consists of relatively simple motifs Each motif consists of a motif s str in the form of a 4 fold figure and a homogeneous background together making up the motif s l See Figure 6 Of course we are allowed to let these motifs be more complex as long as the results keep complying with the distribution of the symmetry elements as given in Figure 17 this distribution being the definition of the given Group In our case the Group P4 The next Figures discuss possible and impossible complexifications of the motifs in a P4 pattern Such a complexification boils down to the addition of new motif units in virtue of which the motif s l becomes more complex Figure 18 The particular complexification of the motif s l as shown here is not possible because the 4 fold rotation axes in the centers of the meshes are destroyed Here it is show for the addition of a 3 fold motif unit in the center of each mesh of the net This motif unit is incompatible with the 4 fold rotation axis which must be present and preserved in the center and corners of every mesh of the square net As can be easily seen the same impossibility applies to the insertion of 6 fold motif units At the centers of the meshes we are allowed to insert 4 fold motif units but it is equally allowable to insert circular motif units discs because a circular disc possesses all rotational symmetries whatsoever so it also possesses 4 fold symmetry Such a disc will be mapped onto itself by a rotation of 90 0 about its center Figure 19 Allowed complexification of the motifs s l of the pattern of Figure 16 by means of the insertion of additional motif units having the form of circular discs We have brought some structure into those discs by letting them be bounded by a black circumference but that does not disturb the symmetries of a pure homogeneous disc The next Figure indicates the boundaries of one of the complexified motifs s l Figure 20 Boundaries of one of the complexified motifs s l indicated Such a motif consists of one 4 fold structure black and four quarter discs The original motif s str suggests that some asymmetric motif units can be inserted at their proper places in addition to the previous inserted new motif units complying with the distribution of symmetry elements defining our Plane Group P4 Figure 21 The original motif s l is further complexified In addition to the previously added motif units new asymmetric motif units green are being added such that the symmetry distribution of the Plane Group P4 is preserved The thin red auxiliary lines are not supposed to belong to the present pattern We can further complexify the motif s l with 2 fold motifs in such a way that the P4 symmetry distribution is still preserved See next Figure and check with Figure 17 Figure 22 Further complexification of the motifs s l by adding 2 fold motif units red at the proper places in the lattice i e such that the P4 symmetry distribution will be preserved The thin auxiliary red and blue lines do not belong to the pattern The next Figure gives the final result of the complexification of the original motifs s l in our P4 pattern This result is the same as that of the previous Figure but now with all auxiliary lines removed except the lines that originally defined the nodes as their intersection points of the square lattice Figure 23 Final result of the complexification of the original motifs s l as in Figure 6 by addition of motif units in such a way that the P4 symmetry distribution as is required for all patterns of the Plane Group P4 is preserved The next Figure is about the boundaries of the complexified motifs s l illustrated for one such motif Figure 24 The boundary of one of the new motifs is indicated and this applies to all those motifs So the pattern is now built up out of square motifs with a complex inner structure These motifs are as before arranged according to the Square Point Lattice The Plane Group to which this pattern belongs is still P4 In the above Figure we outlined a motif s l and we see that its boundaries do not coincide with those of the chemical units i e the boundaries go right through some of these units That this however is not necessary is shown by the following Figure Figure 24a The boundary of the motif s l can be harmonized with those of the chemical motifs Like with respect to the motif s l of Figure 24 the pattern can be built by periodically stacking these motifs s l The motifs s l of the present Figure cannot however be considered as consisting of four group elements The next Figure indicates the group elements in the motif s l highlighted in the previous Figure Each such motif comprises four group elements as was also the case in Figure 13 with respect to the original motifs Figure 25 Each complexified motif s l represents as before four group elements In the highlighted motif these are 1 p p 2 p 3 See also Figure 13 We can now imagine all the depicted motif units as representing different types of 2 D atoms or ions or of small complexes of atoms present in a solution that is about to crystallize Figure 26 Solution containing chemical motifs The solution is supposed to be about to crystallize for the compound consisting of the chemical units symbolized by In addition to these units some alien units are also present The above solution is supposed to be supersaturated with respect to the chemical compound consisting of the four types of chemical motifs chemical units indicated in the subscript of the above Figure This means that the solution has become unstable In order to get to a lower energy state the compound starts to crystallize The quantum mechanical condition determines which atomic arrangement corresponds to the lowest energy state under the given circumstances And this arrangement is supposed to be that of the Plane Group P4 which means that if the chemical units associated with that chemical compound and indicated in the subscript of Figure 26 come together in a symmetrical pattern according to the Plane Group P4 then the whole system will be in a lowest energy state with respect to the given chemical and thermodynamic conditions So a lattice must be formed which should be such as depicted in Figure 23 As we have said earlier quantum theory can as it seems at least in principle predict the lowest energy configuration that must be adopted by the chemical units under the given condition And of course the system will tend to spontaneously adopt just that configuration For simple systems like the one represented by say the crystallisation of Salt NaCl or other simple ionic crystals this will be an easy matter The system will almost directly take up the appropriate arrangement configuration of chemical units And here the guiding process of agents from the Implicate Order are apparently such as to allow a classical chemical description of the process to be a good approximation But in c o m p l e x systems there could be more than one lowest energy state for the given circumstances and we know that compounds always crystallize in the same arrangement in a specified condition And even when there is only one unique minimum energy configuration with respect to the given conditions it could be hard for the complex system to actually realize this configuration The chemical units could get stuck and block each others way and so their movement is impeded resulting in an arrangement corresponding to a somewhat higher energy indeed there exist many so called metastable crystals like for instance diamonds In the present case a crystal would then emerge in which the defects are the rule resulting in a more or less irregular and highly variable arrangement of the chemical units where also internal bonds will be more or less distorted resulting in the mentioned higher energy But we know that in reality crystals generally grow easily and relatively smoothly and that defects although in most cases present are not dominant However when we interpret all natural processes in terms of the dynamics between the Explicate and Implicate Orders the mentioned difficulty for the chemical units to actually find and assume i e realize the proper lowest energy arrangement may disappear simply because of the supposed fact that the chemical units in the solution are as is the case with all explicate entities not separate more or less autonomous particles Via the Implicate Order they are connected with each other in a non local way so they can work in unison Thereby they are guided by a potential that they themselves activate and this potential is assumed to be a certain patch of group theoretic structure enfolded in the Implicate Order We can speculate that this particular potential i e the potential relevant to the chemical situation under discussion is equivalent to a formative or generative field reflecting the P4 distribution of symmetry elements as we gave it in Figure 15 The pattern of which must be imagined to be extended indefinitely We must stress that all explicate physical processes thus also all crystallization processes are directed by formative fields different fields for different processes But most of these processes are such that when we describe and explain them reductionistically and with it by means of the actions of purely local forces we generally obtain very good approximations In these cases we do not need to give primacy to the Implicate Order Because if we did it is expected that the results would be virtually the same But for all cases where we in our explanations are forced to bring in the phenomenon of cooperation i e to bring in a holism prevailing in the given chemical or physical process we must give primacy to the Implicate Order in our description and explanation According to the theory of the Implicate Order time and time orders are secondary as soon as we give primacy to the Implicate Order i e they derive from a timeless but nevertheless dynamic background dynamic in the sense of logical processes in which all is one It is the pattern of degrees of implication that is primary The mentioned group theoretic patch enfolded in the Implicate Order is physically timeless and indeed it reminds us of a Platonic Idea Some mathematicians for example PENROSE 1990 The Emperor s new mind consider mathematical structures to be Platonic Ideas It is however more appropriate to interpret the mentioned group theoretic patch and many other mathematical structures as a Plotinean Idea because a Plotinean Idea is not just a paradigm of how things ideally would have to be in order to be perfect as is a Platonic Idea On the contrary for Plotinus it is causally related via the so called emantion to the material things below Plotinus third century was a Greek Philosopher in the Neoplatonic tradition The theory of the Implicate Order has some essentials in common with Plotinus metaphysics We will say more about it further down and thematize it in a later document But to return to our crystallization process as soon as the solution becomes unstable because of supersaturation it is being organized into its proper form which is the P4 arrangement of the mentioned chemical units The latter s own active energy is modulated by the very small but more subtle directive energy of the generative field such a field is an aspect of the Implicate Order so that the chemical units end up at their proper place and in their proper orientation with respect to each other The small number of possible different chemical bonding patterns of the given set of chemical units restricts the way in which these chemical units can be oriented as they together form the crystal lattice Figure 27 Chemical motif units are being arranged by and in accordance with the P4 pattern of symmetry elements

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  • implicate order 2a
    by rotating the initial motif unit 180 0 about the axis C All those 2 fold rotation axes represent different group elements Although they all are half turns and so all of the same period namely 2 their axes of rotation are different i e those axes lie in different locations And because we must consider the pattern to extend indefinitely every half turn about such an axis will leave the whole structure unchanged i e the whole structure including its details occupies the same space as it did before the rotation had taken place Thus these axes really represent s y m m e t r i e s of the pattern When all the motif units are generated we will see a 2 fold motif consisting of two asymmetric motif units at each node of the lattice as in Figure 1 One such 2 fold motif we already see in the present Figure at the axis I The next Figure depicts the motifs s l i e the black 2 fold motifs each consisting of two asymmetric motif units PLUS their corresponding featureless backgrounds of the P2 pattern of Figure 1 Figure 9 Motifs s l the blue and red parallelograms arranged according to the Oblique 2 D point lattice and as such representing the P2 pattern of Figure 1 The red parallelograms and the blue parallelograms both representing motifs s l are wholly equivalent At every node of the point lattice there is such a motif s l The next Figure emphasizes the group elements of our P2 pattern Figure 10 Each motif s l represents two group elements These group elements are indicated by the red and blue triangles The different colors do not represent any qualitative difference One such half motif s l is chosen as the initial group element denoted by 1 from which the whole pattern can be considered to be constructed by applying to it three generator elements The next Figure shows one such group element Figure 10a One group element from the pattern of Figure 10 Each group element is represented by half a motif s l and thus by one motif s l unit One motif s l consists of two units The next Figures show how these group elements can be generated from the initial group element by applying to it and to the results three generators which are themselves group elements A rotation of 180 0 about the axis indicated in the Figures We call this generator a A translation along the SW NE side of a mesh and with the magnitude of such a side We call this generator s Its inverse will be denoted by s 1 A translation along the horizontal side of a mesh and with the magnitude of such a side We call this generator t Its inverse will be denoted by t 1 Recall that when we write the product say ab where a and b are group elements i e symmetry transformations we mean that we first applied b and then to the result a Figure 11 Three generators are indicated The rotation a is about the point indicated yellow Let us call this point R These generators are at the same time group elements From the initial group element 1 all the other group elements can be generated by applying to it and to the results the generators Figure 12 The group element t 1 is generated from the element 1 by applying the inverse of the generator t Figure 13 The group elements t 1 a and ta are generated from the element a by applying t 1 and t respectively Figure 14 Further group elements are generated by applying the generator s to earlier generated elements Figure 15 Still further group elements are generated by applying the generator s 1 to earlier generated elements Because the group is infinite this process should go on indefinitely The order in which we generate the elements can be different from the one we used above The result will be however totally equivalent So when say we have generated the element s 1 ta we can subject it to the generator a which means rotating it 180 0 about the point R the yellow point in the Figure We then get the element as 1 ta which we had already found earlier as st 1 as we can check in the above Figure so we now know that as 1 ta st 1 Interpretation of the crystallization process in terms of the Theory of the Implicate Order In the previous document we have seen that the group elements are abstract in the sense that their boundaries do not necessarily coincide with those of chemical units The set of group elements forming a certain Group only represents the symmetries involved Also in that previous document we interpreted the crystallization process in terms of the Theory of the Implicate Order However such an interpretation is always incomplete and moreover in some way or another wrong This is because we always have to do with the Whole i e the Whole is always involved in whatever we are investigating Whatever is said about something specific it is in fact always said about the Whole And whatever we say or find out about the Whole there is always something more and something different And this implies that when we try to describe a certain phenomenon in the Explicate Order in our case the crystallization process we must use a plurality of descriptions while for the time being tolerating apparent contradictions between them We must realize that we are engaged in an ongoing process of understanding how things are and behave as just aspects of the Whole Metaphysically the Theory of the Implicate Order can be placed into the Neoplatonic framework of the descend and return of metaphysical levels devised by the third century Greek philosopher PLOTINUS As in the Theory of the Implicate Order in Neoplatonism it is such that if we go down to the fundamentals i e if we go from the material world all the way to The One as the origin of all of Reality things are not becoming ever more simple in the sense of some ultimate particles or ultimate fields for that matter but on the contrary they become more complex but at the same time more unified and especially ever more subtle ending up at an infinite richness without any limits and totally undefinable which implies that we can never arrive at some sort of fundamental and ultimate Theory of Everything aspired by modern physics and cosmology The undefinable and unmeasurable holomovement carrying as its modulation a series of implicate generative orders can be related to Plotinus The One which is pure Unity and the highest of the three hypostases which all are immaterial metaphysical levels of Reality From The One an emanation takes place resulting in the two other hypostases and finally in material things The latter emanation stage takes place by the reception of the immaterial structures into Prime Matter About the latter see the philosophical documents in the First Part of the website accessible by the BACK TO HOMEPAGE link at the end of the present document Because of this emanation the unity of the world is guaranteed there is no unbridgable gap between the material and immaterial domains The material domain of Neoplatonism can in a way be identified with BOHM s Explicate Order Matter according to Plotinus is not an independently existing principle but the point at which the outflow of reality from The One fades away into utter darkness Along with the emanation from The One an increasing fragmentation takes place culminating in the high degree of separateness of objects and processes in the Explicate Order Hence at each stage of this emanation the descent into greater multiplicity imposes fresh limits and restrictions disperses and weakens the power of previous stages and creates fresh needs requiring the development of new faculties previously unnecessary The Second Hypostasis Intelligence Nous as such emanating from The One the First Hypostasis represents the first stage of fragmentation Although its intelligible contents are not separated from each other they are clearly distinguished It is a unity in plurality a multiple organism containing a plurality of Forms These Forms are contemplated or thought by Intelligence in such a way that there is no distinction between subject and object Intelligence is identical to its contents i e it has a sort of self awareness The Forms are the immaterial objective structures which I identify with objective mathematical structures The Third Hypostasis Soul Psychè brings about a still further fragmentation The perfect self awareness of Intelligence based on full identity between subject and object is impossible on the level of Soul A corollary is that Soul s contemplation is confined to mere images reflecting the Forms whereas the objects contemplated by Intelligence are the Forms themselves The Second and Third Hypostases could be identified with BOHM s hierarchy of implicate generative orders In a certain sense The One is not It just creates Being In another sense Prime Matter is not because it is pure potentiality and needs form to be able to exist actually Matter receives this form as intelligible content from Soul and indirectly from Intelligence and ultimately from The One In this way the observational material world comes into being and this world can as has been said be identified with BOHM s Explicate Order In terms of the implicate orders we can further characterize the three hypostases as follows The One is the ultimate implicate order Nous is the explicate order with respect to The One Soul i e the World Soul is the explicate order with respect to Nous Nous is the implicate order with respect to Soul Soul is the implicate order with respect to the material world The material world is the ultimate explicate order It is the very Explicate Order totally concrete and individualized The sequence from The One down or up if one whishes to express it to the material world is an order of increasing fragmentation Soul is the intermediate hypostasis between the intelligible and material worlds We will elaborate further on this Neoplatonic metaphysics in a separate document and will now return to the crystallization process and place it in this broader philosophical context i e in BOHM s Theory of the Implicate Order and the Neoplatonic metaphysics A crystal or a developing crystal or whatever we see in the Explicate Order is just an aspect of the W h o l e Such an aspect is just a relatively autonomous subtotality The same holds for the chemical units involved in crystallization But also everything we distinguish within the Implicate Order s is just an aspect of that same Whole a relatively autonomous implicate subtotality which does or does not become explicate When something becomes explicate we have to do with an act that does not consume anything of the Implicate Order s The latter is with respect to its available resources unaffected by such an act of explication Such an act is undiminished giving it involves no dissipation of the Implicate Order s power among its products With respect to crystallization the elements of the relevant Symmetry Group which are symmetry transformations together having group structure and reflecting the involved symmetry are present in the Implicate Order first of all in a group theoretical abstract sense but after that also present therein as being objectively geometrically interpreted for example as rotations or translations And this geometric interpretation is just a next stage of emanation which as a whole extends all the way down to Prime Matter and consequently a further fragmentation this fragmentation is however never absolute but relative The result of this geometric interpretation is however still abstract but now so in a lesser degree When group elements are objectively geometrically interpreted the relative i e mutual locations in the Explicate Order of the chemical units that represent them are automatically determined The generation of the geometrically interpreted group elements resulting in our case in a symmetry Group takes place within the Implicate Order and is a timeless logical noetic process thinking process of which Soul is its subject while it is an aspect of the content of Intelligence where there is no distinction between subject an object This process and all other such noetic processes is constantly recurring expressing an aspect of the timeless dynamism of the Third and Second Hypostases and ultimately that of The One Holomovement although such a way of speaking is inadequate with respect to The One because it entails its fragmentation Within The One there is not only no separation but also not any distinction And this is not because The One lacks anything It stands above all distinctions Because of the mentioned noetic process of generating group elements taking place within the Implicate Oder the d e s c r i p t i v e o r d e r of Group Theory especially when concerned with Symmetry Groups becomes also an o n t o l o g i c a l o r d e r Explication i e unfoldment into the Explicate Order of a crystal structure according to the noetic generation of geometrically interpreted group elements necessarily implies an additional interpretation that is to say a chemical interpretation of those group elements and now the abstractness is transformed into full concreteness But still all this remains abstract in the other sense that it is just an aspect of the Whole i e a relatively autonomous subtotality which is in spite of that totally integrated into the Whole The possibility of such an abstraction from the Whole is rooted in the laws of the Whole holonomy The explication process can metaphysically be described as the reception of the noetic precursors of chemical units and their symmetric relationships to each other into Prime Matter the ultimate substrate Above we described as an example the case of crystallization according to the Plane Group P2 of imaginary two dimensional crystals The generation of the objectively geometrically interpreted group elements takes as has been said place within the Implicate Order or as one may say within Intelligence and then contemplated by Soul and as such we can see it as an aspect of a dynamic formative field with respect to crystal formation Now as soon as in the Explicate Order a solution or melt becomes unstable with respect to a certain chemical substance that could cystallize from such a melt or solution the appropriate formative field is activated by some non local resonance phenomenon and the weak energy of that field directs i e modulates the higher energies of the system of the relevant chemical units such that they arrange in the particular minimum energy configuration resulting in a periodic structure with a symmetry according to the relevant Symmetry Group What we see in the Explicate Order is the onset of a dynamical process resulting in crystal formation While investigating such a process we sometimes are forced to conclude that some sort of cooperation discussed in the previous document must be at work between the chemical units which behave as a whole This cooperation long range cooperation with respect to the microscopic effective range of the forces at work there then is an indication of the presence of a directing formative field When no cooperation is evident in such a process we theorize that also in this case but now not directly evident a formative field is nevertheless involved In that case a purely mechanical chemical explanation yields a very good approximation to what really happens As we see here we have in explaining cooperation not brought in the model of projection and injection according to the ink in glycerine analogy of BOHM but have instead brought in an explanation and interpretation largely according to SHELDRAKE s hypothesis of formative causation which is very close to the theory of the implicate order of BOHM This illustrates the fact as stated earlier that we must approach the theory of Wholeness from several more or less different viewpoints These different approaches should in some way complement each other The Plane Group P1 The next Figure shows the generation of the group elements of the group P1 The motif s str is an asymmetric Figure comma i e a figure with point symmetry 1 which means that its only symmetry element is a 1 fold rotation axis which in turn means that the figure will only occupy the same patch of space when either rotated 0 0 or 360 0 or an integer multiple of 360 0 And this is equivalent with having no symmetry at all The motif s l i e the comma plus its corresponding background has the shape of a parallelogram just like in the case of the group P2 But we see that the parallelogram representing the motif s l corresponds directly with a group element From a given group element 1 all the other group elements can be generated by applying two generator elements as all group elements they are symmetry transformations These two generators are the translations t horizontal translation and s oblique translation When applied to the element 1 and to the results thereof we will generate all group elements and with it the Group itself and when indeed the group elements are represented by motifs we obtain the whole pattern Figure 16 A 2 D periodic pattern according to the Plane Group P1 can be generated from an initial group element by applying two translations t and s The blue and red parallelograms represent motifs s l and at the same time group elements The colors red and blue do not signify any qualitative difference Also here we must interpret the generation of the Group elements as a dynamic structure present in the Implicate Order

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  • implicate order 3
    indicated and highlighted The two generator elements are A glide reflection g along the glide line indicated by a dashed line A vertical translation t Figure 10 Areas of the Pg pattern of Figure 1 corresponding to the group elements The initial element 1 and the two generators g and t indicated and highlighted purple All elements of the Group Pg can be generated from the initial element by subjecting it repeatedly to the generator elements Figure 11 The group elements g 1 2 g 2 g 1 3 g 3 g 1 g 2 g 3 g 4 are generated from the initial element 1 by repeatedly applying the transformation g The rest of the group elements of the depicted part of our Pg pattern can be generated by applying the vertical translation t and its inverse t 1 to the already generated elements Figure 12 The group elements t 2 g 1 3 t g 1 2 t 1 g 1 2 t 1 g 1 3 t 2 g 1 2 t 2 g 1 tg 1 t 1 t 1 g 1 t 2 t 2 g tg 2 tg t 1 g 2 t 1 g t 2 g 2 t 2 g 3 tg 4 tg 3 t 1 g 4 t 1 g 3 t 2 g 4 are generated from the already obtained group elements by applying repeatedly the vertical translation t and its inverse t 1 The colors blue and red do not signify qualitative differences We have thus generated the group elements of the Group Pg The process must of course be continued indefinitely because the group is infinite The Plane Group P2mm We re now going to generate the group elements of the Plane Group P2mm As before we first let the motif units composing the motifs s str represent the group elements Then we determine the motif units of the motif s l and let them represent the group elements and generate the latter again Figure 13 Placing motifs with 2mm point symmetry in a primitive rectangular 2 D lattice creates a periodic pattern of these motifs representing the Plane Group P2mm The pattern must be conceived as extending indefinitely in two dimensional space The total symmetry content of this Plane Group is given in Figure 14 Figure 14 The total symmetry content of the Plane Group P2mm Solid lines black and red indicate mirror lines Small red solid ellipses indicate 2 fold rotation axes perpendicular to the plane of the drawing The next Figure again gives the P2mm patttern and prepares for letting it be generated Figure 15 Each composed motif of our version of a P2mm pattern consists of four motif units commas partially overlapping such that the symmetry of the resulting composed motif is 2mm As before each motif unit represents a group element One such motif unit is chosen to be the initial motif unit representing the identity element of the group and denoted

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