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  • General Ontology XXIX
    Planimetric Basic Form representing this promorph Examples of holomorphic crystals Non eupromorphic crystal Intrinsic shape Amphitect Hexagon Non eupromorphic crystal Intrinsic shape Amphitect Octagon Autopola Orthostaura Tetraphragma interradialia amphitect polygons here rectangles or rhombi with four antimers and with interradial directional axes Planimetric Basic Form representing this promorph Examples of holomorphic crystals Non eupromorphic crystal Intrinsic shape Amphitect Hexagon Non eupromorphic crystal Intrinsic shape Amphitect Octagon Autopola Orthostaura Diphragma amphitect polygons here rhombi with two antimers Planimetric Basic Form representing this promorph Examples of holomorphic crystals Non eupromorphic crystal Intrinsic shape Amphitect Hexagon Non eupromorphic crystal Intrinsic shape Amphitect Octagon General promorphological category Gyrostaura gyroid i e twisted polygons homogyrostaura regular gyroid polygons Homogyrostaura hexamera Regular gyroid polygons with six antimers Planimetric Basic Form representing this promorph Example of holomorphic crystal Non eupromorphic crystal Intrinsic shape Regular Gyroid Dihexagon Homogyrostaura dihexamera Regular gyroid polygons with 12 similar six by six equal antimers Planimetric Basic Form representing this promorph Example of holomorphic crystal Eupromorphic crystal Intrinsic shape Regular Gyroid Dihexagon Homogyrostaura tetramera Regular gyroid polygons with four antimers Planimetric Basic Form representing this promorph The next two Figures illustrate this basic form still further by giving equivalent geometric figures all representing and expressing the same promorph Example of holomorphic crystal Non eupromorphic crystal Intrinsic shape Regular Gyroid Ditetragon Homogyrostaura ditetramera Regular gyroid polygons with eight similar four by four equal antimers Planimetric Basic Form representing this promorph Example of holomorphic crystal Eupromorphic crystal Intrinsic shape Regular Gyroid Ditetragon Homogyrostaura trimera Regular gyroid polygons with three antimers Planimetric Basic Form representing this promorph Example of holomorphic crystal Non eupromorphic crystal Intrinsic shape Regular Gyroid Ditrigon Homogyrostaura ditrimera Regular gyroid polygons with six similar and three by three equal antimers Planimetric Basic Form representing this promorph Example of holomorphic crystal Eupromorphic crystal Intrinsic shape Regular Gyroid Ditrigon Gyrostaura gyroid i e twisted polygons heterogyrostaura amphitect i e flattened gyroid polygons Heterogyrostaura octomera amphitect gyroid polygons with eight antimers Planimetric Basic Form representing this promorph Example of holomorphic crystal Eupromorphic crystal Intrinsic shape Amphitect Gyroid Octagon Heterogyrostaura hexamera amphitect gyroid polygons with six antimers Planimetric Basic Form representing this promorph Examples of holomorphic crystals Eupromorphic crystal Intrinsic shape Amphitect Gyroid Hexagon Non eupromorphic crystal Intrinsic shape Amphitect Gyroid Octagon Heterogyrostaura tetramera amphitect gyroid polygons with four antimers Planimetric Basic Form representing this promorph Examples of holomorphic crystals Non eupromorphic crystal Intrinsic shape Amphitect Gyroid Hexagon Non eupromorphic crystal Intrinsic shape Amphitect Gyroid Octagon Heterogyrostaura dimera amphitect gyroid polygons with two antimers Planimetric Basic Form representing this promorph Examples of holomorphic crystals Non eupromorphic crystal Intrinsic shape Amphitect Gyroid Hexagon Non eupromorphic crystal Intrinsic shape Amphitect Gyroid Octagon Heterostaura Allopola half amphitect polygons bilateral forms Allopola amphipleura bilateral forms with 3 5 6 7 8 antimers Allopola octamphipleura half amphitect polygons with eight antimers Planimetric Basic Form representing this promorph Example of holomorphic crystal Eupromorphic crystal Intrinsic shape Bilateral Octagon Allopola hexamphipleura half amphitect polygons with six antimers Planimetric Basic Form representing this promorph Examples

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  • General Ontology XXIXa
    maximal distance from each other and successive clusters would be arranged in the spaces as seen from above of the previous cluster resulting in a pattern with alternating whorls The same goes for leaves of higher plants in decussate phyllotaxis See for phyllotaxis and the symmetries and promorphs it can produce next document Another way of formation of whorls as it probably goes on in some giant unicellular algae Dasycladales can be described by a growing tip of the one cell of which the alga consists that flattens and then forms a ring or annulus which is as such unstable and therefore transforms in virtue of the slightest perturbation into a stable state by the transformation of it i e of this annulus into of a crown of spikes as can also be seen in a splash which as such then represents a simple fluid dynamics analogue See for whorl formation in these algae as well as in higher plants GOODWIN B How the Leopard Changed Its Spots 2001 As in crystallization also in these latter two cases whorl formation in higher and in lower plants we can say that we have to do with a form potential field which is as such unstable and consequently slides off into a stable configuration which is a whorl So in crystallization initially there is maximal symmetry solution or melt without crystals And this symmetry breaks resulting in the point symmetry of the crystal i e a symmetry according to one or another of the 32 three dimensional Crystal Classes And for two dimensions we first have D infinity maximal symmetry which breaks to D n or C n symmetry where n 1 2 3 4 or 6 In the same way we must assume that the relevant excitable medium of a generating plant whorl comparable to a supercooled melt or a supersaturated solution has seen from above D infinity symmetry which breaks to D n or C n where n could in principle be any whole number although the number of petals forming a whorl is often either 2 or 3 4 5 8 13 etc If we neglect the number four which we see in all Crucifer flowers then we have the sequence 2 3 5 8 13 etc which belongs to the Fibonacci Series which can be obtained by starting with 1 1 and then adding numbers that are equal to the sum of the previous two numbers So the number of petals stands in a way in connection with this number Series Even closer zeroing in on the comparison between crystals and organisms The initial annulus that in the unicellular algae mentioned above appears before the whorl arises and being highly comparable to the annulus that rises up in a splash corresponds to the supercooled melt or supersaturated solution in the vicinity of the crystal embryo Both are unstable and lead to a symmetry breaking process resulting in some kind of whorl i e a radiate structure The formal transition from the splash where the spiked crown develops from the annulus that rises up after impact to the plant whorl especially as it occurs in the mentioned unicellular algae or maybe even to all whorled organisms or parts thereof involves a complexity jump and as such can represent the transition from the Inorganic to the Organic Layer of Being While crystallization is a process tending toward thermodynamic equilibrium and thus to a static condition the organic form generation is a process involving non linearity energy flow through the system a displacement from thermodynamic equilibrium and chaotic patterns of fluctuation at bifurcation and transition points they nevertheless have much in common Both involve a medium that initially is or becomes unstable They then slide off onto a stable state which involves the formation of a definite pattern But whereas in inorganic systems there generally is just one level involved or at most a microscopic level and a macroscopic level and maybe one level in between in organic systems many more levels are intercalated between the ultimate lowest level and the appearance of organic macroscopic structures i e these macroscopic structures are not as they are in crystals direct consequences of lower or lowest levels but very indirect ones This is because many regulatory processes intervene going about in efficient cooperation and resulting in robust structures that are stable against a broad range of external perturbations whereas in crystallization a change in temperature pressure concentration etc could terminate and even reverse i e undo the pattern generating process Now it could be that as already the eminent philosopher Nicolai Hartmann proposed in 1940 Der Aufbau der realen Welt the formal transition from inorganic to organic involves the appearance of a novum a categorical novum as he calls it that is to say the emergence of one or more totally new p r i n c i p l e s o f b e i n g i e new ontological categories While on this subject more than a whole book is required to evaluate the pros and cons of such a position let us assume this position to be basically correct We have argued for such a position and proposed some additions and amendments in Part I of the present Series of documents Let us assume that there appears one new category i e a new fundammental ontological principle when formally going from the Inorganic Physical Layer to the Organic Layer This new category categorical novum is undoubtedly very complex and not directly definable So we can only give it a name the state of something BEING ALIVE loosely definable as an integrated all out cooperation between that something s parts in such a way that the whole is in some way prior to its parts This category then generally influences many inorganic categories as they formally pass over from the Inorganic to the Organic Layer of Being For instance the category to be an individuum passes over from the Inorganic to the Organic where we find it respectively at the macroscopic level of the Inorganic Layer as crystal individuals in the case of single not twinned crystals and at the macroscopic level of the Organic Layer as living individuals including bacteria an other unicellular organisms Although both types of beings i e crystals and organisms can indeed be considered as genuine individuals the way in which they are individuals is totally different Where this way is more or less static in inorganic individuals without active self maintenance it is eminently dynamic and regulatory in organic individuals which actively maintain themselves and which finally culminate in self conscious beings So although the category of being an individual or possessing individuality does not when coming from the Inorganic stop at the boundary where the Organic begins but passes over from the Inorganic Layer to the Organic Layer it is significantly transformed as a result The cause of this transformation can be found in the presence of the categorical novum the state of being alive in the Organic Layer and absent not only in the Inorganic Layer but also in the Mathematical Layer In fact the essence of this novum totally consists in or is totally exhausted by all the transformations themselves that categories undergo when while coming from the Inorganic crossing the boundary between the latter and the Organic Layer In fact these categories are not transformed into other categories which means that they are not actually trans formed but only over formed such a category while keeping its identity as to what particular category it is it is nevertheless changed when passing into a higher Layer of Being And all these over formed c a t e g o r i e s lie at the base of the corresponding jumps in complexity when b e i n g s cross that boundary as can be visualized in crystals formally not evolutionarily passing over into organisms As with being an individuum the same can be said about being in a state of growing being in a state of regeneration etc In fact all crystal analogies See Part IV can be seen as transitions from the Inorganic to the Organic involving complexity jumps and categorical over forming However we must emphasize that this is just a hypothesis not a proven fact GOODWIN How the Leopard Changed Its Spots p xiv writes I take the position that organisms are as real as fundamental and as irreducible as the molecules out of which they are made Although this statement does not propose the introduction of some immaterial and independent component in organisms that is absent in inorganic beings i e it certainly denies a form of dualism that was assumed in earlier theories it nevertheless maintains that organisms are irreducible And the latter is not meant only epistemologically i e referring to our inability to derive organisms from purely inorganic basic structures which is up to now true but also ontologically i e it means that organisms are all by themselves not ultimately equivalent to inorganic processes and structures The truth of the latter is not at all proven and there are many biologists maintaining that organisms are nothing more than rather very complex and integrated inorganic structures having a behavior that we call alive and that have evolved from inorganic beings or processes It is easy to state that organisms are holistic entities fundamental in their own right But it is very hard to make that explicit After all our planet our planet began as an aggregation of physical matter The ensuing decrease of temperature of the Earth s surface made the existence of complex molecules possible And because an extraterrestrial origin of organismic life is highly improbable at most certain crucial but still non living chemical compounds could be of extraterrestrial origin and if it were true the problem is not solved but shifted to some other place in the universe So where does life ultimately come from The only plausible but still very non specific answer is from a particular complexification of inorganic matter This complexification not only led to life but also to the appearance of consciousness as we find it in higher animals and in humans So it seems certain that there must be ontological continuity between inorganic matter and living and finally conscious matter In Part I we have argued that indeed the formal transition from the Inorganic to the Organic is not as dramatic and fundamental as many philosophers including Nicolai Hartmann want it to be The novum that appears is only in a much more modest sense a novum Many of the categories that pass from the Inorganic to the Organic are over formed in virtue of its influence Of course there are also many categories that pass over from the Inorganic to the Organic without being over formed at all They are not affected by the novum Evidently categories like being spatial and being temporal i e the categories of Space and Time happily pass over from the Inorganic to the Organic without being over formed And also our principal objects of study but serving a general ontology viz having an intrinsic shape A having an intrinsic symmetry B and having a promorph C pass over into the Organic Layer without being over formed They even are already present in the Mathematical Layer where many more of them are present that can symbolically be indicated by D E F G etc In order to exist in the material Layers i e Inorganic and Organic they must satisfy energetic stability The fact that when passing over into higher Layers of Being intrinsic shape intrinsic symmetry and promorph are not over formed is because they are as such already very specific in contrast for example to individuality growth and the like They do not admit of higher and lower forms C 3 symmetry for example can only manifest itself in precisely the same way in whatever Layer it appears i e it is exactly the same in the Mathematical in the Inorganic Physical and in the Organic Layer There is no higher C 3 symmetry Likewise shape can only mean exactly the same in all these three Layers The promorph which in fact is just a geometric pattern is in the same way present in all these three Layers Let s give an example to clarify this A mathematical figure say a pyramid can have a definite shape a definite symmetry and a definite promorph Consider the following mathematical figure Figure above Slightly oblique top view of an amphitect octogonal pyramid The lines originating at the pyramid s apex and ending up at A B C D E F G and H are the eight ribs of the pyramid This pyramid is in the present example provided with two additional lines namely the ones that also originate at the pyramid s apex but end up at X and Y So the present three dimensional mathematical figure depicted here consists of eight lines constituting an amphitect octogonal polygon the base of the pyramid eight lines representing the pyramid s ribs running from the apex to the corners of the octogonal polygon and two lines connecting the pyramid s tip with the centers of the lines BC and GF It is this mathematical figure we can call it a mathematical solid that we re now going to examine with respect to shape symmetry and promorph The next Figure depicts the same mathematical solid amphitect octagonal pyramid with two extra lines not ribs as the previous Figure but viewed from an even more oblique angle Figure above A little more tilted top view of the amphitect octogonal pyramid of the previous Figure As in that previous Figure the sides of the pyramid are indicated by the colors green and yellow The next Figure depicts the same but now emphasizing an interpretation of this mathematical solid as consisting not of eight but of two antimers Figure above The same mathematical solid as was depicted in the previous Figure The lines running from the pyramid s tip to the points X and Y suggest an interpretation of this particular mathematical solid as possessing not eight but two antimers Generally an amphitect octogonal pyramid i e an amphitect 8 fold pyramid is interpreted as having eight antimers The shape In mathematical figures the distinction between intrinsic and extrinsic is irrelevant of this solid is an amphitect octogonal pyramid The symmetry of this solid is according to the dihedral group D 2 because it has two mirror planes perpendicular to each other implying a 2 fold rotation axis coinciding with the line of intersection of the two mirror planes The promorph of this solid i e of the particular solid as drawn above consisting of the two lines connected with the points X and Y the eight lines connected with the points A B C D E F G and H and the eight lines forming the base of the pyramid is a rhombic pyramid with two antimers and as such belonging to the Heterostaura Autopola Orthostaura diphragma This promorph is depicted in the next Figure Figure above Slightly oblique top view of a four fold amphitect pyramid rhombic pyramid as the basic form of the Autopola orthostaura diphragma This configuration as such possesses t w o antimers These a n t i m e r s are indicated by colors yellow blue It is the promorph of the mathematical solid amphitect octogonal pyramid as depicted earlier To express the presence of just two antimers we could call the pyramid representing the promorph and depicted in the present Figure a two fold amphitect pyramid or an rhombic pyramid with two antimers All this i e an amphitect octogonal pyramidal shape a D 2 symmetry and a promorph belonging to the Autopola Orthostaura diphragma represented by a two fold amphitect pyramid as such present in the Mathematical Layer is also represented by objects in the material Layers inorganic and organic because there can be and indeed are conditions that allow them to be energetically s t a b l e configurations of material elements Representation by objects in the I n o r g a n i c Layer of Being of the shape symmetry and promorph of the amphitect octogonal pyramid as it was given above The D 2 s y m m e t r y as represented in the Inorganic Layer The D 2 s y m m e t r y of our mathematical solid is found as intrinsic point symmetry in the Inorganic Layer for example in crystals of the mineral Hemimorphite Zn 4 Si 2 O 7 OH 2 H 2 O See next Figure Figure above A crystal of the mineral Hemimorphite The actual color of the crystal is white in some cases with faint bluish or greenish shade Also yellow to brown Transparent to translucent After BRUHNS W Kristallographie 1912 This crystal belongs according to its intrinsic D 2 point symmetry to the Rhombic Pyramidal Class mm2 of the Orthorhombic Crystal System Representation of the p r o m o r p h belonging to the Heterostaura Autopola Orthostaura diphragma in the Inorganic Layer Because I do not have relevant information as to the morphology of the translation free residue which has D 2 symmetry of a Hemimorphite crystal I cannot assess the number of antimers and so the assessment of the p r o m o r p h of these crystals can for the time being only point to a more general promorphological category namely that of the Heterostaura Autopola amphitect pyramids Of course there is nothing that would forbid beforehand the possible existence of D 2 three dimensional crystals having shapes like we see in Hemimorphite crystals or having other shapes or even having the same shape as our mathematical amphitect octogonal pyramid that promorphologically belong to the Heterostaura Autopola Orthostaura diphragma meaning that in that case the promorph of our mathematical amphitect octogonal pyramid as the latter was drawn earlier is indeed represented by objects in the Inorganic Layer of Being Representation of the amphitect octogonal pyramidal s h a p e in the Inorganic Layer The s h a p e of our above mathematical solid namely that of an amphitect octogonal pyramid can be expected among the same Class of Orthorhombic crystals as to which the above hemimorphite crystal belongs viz the Rhombic Pyramidal Crystal Class mm2 A crystal having this shape is made up of a combination of four crystallographic Forms viz one rhombic pyramid two domes and one pedion Each Form is generated from some initially given crystal face in virtue of the action of the symmetry elements of the crystal depending on the orientation of such an initial face with respect to the crystallographic axes a b c These latter are taken such as to align with the crystal s symmetry elements mirror planes and rotation axis The c axis coincides with the line of intersection of the two mirror planes and thus coincides with the crystal s 2 fold rotation axis The a axis lies in one mirror plane while the b axis lies in the other and both are perpendicular to the c axis See next Figure Figure above Position of the crystallographic axes a b and c of a three dimensional crystal oblique top view having the shape of an amphitect octogonal pyramid and belonging to the Rhombic Pyramidal Class mm2 and thus having D 2 intrinsic symmetry of the Orthorhombic Crystal System The c axis connects the apex of the pyramid with the center of its base where the latter is an amphitect octagon We will now list and depict the four crystallographic Forms indicated by red color that together constitute our amphitect octogonal crystal Figure above The upper rhombic pyramid red which is one of the Forms of the Rhombic Pyramidal Crystal Class that participates in the combination of Forms making up the crystal having the shape of an amphitect octogonal pyramid consists of four faces the pyramid is open at its base generated from one oblique face in virtue of the symmetry elements of this Crystal Class two mirror planes perpendicular to each other and a 2 fold rotation axis along their line of intersection Figure above The upper first order dome red another Form of the Rhombic Pyramidal Crystal Class that participates in the combination of Forms is a Form consisting of two faces parallel to the crystallographic a axis It is generated from one initially given face intersecting the crystallographic c axis i e the axis of the pyramid containing its apex and parallel to the a axis The second face of this Form is generated by the mirror plane containing the a axis and the c axis Figure above The upper second order dome red that participates in the combination of Forms is also a Form consisting of two faces but now parallel to the crystallographic b axis It is generated from one initially given face intersecting the crystallographic c axis i e the axis of the pyramid containing its apex and parallel to the b axis The second face of this Form is generated by the mirror plane containing the b axis and the c axis Figure above The lower pedion is a Form consisting of one face only This face is parallel to the a and b axes and consequently perpendicular to the c axis It closes the pyramid and forms its base Given this face no more faces are produced from it by the crystal s symmetry elements So this was the s h a p e of the above mathematical solid depicted earlier as that shape can be represented in the Inorganic Physical Layer as an Orthorhombic crystal of the Class mm2 Rhombic Pyramidal Class consisting of nine faces and as such a combination of four crystallographic Forms Of course such a crystal also represents D 2 symmetry in the Inorganic Layer The crystal example that has been given earlier hemimorphite crystal representing only i n t r i n s i c p o i n t s y m m e t r y D 2 has a more complex shape than that of an amphitect octogonal pyramid This particular hemimorphite crystal as depicted above consists of six crystallographic Forms together making up 15 crystal faces Let s list the Forms that participate in constituting this crystal The face c is an upper pedion It is a Form consisting of one face only The face d is one of the two faces making up an upper first order dome The face m is one of the two faces making up an upper second order dome The faces p p are two of the four faces making up a third order i e vertical prism The face b is one of the two faces making up a pinacoid which is a Form consisting of two parallel faces The faces s s are two of the four faces making up a lower rhombic pyramid These six Forms combine to give our hemimorphite crystal as it was depicted above See with respect to crystallographic Forms of three dimensional crystals First Part of Website In Special Series The Morphology of Crystals and Orthorhombic Crystal System II Class mm2 Representation by objects in the O r g a n i c Layer of Being of the shape symmetry and promorph of the amphitect octogonal pyramid as it was given earlier Let s start with the symmetry and the promorph Representation of D 2 symmetry and of the Heterostaura Autopola Orthostaura diphragma in the Organic Layer As for occurrence in the Organic Layer we can as an example advance the flower of the plant Circaea lutetiana enchanter s nightshade Family Onagraceae Oenotheraceae that indeed has intrinsic D 2 symmetry and a promorph belonging to the Heterostaura Autopola Orthostaura diphragma See next Figures Figure above Flower of Circaea lutetiana It has two sepals two incised petals two stamens and one pistil It is a two fold structure After WETTSTEIN R Handbuch der Systematischen Botanik 1924 The next Figure shows a promorphological analysis of this flower as done by Ernst Haeckel in his Generelle Morphologie der Organismen 1866 where he first proposed Promorphology as the Science of Basic Forms in organisms Figure above Promorphological analysis of the flower of Circaea lutetiana After HAECKEL E Generelle Morphologie der Organismen 1866 Representation of the amphitect octogonal pyramidal s h a p e by objects in the Organic Layer While our presently discussed shape can occur as crystals in the Inorganic Layer of Being it will be hard to find it represented as the intrinsic shape of entities of the Organic Layer This is because our shape the amphitect octogonal pyramid consists of straight lines and plane surfaces which are relatively rare among organisms because of their semi liquid state allowing surface tension to play some role i e allowing it to partake in the generation of shape Generally such a shape consisting of straight lines and plane surfaces is to be expected only as an extrinsic shape as for instance in certain cells making up certain organic tissues The pyramidal shape has then come about by extrinsic forces with respect to such a cell But maybe we can find it as intrinsic shape in certain pollen grains or other unicellular organic entities such as Radiolarians One could also think of some virusses having this shape when they are in their crystalline state But virusses are especially in this state not really alive they are so to say fragments of life presupposing genuine life for their bioactivities especially their reproduction But of course s h a p e itself i e shape just as shape including intrinsic shape occurs everywhere in the organic realm So far so good with respect to the occurence of certain shapes symmetries and promorphs in the Mathematical Layer of Being and their representations in the Inorganic and Organic Layers Later in a new Series of Documents we are internded to present many more examples The above had illustrated that the categories of Shape Symmetry and Promorph can pass from the Mathematical Layer into the Inorganic and from there into the Organic Layer without being over formed The concepts expressing them have exactly the same meaning in all three Layers But the way Shape Symmetry and Promorph are generated is different in these Layers In the Mathematical Layer they are just present as logically consistent entities not dependent on stability and not actually generated They formally derive from simpler structures In the Inorganic Layer they appear in three dimensional crystals and are generated by relatively simple and straightforward processes where atomic and molecular entities aggregate as soon as their dispersed state becomes energetically unstable For intrinsic symmetries to occur in the realm of crystals there are physical constraints involving energy and therefore stability Common to all three Layers is the mathematical constraint that strict periodic patterns cannot contain 5 7 8 9 10 11 12 13 14 fold rotation axes Such rotation axes can and are it is true observed in organisms but organisms are not periodic patterns And such axes especially 5 and 10 fold do sometimes occur in certain crystals but the latter too lack strict periodicity i e their internal structural pattern is not strictly periodic and therefore they are called quasi crystals The same goes for liquid crystals which also show only partial periodicity Finally in the Organic Layer Shape Symmetry and Promorph are co generated by very complex and diverse processes not simple aggregations or regular stackings processes in the context of non linear dynamical systems far from thermodynamic equilibrium dynamical systems involving genes and their products complex feedback loops etc all cooperatively integrated into a single whole which is the organism itself So Shape Symmetry and Promorph as they are in themselves do not teach us about the transition from the Mathematical to the Inorganic and from there to the Organic Layers Only their generation their formation does They are categories If Then constants that go unchanged from one Layer to the other In the present Series of Documents we emphasize the aspect of continuity between the three Layers by studying Shape Symmetry and Promorph in crystals and organisms For the latter this is done extensively in Second Part of Website Basic Forms Further we will study the continuity and difference between the Inorganic Physical Layer and the Organic Layer by evaluating crystal analogies There we will see that many properties that we see in crystals return in organisms However they there appear in a sort of elevated state i e they appear in a much more complex fashion than they do in crystals embodying in this way the complexity jump that occurs at the formal transition between the Inorganic and the Organic Until now we have studied real crystals by way of imaginary two dimensional crystals because three dimensional crystal structure is hard to depict on a two dimensional screen The geometrical structure of two dimensional crystals is easier to grasp and reveals most of the structural principles of three dimensional crystal structure Moreover in themselves two dimensional crystals are true inhabitants of the Mathematical Layer of Being as long as we stick to their geometry shape symmetry and promorph without interpreting their constituents as atoms or molecules In fact we do so interpret but after that discard it again This we do to maintain the difference that obtains between two dimensional crystals and other geometric plane figures But as soon as we want to fully evaluate all existing crystal analogies we will have to deal with stability questions and these necessarily involve energy considerations And this can of course only be done with real i e three dimensional crystals Therefore we will study their structure as well but in a concise and algebraic fashion Group Theory and only in a very limited way in a geometric fashion This concludes our preliminary Ontological Discussion concerning the role that Shape Symmetry and Promorph play in the layered structure of Being But before we go to the next document we d like to think an rethink the idea of a layered structure of Reality as presented in this website because it could entail some remarkable consequences as to the overall and global structure of the whole of Existence Reality the World or whatever we care to name it On the Transformation of the Organic into the Inorganic The Inorganic as the result of a h o l i s t i c s i m p l i f i c a t i o n of the Organic The theory of Layers of Being as presented on this website and inspired by and partly based on the philosophy of Nicolai HARTMANN can still be discussed further as to its general validity or falsity when we concentrate on the division between the Inorganic and the Organic These can be related to each other in four different ways The transition from Inorganic to Organic takes place through a gradual and specific increase of material complexity This implies that the Organic can completely be derived as in a scientific theory from the Inorganic or in other words the Organic can be completely reduced as in a scientific theory to the Inorganic We can change from Inorganic to Organic by what we could call a reductionistic complexification Such a position is called r e d u c t i o n i s m as opposed to holism Because of the gradual nature of the complexity increase the reduction is not only perfectly possible for nature meaning that the Organic IS in fact nothing more than complex Inorganic or in other words the Organic is ontologically reducible to the Inorganic but is also within the not too distant future possible for us as knower i e the Organic is also epistemologically reducible to the Inorganic The transition from Inorganic to Organic takes place through a complexity jump as was extensively discussed in Part I of the present Series of Documents This implies that the Organic is ontologically reducible to the Inorganic but not epistemologically so Also here we can say that we can change from Inorganic to Organic by a reductionistic complexification The vast sudden and dramatic increase in specific complexity cannot be unfolded by science because many traits and traces have become lost or cannot be disentangled Here we have to do with the Reductionist Nightmare as described by I STEWART J COHEN in Figments of Reality 1997 The complexity jump entails novelty to appear as we go from Inorganic to Organic But this novelty is only a novelty in a moderate degree As such it is not present in the Inorganic but it nevertheless originates from the latter So it is not a full fledged ontological novelty The transition from Inorganic to Organic takes place through the appearance of genuine and ontological novelty i e through the appearance of a totally new category or principle the categorical novum as was proposed by Nicolai HARTMANN in Der Aufbau der realen Welt 1940 This implies that the Organic is neither epistemologically nor ontologically reducible to the Inorganic or in other words the Organic can in no sense epistemologically or ontologically be completely derived from the Inorganic and such a position is called h o l i s m as the opposite thesis with respect to reductionism mentioned above This would mean that the Organic could not historically have evolved from the Inorganic And when we rightly so refuse to assume that the Inorganic and the Organic have nothing to do with each other i e when they are contingent with respect to each other we must assume that the Inorganic has evolved from the Organic because then we don t have the problem of the NOVUM especially where it came from i e the problem that it came out of the blue However we cannot assume this because it is certain that the earlier stages of the Earth and also of the universe as a whole were entirely inorganic In those early epochs temperatures were too high to sustain stable complex molecules and according to recent cosmological theories the very early stage of the universe could not even sustain stable atoms let alone complex molecules So physically or let us say materially and historically the Inorganic has not evolved from the Organic But however this may be in some way the Inorganic must have been evolved from the Organic if we are not willing to accept the appearance of a genuine NOVUM Indeed when the Inorganic evolves from the Organic only certain typical aspects of the Organic need to be subtracted to result in the Inorganic But as has been just found out this has not actually happened in the observable world And this is precisely where the Implicate Order comes in The Theory of the Implicate Order is extensively considered in Third Part of Website If the reader wants to appreciate the ensuing discussion he or she should read especially the first page below the Table of Contents of this Third Part of Website beginning with the Section The Theory of the Implicate Order s and Wholeness Introduction and then all the way down to the end of that document And also as can be clicked in the Table of Contents The Ink in Glycerin Model The theory boils down to the assumption that all things apparently separated in space and or time are manifestations of one single dynamic whole like the waves of an ocean These things are projections from that whole which is called the Implicate Order into their states of separatenes which collectively is called the Explicate Order The mentioned things can also return from the Explicate unfolded Order to the Implicate enfolded

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  • General Ontology XXIXb
    of a plant with spiral phyllotaxis where primordia successively appear R 0 is the radius of the region that corresponds to the center of the meristem around which the leaf primordia are generated The big vertical arrow symbolizes vertical growth of the branch or stem of the plant while the laterally depicted arrows indicate the spatial expansion of the system of appeared primordia i and j are pairs of successive Fibonacci numbers i e such a pair of successive Fibonacci numbers is indicated as i j The symbols n n i n j n i j stand for numbers indicating the order of appearance of primordia along the generative spiral during growth However they d better be symbolized by n n i n j n i j We see that consecutive numbers in one and the same family of secondary spirals display a constant difference between them So for the anticlockwise family n i n i which is a Fibonacci number n i j n j i which is the same Fibonacci number For the clockwise family n j n j which is the second Fibonacci number n i j n i j which is the same Fibonacci number So here we have a case of i j phyllotaxis The next Figure gives the pattern of primordia leaves that in top view shows 13 21 phyllotaxis i e there are thirteen secondary spirals going one way and twenty one the other Figure above The spiral pattern of primordia leaves obtained when i 13 and j 21 i e a case of 13 21 phyllotaxy If the primordia appear s l o w l y then by the time the next primordium appears the only primordium that has any influence on it with respect to repelling each other is the immediately preceding one The other primordia are too far away to have any effect As a result a newly appeared primordium is repelled to a position 180 0 away from the previous one so that the pattern produced is like alternate or distichous phyllotaxis as in maize As the r a t e o f a p p e a r a n c e of new primordia is i n c r e a s e d a new primordium experiences repulsive forces from more than one previous primordium and the pattern changes The initial simple symmetry of the alternate mode gets broken and a spiral pattern begins to emerge If the rate of appearance of primordia is r a p i d so that there is strong interaction between the primordia then a stable pattern emerges rapidly and successive primordia quickly settle into a divergence angle of 137 5 0 the secondary spirals obeying the normal Fibonacci series meaning that the numbers in the pair i j representing the two constant differences between the numbers of the members indicating their order of appearance of each of the two types of secondary spirals and as such characterizing the phyllotaxis are consecutive Fibonacci numbers for example 8 13 or 13 21 and not say 1 3 and certainly not say 1 4 because 4 is not a Fibonacci number All these findings are based on models one of which is described in GOODWIN B 2001 How the Leopard Changed Its Spots as well as in STEWART I 1995 Nature s Numbers Whether the emerging phyllotaxis is distichous i e alternate or astichous i e spiral depends on the rate of appearance of successive primordia Above some critical value of this rate distichous phyllotaxis transforms to astichous phyllotaxis Further change of this rate while remaining above the critical value determines which kind of spiral phyllotaxis we get in terms of consecutive Fibonacci numbers i e we either get 1 2 phyllotaxis 2 3 phyllotaxis 3 5 phyllotaxis 5 8 phyllotaxis etc And while the rate of appearance of primordia increases as such representing different individual cases the system of leaf arrangement quickly comes close to a 137 5 0 dovergence angle i e if no particular circumstances happen to prevail then this angle is being approched more and more for every individual case Only when such particular circumstances happen to prevail we not only get other divergence angles but also pairs consisting of the two constant differences characterizing the two families of secondary spirals that consist of two non consecutive Fibonacci numbers or even contain non Fibonacci numbers But all these are minority classes representing less robust i e less stable phyllotaxy patterns Over 80 percent of the 250000 or so species of higher plants have spiral phyllotaxis with respect to leaves as contrasted with the phyllotaxis of flower organs like sepals petals etc We now can easily describe the third type of phyllotaxis namely whorled phyllotaxis polystichous phyllotaxis This would be obtained if more than one primordium were to appear at any one time in the growing meristem growing tip of stem with a rate of appearance below the above mentioned critical value so that only the influence of the previous cluster of primordia were experienced Then each cluster of primordia would take up positions at maximal distance from each other and successive clusters would be arranged in the spaces of the previous cluster because of their repellence while also the members of such a cluster try to stand apart from each other as far as possible also because of their mutual repellence The result is whorled pattern The number of primordia that appear simultaneously determines the radiateness of the resulting whorl i e its homotypic number As we know this number is often one from the Fibonacci series but sometimes also other numbers in particular the number 4 as we see in Crucifer flowers The minimum number of elements leaves or flower organs a single whorl can have is two as we see in Fuchsia See in the Figure depicted earlier left image And this implies four stichies because the whorls alternate So this paricular case of polystichous phyllotaxis should be called tetrastichous phyllotaxis Whorled phyllotaxis is most common in flowers and less so in the case of arragement of genuine leaves So all the patterns viz distichous i e alternate polystichous i e whorled and astichous i e spiral phyllotaxis can be generated simply by changing growth rates and numbers of leaves or sepals petals etc generated at any time These are presumably the main parameters that differ among plant species GOODWIN from whom we have most of the information on phyllotaxy writes about this the following ibid p 132 where we insert comments between square brackets So we get an interesting conjecture the frequency of the different phyllotactic patterns in nature may simply reflect the relative probabilities implied by the degree of dynamic stablity of the morphogenetic trajectories of the various forms and have little to do with natural selection That is to say all the phyllotactic patterns may serve well enough for light gathering by leaves and so are selectively neutral Recall that flower organs are derived from genuine leaves Minimal covering of the leaves however is only the case where the phyllotaxis is spiral with a divergence angle of 137 5 degrees Then it is the size of the domains of stability in the generative space of these generic forms that determines their differential abundance This is not to deny that the forms taken by organisms and their parts contribute to the stability of their life cycles in particular habitats which is what is addressed by natural selection It is simply to note that an analysis of this dynamic stability of life cycles can never be complete without an understanding of the generative dynamics that produces organisms of particular forms because their intrinsic stability properties may play a dominant role in determining their abundance and their persistence The objective is not to separate these different aspects of life cycles but to unify them in a dynamic analysis that puts natural selection into its proper context it is in no sense a generator of biological forms but it maybe involved in testing the stability of the form and weeding out unstable forms that weaken the firmness of an organism Before we are going to determine the promorphs that are involved in these three types of phyllotaxis we first consider briefly a comparison between the described process generating radiate organic structures radiate in a broader sense i e not only single whorls according to these three types of phyllotaxis in higher plants and the process that generates single radiate crystals As we have seen the appearance of radiate structures in higher plants is the result of a dynamical system which has many potential trajectories to follow Which trajectory will actually be followed depends on the parameter settings of the system where these settings could ultimately come from the genes and initial conditions Indeed the emergence of radiate structures in plants to which we recken the structures that result from all three types of phyllotaxis is what comes naturally A medium initially possessing a high degree of symmetry becomes unstable when growth leads to the generation of unstable initial morphologies which are therefore transient morphologies or unstable initial physiological states which means that even very small perturbations will shove off the system to a new morphology new structure that represents a stable configuration under the circumstances This new morphology generally has a lower symmetry than that of the original morphology In phyllotaxis the ultimate elements that are going to interact with each other resulting in a stable spatial distribution of them viz a phyllotaxis of leaves sepals petals etc as they are attached on a main axis and thus forming a radiate structure come from within the growing plant i e they are generated in and by the meristem while of course ultimately coming from without as nutrient material One by one or cluster by cluster they originate from a central well so to say and then migrate outward and repel each other or in other words they dynamically interact They keep so interacting as long as they are under tension i e as long as they are being pushed and pulled around by each other Ultimately these elements will settle onto a stable pattern where all tension has vanished This pattern is apart from totally irregular flowers and apart from bilateral flowers a radial structure regular or amphitect And it is on these strucures that we will concentrate when comparing organic pattern formation with inorganic pattern formation See next Figure Figure above A species of spiral phyllotaxis namely 8 13 phyllotaxis Successive dots representing leaves or their primordia arranged at angles of 137 5 0 to each other along a tightly wound spiral not shown naturally fall into two families of loosely wound spirals that are immediately apparent to the eye Here there are 8 spirals in one direction and 13 in the other consecutive Fibonacci numbers After STEWART I 1995 Nature s Numbers The above structure which is the result of 8 13 phyllotaxis in the individual development of some higher plant clearly is a regularly radiate structure In fact however we see two radiate structures i e two interlaced radial structures One of them has C 8 symmetry while the other has C 13 symmetry As the Figure suggests we could say that the elements of the structure representing for example the florets of a giant sunflower or the scales of a pineapple are organized according to a deformed lattice while in itself it is definitively not a true lattice because absolute periodicity is absent it is a distorted lattice In addition to it being so distorted it is also more or less folded it is a disorted folded two dimensional lattice or net See Figure above while that of real crystals is three dimensional and not folded and intrinsically not distorted in any other way This more or less folded two dimensional net of spiral phyllotaxy represents one layer of elements as the only layer of elements that is present And this layer when interpreted as a physical layer is of course three dimensional In crystals we have a regular stacking of a great many of two dimensional nets And also here when such a net is interpreted as a physical layer of elements it is three dimensional Further we should add that the lattice according to which sunflowers pineapples branches having their leaves according to spiral phyllotaxis etc are structured is macroscopic while the lattice of crystals is microscopic In crystallization as representative for inorganic pattern formation something similar as described for the formation of organic whorled structures is going on An initially highly symmetric medium becomes unstable in virtue of supercooling or supersaturation It then is highly sensible to even small perturbations which will shove off the system to a new morphology a structure a morphology that represents a stable configuration of the system under the circumstances The symmetry of the resulting structure is lower than the initial symmetry and this new structure will either be radiate regular or amphitect or non radiate Isometric crystals Monoclinic crystals of the Domatic Class and Triclinic crystals We will concentrate on the former i e on those crystals that represent inorganic radiate structures And because we want a comparison with structures that emerge as a result of phyllotaxis in plants we only consider those radiate crystals whose main axis is as it is in plants heteropolar Tectology of higher plants An individual higher plant is in fact a colony of lower organic individuals See for this Second Part of Website Tectology Series As seen from the whole plant which as such is a six order individual or cormus it is made up of a number of next lower individual entities namely the fifth order individuals called persons personae each of which is tectologically equivalent to a single non colonial animal Such a person itself is in turn built up of next lower individuals antimers metamers organs in the morphological sense and cells In higher plants there are two types of person viz asexual and sexual persons The former are just shoots and off shoots bearing leaves while the latter are flowers Some flowers as in the Family of Composites are themselves made up of a multitude of tiny flowers or florets as we see them in sunflowers Each such a flower is a person These florets are highly integrated resulting in one apparent flower which itself then is already a cormus now consisting of sexual persons only We could say that such a composite flower is a lower order cormus But in a way we also can regard it as a higher order sexual person When we do Promorphology of plants we consider asexual and sexual persons separately because the one and the other differ very much in structure shape and promorph So we consider either single shoots main shoot or off shoot single flowers or composite flowers as to their shape symmetry and promorph Promorphology of Plants with Spiral Phyllotaxis In our Promorphological System of Basic Forms Second Part of Website Basic Forms Heteropola Homostaura we considered structures made up by spiral phyllotaxis provisionally under the promorphological category of the Heteropola Homostaura regular single pyramids Now we shall reconsider and finally assess their true promorphological status A structure that is built up according to spiral phyllotaxis morphologically or geometrically for that matter consists of two interlaced families of spirals according to which elements leaves are while changing size repeated non periodically Mathematically such a structure is constructed by letting elements be repeated by having them stand apart by an angle of 137 5 degrees along a single tightly wound Archimedian spiral while dynamically such a structure is generated by successive appearance of elements at the center of the incipient structure elements that repel each other and as a system of elements expand Promorphology tries to express geometrically i e by using a geometric plane figure or solid the intrinsic symmetry and the number of antimers of an intrinsic material object or a part of it How can we geometrically express the intrinsic symmetry and the number of antimers of an intrinsic object that is made up according to spiral phyllotaxis To answer this question we must consider the object s structure from a morphological or geometrical viewpoint We then see the two interlaced families of spirals indicating that morphologically we have in fact to do with two structures that are in some way superimposed upon each other What structures Well it is clear that each composing structure in itself is a regular i e not amphitect gyroid structure having C n symmetry And this can promorphologically be represented by a regular gyroid pyramid the base of which is a regular gyroid polygon In say 8 13 phyllotaxy we have eight spirals going one way and thirteen going the other way So the promorph of this structure can be regarded as consisting of two interlaced regular gyroid pyramids one eight fold and one thirteen fold Generally a b phyllotaxy makes up a structure of which the promorph consists of two interlaced regular gyroid pyramids one a fold one b fold The next series of drawings consider the promorph involved in structures made up according to 5 8 phyllotaxy We then have to do with two regular gyroid pyramids one five fold one eight fold which can for the time being be adequately represented by their respective bases which are regular gyroid polygons one five fold and one eight fold See next Figures Figure above A five fold regular gyroid polygon representing a five fold regular gyroid pyramid The spikes emphasize its gyroid nature It represents one of the two components making up the promorph of an object whose structure is constituted according to 5 8 phyllotaxy And without auxiliary lines Figure above A five fold regular gyroid polygon And as a pyramid Figure above A five fold regular gyroid pyramid Oblique top view Figure above An eight fold regular gyroid polygon representing an eight fold regular gyroid pyramid The spikes emphasize its gyroid nature It represents the second of the two components making up the promorph of a structure that is made up according to 5 8 phyllotaxy And without auxiliary lines Figure above An eight fold regular gyroid polygon And as a pyramid Figure above An eight fold regular gyroid pyramid Oblique top view When superimposing the two polygons representing pyramids while retaining the auxiliary lines we get Figure above Superimposition of a five fold regular gyroid polygon onto an eight fold regular gyroid polygon each representing the corresponding gyroid pyramid resulting in the promorph of a structure or object that is made up according to 5 8 phyllotaxis The next Figures refine and complete the construction of this composite gyroid polygon Figure above Superimposition of a five fold regular gyroid polygon onto an eight fold regular gyroid polygon each representing the corresponding gyroid pyramid resulting in the promorph of a structure or object that is made up according to 5 8 phyllotaxis Figure above Superimposition of a five fold regular gyroid polygon onto an eight fold regular gyroid polygon each representing the corresponding gyroid pyramid resulting in the promorph of a structure or object that is made up according to 5 8 phyllotaxis Advanced stage of the construction Figure above Superimposition of a five fold regular gyroid polygon onto an eight fold regular gyroid polygon each representing the corresponding gyroid pyramid resulting in the promorph of a structure or object that is made up according to 5 8 phyllotaxis Final stage of the construction The plane composed figure as depicted here green blue as such represents the two dimensional analogue of the promorph of a three dimensional structure or object made up according to 5 8 phyllotaxy or equivalently it represents the promorph of a two dimensional figure made up according to 5 8 phyllotaxy The promorph two or three dimensional belongs to the Dihomogyrostaura penta kai octomera Dihomogyrostaura 5 8 mera itself a species of Homogyrostaura or regular gyroid polygons or pyramids Figure above Same as previuos Figure but now drawn as a slightly oblique top view of the corresponding composite gyroid p y r a m i d and as such representing the promorph of a three dimensional structure or object made up according to 5 8 phyllotaxy The promorph belongs to the Dihomogyrostaura penta kai octomera Dihomogyrostaura 5 8 mera Let us explain this remarkable promorph still further Recall that the promorph of some object is the geometric figure or solid that geometrically primarily by its shape fully expresses the intrinsic point symmetry of that object and its number of antimers In the present case it is a composite promorph consisting of two regular gyroid pyramids which can most conveniently be represented by their respected bases which are consequently regular gyroid polygons The first polygon is a regular gyroid five fold polygon representing a promorph that belongs to the Homogyrostaura pentamera Figure above Regular gyroid five fold polygon representing a promorph that belongs to the Homogyrostaura pentamera It has five antimers indicated by the colors blue green and yellow The second polygon is a regular gyroid eight fold polygon representing a promorph that belongs to the Homogyrostaura octomera Figure above Regular gyroid eight fold polygon representing a promorph that belongs to the Homogyrostaura octomera It has eight antimers indicated by the colors green and yellow When we now superimpose these two promorphs which here means that they should penetrate each other then we get the composed polygon already depicted earlier As such we cannot say that this promorph and its three dimensional analogue the composed pyramid and with it the material objects of which it is the promorph has a definite and single number of antimers Its number of antimers is in fact its number s of antimers because there are two such numbers involved in the present case five and eight We have an interpenetration of a regular gyroid five fold polygon representing as a promorph the Homogyrostaura pentamera with a regular gyroid eight fold polygon representing as a promorph the Homogyrostaura octomera The resulting promorph we then asses and indicate as belonging to the Dihomogyrostaura penta kai octomera where Dihomogyrostaura means two homostaura while penta kai octamera means having five plus eight parts The name of the category of the present promorph can also be written as follows Dihomogyrostaura 5 8 mera The structure as such i e the object that is structured according to 5 8 phyllotaxis and with it its promorph has no mirror lines planes no rotation axes except the trivial 1 fold axis and no center of symmetry which means that it is totally asymmetric and as such belongs to the symmetry group C 1 Only its constituents the two families of spirals have each for them selves positive symmetries namely respectively C 5 and C 8 symmetry Their interlacing destroys all symetry As for our example concerning 5 8 phyllotaxis and the promorph it implies the same goes for all cases of a b phyllotaxis So let us take a look at an object that is structured according to 8 13 phyllotaxy as we had it already depicted earlier and which is here again depicted with reversed colors Figure above An object that is structured according to 8 13 phyllotaxis Successive elements represented by dots are arranged at angles of 137 5 0 to each other along a tightly wound spiral not shown They naturally fall into two families of loosely wound spirals that are immediately apparent to the eye Here there are 8 spirals in one direction and 13 in the other which are consecutive Fibonacci numbers Adapted after STEWART I 1995 Nature s Numbers We can let the above Figure express an object of which the elements are arranged according to a two dimensional distorted but not folded net as we can see such an object in the case of a sunflower where the florets are arranged according 8 13 phyllotaxy in a more or less plane surface The next Figures depict the respective regular gyroid polygons viz the 13 fold and the 8 fold Figure above A regular gyroid 13 fold polygon which is a component of the promorph of the above 8 13 phyllotaxy object a promorph that consists of two interlaced polygons and of two corresponding pyramids in a three dimensional case The next Figure finishes the construction of this polygon Figure above Completed constriction of a regular gyroid 13 fold polygon which is a component of the promorph of the above 8 13 phyllotaxy object a promorph that consists of two interlaced polygons and of two corresponding pyramids in a three dimensional case The symmetry of this polygon and also of the corresponding pyramid is according to the Cyclic Group C 13 The next Figure shows that this regular gyroid 13 fold polygon can represent a promorph that belongs to the Homogyrostaura 13 mera Figure above A regular gyroid 13 fold polygon representing a promorph that belongs to the Homogyrostaura 13 mera It has 13 antimers green yellow blue The second component of the promorph of the above depicted object with a structure according to 8 13 phyllotaxy is a regular gyroid 8 fold polygon that was already depicted earlier and which can represent a promorph that belongs to the Homogyrostaura octomera as such already depicted above These two gyroid polygons now combine resulting in the composed promorph of our object a promorph that consequently belongs to the Dihomogyrostaura 8 13 mera or equivalemtly Dihomogyrostaura octa kai decakaitrimera Figure above Superimposition of a regular gyroid 8 fold polygon onto a regular gyroid 13 fold polygon resulting in a composed polygon that represents the promorph of our 8 13 phyllotaxy object The completed construction of this composed promorph is given in the next Figure Figure above Superimposition of a regular gyroid 8 fold polygon onto a regular gyroid 13 fold polygon resulting in a composed polygon that represents the promorph of our 8 13 phyllotaxy object Construction completed So indeed because our 8 13 phyllotaxis object Figure above consists of two interlaced structures of which the respective promorphs are Homogyrostaura octomera and Homogyrostaura 13 mera the promorph of the object as a whole belongs to the Dihomogyrostaura 8 13 mera Dihomogyrostaura octa kai decakaitrimera So with this promorph generally taken as Dihomogyrostaura a b mera or just Dihomogyrostaura we cover about 80 percent of the 250000 or so species of higher plants with respect to the promorphology of their asexual persons So from an organic viewpoint the Dihomogyrostaura are an important promorphological category One might wonder while inspecting our Promorphological System of Basic Forms Second Part of Website why we haven t promorphologically evaluated the above spiral phyllotaxis objects i e objects that have their structure built up according to spiral phyllotaxis especially as we see them in many asexual persons of higher plants i e shoots and off shoots bearing leaves according to this type of phyllotaxis as belonging to the Spiraxonia instead of to some species of Stauraxonia namely Dihomogyrostaura as we have done on the basis of the presence of a generative spiral The reason for this is that in Spiraxonia we consider the main body axis as being itself spirally wound and in addition to that at the same time forming a continuous spiral like we see in Ammonites many shell bearing snails etc In plant shoots having spiral phyllotaxis the body s main axis is not spirally wound but intrinsically straight Moreover the elements that form the spiral in plants are discrete We see separate leaves attached to the straight main axis of an asexual person climbing up in a spiral fashion according to the generative spiral So because of these facts straight main axis discretenes of elements we assign asexual plant persons with spiral phyllotaxis and also all other similarly built objects promorphologically to the Dihomogyrostaura Stauraxonia Promorphology of Plants with Whorled Phyllotaxis Whorled phyllotaxis or equivalently polystichous phyllotaxis is observed in the asexual persons of some higher plants and especially in many sexual persons i e single flowers Above left image it was depicted with respect to asexual persons of Fuchsia Let us consider this particular example The next Figure gives a schematic representation of how the leaves are arranged along the stem in the case of phyllotaxis as we see it in Fuchsia which is such that during growth two leaves appear at any one time These two leaves then form a two fold whorl Then a little higher up the growing stem another pair of leaves appears Because they are younger they are smaller Seen from above they fill in the gaps left open by the leaves of the first pair So we now have two whorls each consisting of two leaves which differ in size and orientation Then still higher up the stem a third whorl appears also consisting of two still smaller leaves This goes on until the shoot finally terminates in a flower So the result is that the asexual person consists of a stem that is provided by a series of alternate 2 fold whorls Here we schematically depict two such consecutive whorls Figure above Diagram of two consecutive 2 fold whorls top view of a plant with whorled or equivalently polystichous phyllotaxis In the present special case we have to do with tetrastichous phyllotaxis because there are four straight lines running up along the stem and connecting leaf attachment points The leaves A 1 and A 2 emerged from opposite points on the stem They reside at the same level in the vertical direction plant s root plant s tip A little higher up the leaves B 1 and B 2 emerge They are a little smaller than A 1 and A 2 and seen from above filling in the space between the latter B 1 and B 2 are with respect to each other at the same vertical level but at a different level than that of A 1 and A 2 If we want to delineate the antimers of the just described structure tetrastichous phyllotaxis we must consider the whole asexual person i e a whole shoot without flower of the plant consisting of a whole series of consecutive 2 fold whorls The cental axis of the shoot whose stem is a narrow cone is then its main body axis i e the axis in which the antimers of the shoot meet The next Figure indicates the domains of these antimers There are four of them Each such an antimer then consists of a vertical series of leaves and the corresponding longitudinal part of the stem decreasing in size as we go up along the shoot Figure above The domains of the four antimers 1 2 3 4 indicated in the above diagram of two consecutive 2 fold whorls top view of a plant with tetrastichous phyllotaxis Each antimer consists of a vertical series of leaves and the corresponding longitudinal part of the stem equally oriented but getting smaller as we go up along the stem to which these leaves are attached We can see that the four antimers are each for themselves mirror symmetric and are radially arranged From the above it is clear that such a shoot has D 2 symmetry while having four radially arranged mirror symmetric antimers So its promorph must then be according to the Autopola Orthostaura Tetraphragma radialia This promorph i e the geometric solid that expresses the mentioned features geometrically is depicted in the next Figure Figure above Promorph of an asexual person of a plant with tetrastichous phyllotaxis It is a rhombic pyramid with four radially arranged antimers here represented by its base which is a rhombus with four radially arranged antimers The antimers are indicated by the colors green and yellow And as a pyramid Figure above Promorph of an asexual person of a plant with tetrastichous phyllotaxis whorled phyllotaxis with 2 fold whorls It is a rhombic pyramid oblique top view with four radially arranged antimers The antimers are indicated by the colors green and yellow Radial and interradial planes indicated by red lines and by the symbols R and IR The two radial planes contain the main axis as their line of intersection and stand perpendicular to each other Whorled or polystichous phyllotaxis is especially common in flowers There we often have whorls consisting of more than two elements for instance 3 4 or 5 A single flower built according to polystichous phyllotaxis normally consists of four whorls a whorl of sepals a whorl of petals a whorl of stamens and a whorl of carpels The first two whorls and especially the second are the most conspicuous Sepals petals stamens and carpels must be considered to be transformed leaves The latter are their ground state Therefore we can still speak of phyllo taxis in flowers phyllous Greek means leaf Often the number of sepals is equal to that of petals where the petals when seen from above fill in the space between the sepals As an example we will consider a flower of the plant Arabidopsis thaliana as it is extensively discussed with respect to its formation by GOODWIN B How the Leopard Changed Its Spots pp 134 An individual Arabidopsis plant has in addition to leaves many flowers The latter are as its leaves arranged according to spiral phyllotaxis Each individual flower however has its components i e flower organs which are modified leaves arranged according to whorled phyllotaxis Such a flower consists of four whorls first a whorl of four sepals then a whorl of four petals whose positions are located in the gaps of the sepals The third whorl consists of six stamens the organs that produce pollen while the fourth one is two carpels which fuse to form the gynoecium with a two chambered ovary the female part of the flower with the eggs The next Figure depicts all this schematically in the form of a diagram Figure above Diagram of the four whorls of a flower of Arabidopsis top view One should consider each whorl to be positioned at a different level as we go up along the stem despite the fact that these differences are small If we go up along the stem or in other words along the flower s main axis the order of these whorls is Sepals Petals Stamens Carpels When we look to the above diagram depicting the organization of an Arabidopsis flower we see that not all whorls have the same number of elements We have four sepals and four petals but six stamens and only two carpels Moreover while the first two whorls can promorphologically be assessed as Homostaura i e as regular pyramids the third and fourth whorls must be assessed as Heterostaura Autopola i e as amphitect pyramids It is therefore not very well feasible to bring the whole flower under one common promorphological category without losing important morphological information The symmetry of the flower as a whole must be assessed as D 2 symmetry because it possesses two perpendicular mirror planes with a 2 fold rotation axis in their line of intersection and no other symmetry elements The regularly four fold symmetry D 4 of the first two whorls cannot at the same time be the four fold symmetry of the whole flower because it is overuled by the presence of two other whorls having only D 2 symmetry So promorphologically the flower as a whole should then belong to the Heterostaura Autopola amphitect pyramids and the number of antimers is then as such unclear Consequently when doing promorphology it is better to consider the whorls separately except where some whorls share the same promorph In our present case we can consider the first two whorls four sepals four petals together The promorph of this pair of whorls is then unequivocally belonging to the Homostaura Isopola tetractinota

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  • General Ontology XXIXc
    But in each individual snow crystal we see Whatever the morphology of an extension is this morphology is repeated six times around the 6 fold axis Sometimes we see snow crystals with only three extensions and they need special crystallographic studies in order to understand them which is done in Part XXIX Sequel 19 Sequel 20 Sequel 21 and Sequel 22 Perhaps H 2 O is crystallographically polymorphic even with respect to just snow Because naturally grown crystals never experience a perfectly uniform growing environment they always show irregularities with respect to their intrinsic symmetry That s why the six outgrowths are seldom precisely identical Instead they are similar but very similar indeed So intrinsically these six outgrowths are precisely identical The next Figure depicts a hexagonal two dimensional crystal itself not necessarily a 2 dimensional projection of a snow crystal which is in certain respects well comparable with our two dimensional version of a snow crystal or we can say with a projection onto the equatorial plane of a three dimensional snow crystal The crystal is provided with fictitious atomic motifs that have the symmetry and distribution compatible with the plane group P6mm and with the point group 6mm D 6 Of course as such this crystal has an extremely small size but this is immaterial for the present discussion These motifs should not be confused with inclusions of all kinds as seen in snow crystals because these inclusions are macroscopical The motifs and their distribution express the two dimensional crystal s internal symmetry P6mm and also its point symmetry 6mm Figure above Two dimensional hexagonal crystal belonging to the plane group P6mm and to the point group 6mm or equivalently to the D 6 symmetry group The motifs having themselves D 3 symmetry and their distribution express the crystal s internal symmetry and also its point symmetry As such this crystal is comparable with a projection onto the equatorial plane of a tabular snow crystal The next Figure depicts the hexagonal point lattice indicated by lattice connection lines of the just depicted two dimensional crystal Figure above A pattern according to the plane group P6mm from which the just depicted two dimensional crystal can be constructed by deriving one crystallographic closed Form consisting of six equivalent faces each representing an equivalent atomic aspect Point lattice indicated by connection lines Each rhombic lattice mesh contains two D 3 motifs s str that are rotated 60 0 with respect to each other The pattern must be imagined to be extended indefinitely over the plane The next Figure gives the set of symmetry elements of the above depicted two dimensional crystal Figure above Symmetry elements of the two dimensional hexagonal crystal of the Figure above belonging to the plane group P6mm and to the point group 6mm or equivalently to the D 6 symmetry group It has six mirror lines and one 6 fold rotation axis at their point of intersection Figure above Each atomic aspect that can as such be presented to the growing environment of the crystal and which represents a possible crystal face is translationally being repeated throughout the crystal i e given some particular atomic aspect we see copies of it parallel to it This holds for all possible atomic aspects of the crystal One such atomic aspect is highlighted yellow coloring of relevant parts of the motifs dark blue lines As we compare our two dimensional crystal previous Figures with a tabular snow crystal with six similar outgrowths we see that it does not possess real and unique macroscopical boundaries of or intermediate regions between some equivalents of the outgrowths This is illustrated by the next three Figures Figure above Two dimensional crystal of the previous Figures It doesn t show genuine boundaries or intermediate areas between radially repeated structures as we do see them in many snow crystals that have outgrowths Figure above Also in alternative positions there are no genuine boundaries in the two dimensional crystal of the previous Figures that could suggest the radial repetition of six similar or equal parts This despite the fact that the suggested boundaries do not cut through motifs as the next Figure shows Figure above Same as previous Figure The suggested boundaries are not unique boundaries because each of the six boundaries is repeated by parallel copies as is shown for one such boundary The next Figure depicts snow crystal 5 and shows that the morphology of an outgrowth is not translationally repeated and is thus not comparable with an atomic aspect Figure above Snow crystal 5 from the series of snow crystals depicted earlier The morphology of the outgrouwths is not translationally repeated i e the crystal does not contain parallel copies of its outgrowths Above we depicted the hexagonal point lattice on which the two dimensional hexagonal crystal was based The plane group symmetry was according to the group P6mm This symmetry was accomplished by placing two D 3 motifs into each rhombic lattice mesh motifs that are rotated with respect to each other by 60 0 But precisely the same plane group symmetry can also be obtained by placing a D 6 motif at each lattice node of that same hexagonal point lattice Figure above Pattern representing the plane group P6mm The motifs s str blue are interpreted as D 6 structures placed at the lattice points i e having their centers coincide with the lattice points The pattern must be imagined to extend indefinitely over the plane Let us analyse all the above concerning the status of the six similar and symmetrically arranged outgrowths in the above depicted snow crystals more closely The structure of a crystal is Point lattice plus chemical filling in This in fact means that certain chemical units are arranged according to a certain lattice which itself does not belong to the actual crystal structure but indicates the arrangement of the mentioned chemical units resulting in the possibilitiy of distinguishing a unit cell that is periodically repeated The chemical motifs have a certain morphology and a certain symmetry The Space Group 3 dimensional or Plane Group 2 dimensional is Point lattice plus chemical filling in with motifs only considered with respect to their symmetry When we eliminate all translational elements from the Space Group or Plane Group symmetry we obtain the corresponding point group symmetry So the different features of crystals relevant to our discussion are related to each other in the following ways The Point Group is the translation free residue of the Space Group or Plane Group symmetry But also objects without periodic internal structure and thus not belonging to whatever Space Group or Plane Group can have a symmetry according to a Point Group which in fact means that the object s symmetry operations reflection rotation etc are such that at least one point remains in place i e it remains where it was before any such operation is applied to the object The intrinsic shape of a crystal is determined by the pattern of growth rates which in turn are determined by the possible atomic aspects of the crystal that can be presented to the growing environment An atomic aspect can be symmetric or asymmetric but this does not influence the intrinsic shape of the crystal When it is asymmetric it influences the point symmetry by eleminating a mirror reflection The geometric symmetry of a crystal is the symmetry of its intrinsic shape The crystallographic point group symmetry of a crystal is either the same as its geometric symmetry or lower and in that case it is according to a genuine subgroup of the group describing the geometric symmetry of the crystal This crystallographic point group symmetry point symmetry of the crystal is in fact the symmetry of the pattern of the crystal s possible atomic aspects presented to the growing environment in combination with the symmetry of each of these aspects themselves These aspects represent possible crystal faces Same aspects imply same growth rates under uniform growing conditions concentration of nutrient material temperature pressure etc So the crystallographic point symmetry of the crystal is determined by its atomic aspects which in turn are determined by the crystal s internal structure But these aspects are in fact not morphological units but some chemical state of affairs connected with certain directions within the crystal That s why any particular atomic aspect is repeated translationally within the crystal i e it has many parallel copies within the crystal and is thus not a unique or only radially repeated morphological unit See Figure above So the crystallographic point symmetry of a crystal is not a morphological symmetry as we see it morphological symmetry in o r g a n i s m s or their parts for instance flower whorls Radiolarian skeletons jelly fishes star fishes etc A particular morphological point symmetry of a crystal say it is a D 6 symmetry is always the same as its crystallographic point symmetry and consequently also D 6 symmetry because the relevant morphological units expressing morphological point symmetry are repeated wholly in accordance with the crystal s crystallographic point symmetry However these morphological units are macroscopical substructures of the crystal and are only repeated r a d i a l l y and in this way expressing morphological point symmetry i e they are in any case not translationally repeated As an example of the mentioned morphological macroscopic unit determining the point symmetry of the object also to be morphological point symmetry we can mention outgrowths as in hopper growth in crystals that underwent rapid growth under uniform conditions We could then be tempted to visualize a crystallographic Form of a crystal i e a set of equivalent faces also as such a morphological unit But this would be incorrect because the crystal face is in fact a certain atomic aspect that could as such be presented to the growing environment and we know that this aspect is not only radially repeated for instance as the six sides of a regular hexagonal prism but also translationally so One such individual atomic aspect is part of a whole set of parallel aspects within the crystal while another individual version of this same aspect in the same crystal but now making an angle with the former also is part of a whole parallel set of aspects i e of another set making an angle with the first set But these sets completely interpenetrate each other resulting them not to be independent structures and thus not representing independent macroscopical morphological units See next Figure Figure above Interpenetration of atomic equivalent aspects shown for two of them Each atomic aspect that can as such be presented to the growing environment of the crystal and which represents a possible crystal face is translationally being repeated throughout the crystal i e given some particular atomic aspect we see copies of it parallel to it The Figure shows two out of six equivalent atomic aspects of the crystal interpenetrating One such individual atomic aspect is highlighted by yellow coloring of relevant parts of the motifs and dark blue lines while the other equivalent aspect is highlighted by dark blue coloring of the relevant parts of the motifs and dark blue lines Well such a morphological point symmetry is clearly expressed in the complex in contrast to simple snow crystals i e in snow crystals possessing large similar outgrowths or extensions from some central region We have depicted such crystals above crystal 2 9 They show hexastichous phyllotaxis although having only one whorl Clearly they have six antimers based on macroscopic structures like it is the case in all organisms that have antimers As has been said the crystallographic point symmetry is the symmetry of the pattern of atomic aspects plus their symmetries Above we considered a two dimensional hexagonal crystal having D 6 point symmetry 6mm and P6mm plane group symmetry The atomic aspects as represented by the faces were each for themselves mirror symmetric The crystal in terms of its set of faces can be seen as constructed by such a symmetric atomic aspect that is radially repeated six times Hence the crystallographic point symmetry is D 6 six mirror lines one 6 fold rotation axis Let us now consider the case of a two dimensional crystal that also has an intrinsic shape according to a regular hexagon and thus also consists of six equal atomic aspects related to each other by a 6 fold rotation axis but which atomic aspects are not mirror symmetric We then see that this entails that the crystallographic point symmetry of the crystal is not D 6 but C 6 See next two Figures Figure above A two dimensional hexagonal crystal having a crystallographic point symmetry according to the group C 6 crystallographically denoted by 6 and a plane group symmetry according to the group P6 The fictitious motifs each have a symmetry according to the group C 3 Their arrangement however ensures that the point symmetry of the crystal is C 6 The 6 fold rotation axis as the crystal s only symmetry element with respect to its point symmetry is indicated The next Figure indicates the atomic aspects representing the faces of the just depicted crystal These aspects are not mirror symmetric causing mirror lines to be absent Figure above The six asymmetric atomic aspects highlighted by yellow coloring of parts of motifs of the above depicted two dimensional hexagonal crystal point symmetry C 6 plane group symmetry P6 Because of the asymmetry of the atomic aspects the crystal does not possess mirror lines So now we have clearly seen that the crystallographic point symmetry of a crystal depends on the pattern of atomic aspects of which their own symmetry must also be taken into account The nature of the atomic aspects as presented to the growing environment determines the relative growth rates of the corresponding crystal faces So in a hexagonal D 6 crystal six equal atomic aspects as well in a hexagonal C 6 crystal also six equal atomic aspects we have to do with a regular pattern of six equal growth rates These are growth rates that are perpendicular to the corresponding crystal faces that have actually developed In addition to the mentioned six atomic aspects there are also other atomic aspects present They regularly alternate with the atomic aspects mentioned earlier But these atomic aspects represent faces that grow faster than the faces we ve just discussed which implies that these faster growing faces quickly grow themselves out of existence and are not present anymore in the fully grown crystal However these alternating atomic aspects still represent six equal growth vectors originating from the center of the crystal to its corners And when hopper growth takes place See Figure above illustrating a four fold case which occurs when the absolute growth rate of the crystal is very high the outgrowths that start from the corners of the hexagon or from the six edges of the hexagonal prism will grow with the same speed i e every outgrowth increases with a certain speed which is the same for all six of them So what we expect to see in a very fast growing snow crystal is the emergence of six outgrowths because there are six corners and it is at corners where outgrowths will emerge in such a fast growing crystal And especially we will expect that these outgrowths when the growing medium is uniform have the same spatial extensions But what we cannot expect is a same m o r p h o l o g y of them i e a repetition of six extensions outgrowths that are also equal morphologically Instead we would expect that snow crystals like the following can occur Figure above A fictitious snow crystal that could be expected to occur if the outgrowths solely depended on equal growth rates We see six outgrowths that have about the same spatial extension but which show different morphologies In fact we see three different morphological types of outgrowth destroying the crystal s morphological point symmetry But in nature we do not encounter such snow crystals We only see crystals where not only the spatial extension of the outgrowths is equal but also their entire morphology So this degree of equality of the outgrowths does not solely depend on equal growth rates and thus not solely on a pattern of atomic aspects which in turn means that the complete equality of the outgrowths does not solely depend on the crystallographic point symmetry as defined above of the crystal What then is the extra factor that co determines this morphological equality of the outgrowths Well this factor seems to be of a g l o b a l nature with respect to the space that the crystal finally occupies and thus not of a local nature It must be some restoring force that begins to operate when some fixed form equilibrium is broken by the appearance of an outgrowth on one of the corners Or differently expressed the six outgrowths emerge simultaneously while constantly keeping in touch with each other as to their morphology in such a way that a same morphology develops in all six of them What we see in such snow crystals we also see in organic forms Consider for example your right and left hands They are different from everyone else s in both the pattern of lines on the palms and the pattern of ridges on the finger tips Yet they are very similar to each other i e the right and left hands of one person just like the outgrowths of an individual snowflake are similar to each other This suggests that within the developing organism or a growing crystal some morphic resonance takes place between similar structures i e there is some sort of correlation between their development See SHELDRAKE R The Present of the Past 1988 p 132 It is as though this correlation guarantees the symmetrical development of structures or in other words it ensures the maintenance of some form equilibrium Concerning snow crystals there seems to be some long range connection between the developing outgrowths And this connection is not just some sort of connection but a very subtle one because it involves not just some quantitative feature to be repeated but a complex morphology i e the complex morphology of an outgrowth of a snow crystal We could surmise that a given particular morpholgy of such an outgrowth depends on some specific initial condition at the site of the crystal where the outgrowth starts to grow and that such a specific condition is symmetrically present at all six corners of the initial hexgonal plate of the growing snow crystal in virtue of the intrinsic point symmetry of that hexagonal plate But as we will show below there is an almost infinite number of different outgrowth morphologies to be seen in different snow crystals So when such different morphologies of outgrowths only depended on different intial conditions at the sites of their origin these morphologies should then be utterly sensitive to such initial conditions i e already very small differences of such conditions must result in different morphologies They do not relate to gross meteorological differences because the star shaped snow crystals all occur in the front portion of the snowstorm i e they are not scattered all over it and so over many different meteorological conditions temperature saturation etc But the condition within such a portion of snowstorm within which the growing snow crystals tumble about is likely to be very uniform at least on average Moreover infinitesimal differences in the mentioned initial conditions are already morphologically accounted for because the six outgrowths on a single snow crystal are seldom exactly alike So within the front part of a particular snowstorm we could expect that all the snowflakes bear similar outgrowths if we go from snowflake to snowflake because of the mentioned uniform conditions with minor differences due to corresponding minor differences in conditions But this is not observed As far as my knowledge goes wherever looked for no two star shaped snowflakes are found to be semi identical All are significantly different as to the morphology of their outgrowths The morpholgy of an outgrowth cannot be sensitive to already very small differences of conditions because these very small differences must already be present at the six corners of the hexagonal plate which are in the case of rapid growth expected to contain several crystallographic defects while the outgrowths of any particular snow crystal are still very similar In fact we should maintain that the appearance of one or another morphology of outgrowth does nevertheless depend on very small differences in conditions crystallographical and meteorological which is why there occur so many different morphologies but that the determining influence of these differences of conditions are within any particular individual snow crystal o v e r r u l e d by a h o l i s t i c f a c t o r that safeguards some form equilibrium So although very small differences of local conditions determine a particular outgrowth morphology the mentioned holistic factor guarantees that that particular morphology will be present in all six outgrowths of that crystal At the site of another crystal individual slightly different conditions prevail resulting in a different outgrowth morphology but again form equilibrium is maintained by distributing this same morpholgy to all other outgrowths of that crystal As far as I know no such factor is found in physical or chemical investigations We must take into account that this is a genuine holistic factor which will undoubtedly also play a role in the individual development of organisms Such a factor which is global in character not local connects the parts of the crystal i e it keeps all the parts more or less informed about each other And in terms of the Theory of the Implicate Order such a holistic connection cannot take place in the Explicate Order The parts of the crystal including its remote parts are connected to each other via the Implicate Order Why this is so we cannot tell but some facts point to it In fact any discovery of a holistic aspect in nature points to the existence of some irrational element in Nature which means that there are certain absolute limits as to how far Natural Science can proceed and succeed to uncover the structure of Reality It is now time to discuss the snowflake not only in terms of some analogous two dimensional crystals that while possessing the symmetry of a projection of the snowflake D 6 have fictitious microscopic motifs but in terms of the real structure of solid H 2 O under normal conditions To begin with an ice crystal and thus also a snow crystal consists not of a periodic arrangement of ions in the usual chemical sense of electrically charged atoms or atomic complexes or of neutral atoms but of periodic arrangements of molecules Such a water molecule consists of three atoms viz one oxygen and two hydrogen atoms Almost if not all anomalies of water especially its high boiling point and the fact that ice is less dense than liquid water are caused by the fact that the two hydrogens are not neatly placed at either side of the much larger oxygen but making an angle with each other of 104 5 0 Why this is so is explained by Quantum Mechanics Without loosing anything important let us paraphrase this explanation See BALL P H 2 O A Biography of Water 1999 The true shape of water is not a V but approximately a tetrahedron A regular tetrahedron is a mathematical regular solid consisting of four faces that have the shape of an equilateral triangle The water molecule looks kinked because we are seeing in a diagram of the molecule only two of the four corners of the tetrahedron The other two are occupied not by atoms but by pairs of electrons called lone pairs These are electrons from the oxygen atom s complement which don t partake in the bonding between the molecule s atoms but which nevertheless have to go somewhere They pair up as electrons in atoms are wont to do and take up residence about as far from each other and from the hydrogen atoms as they can get The tetrahedral arrangement affords the greatest distance between each of these four entities the two hydrogen atoms and the two lone pairs If it was a perfect tetrahedral arrangement the angle of the kink i e the angle between the two hydrogen atoms would be 109 5 0 The difference between that and 104 5 0 which is actually observed is the consequence of the slightly stronger aversion of the lone pairs for each other than for the hydrogen atoms so that the latter are pinched together BALL 1999 Ibid p 155 156 Well generally some atoms hold on more tightly to the bonding electrons than others In its greed for electrons oxygen is surpassed only by fluorine So in the water molecule oxygen hogs the electrons As a result the oxygen atom in a water molecule acquires a negative charge and the hydrogens are left positively charged It has been proposed that these charges in the water molecule give rise to an electrical force of attraction between neighboring molecules in which the hydrogen atoms of one molecule point towards the oxygen atom of another This attraction can be regarded as a kind of chemical bond about ten times stronger than the van der Waals forces that hold regular liquids together but ten times weaker than the bonds that link hydrogen and oxygen atoms into discrete molecules This is called a hydrogen bond BALL 1999 Ibid p 157 The hydrogen bond is really a bond between a hydrogen atom and a lone pair of electrons This means that a water molecule can form four hydrogen bonds the molecule s two hydrogens form two bonds with neighboring oxygens while the molecule s two lone pairs interact with neighboring hydrogens In fact there is in this bond in addition to the electrostatical element also an element of electron sharing as the latter is the case in covalent bonds So water molecules attract each other more strongly than in most other liquids And this hydrogen bond also links the water molecules together in ice crystals formed when temperatures are low There these molecules form a regular hexagonal network i e they form the crystal lattice of snowflakes and other forms of ice For all this see next Figures Figure above Water molecules have an affinity for one another through the hydrogen bond dotted A positively charged hydrogen nucleus which belongs to one oxygen atom can still be attracted to the negatively charged electron cloud of a nearby oxygen atom in another water molecule In ice this attraction causes the molecules to form a crystalline array similar to the structure of diamond but not as strong This very open structure gives ice its low density and explains why ice floats on water After GRIBBIN J In Search of the Double Helix 1985 1987 REMARK Above we stated that the crystallographical point symmetry of ice or snow crystals is 6 m 2 m 2 m i e the most symmetric Class of the Hexagonal Crystal System as it is given by NESSE W Introduction to Mineralogy 2000 I am not sure how well established this is because there are snow crystals with only three fold symmetry suggesting a lower symmetry Class of the Hexagonal System In x ray studies only the oxygen atoms seem to count BENTLY HUMPHREYS Snow Crystals 1931 But because oxygen atoms are much larger than hydrogen atoms the latter may be disregarded which means that only the arrangement of oxygen atoms creates the symmetry of the ice or snow crystal Figure above Left image a The water molecule is bent with the two bonds between oxygen and hydrogen splayed at an angle of 104 5 0 These bonds are normal chemical bonds Relative sizes of atoms of atoms are in reality different than displayed here The oxygen atom is much larger than the hydrogen atom Right image b In fact a water molecule forms approximately a tetrahedron the four corners of which can form bonds hydrogen bonds with other tetrahedra i e with other water molecules The four corners consist of two hydrogen atoms H at one side of the oxygen atom large sphere and two lone pairs of electrons at the other side At the molecular scale the structure of water is imprinted with this tetrahedral geometry After BALL P H 2 O 1999 Figure above Ice structure viewed down the c axis i e down the 6 fold rotation axis of a single crystal Water molecules consisting of one oxygen large sphere and two hydrogens small spheres are arranged in a hexagonal array Hydrogen bonds between water molecules are shown with thinner lines The positions of the H hydrogen and O oxygen are shown accurately but the sizes of the atoms are not See next Figure After NESSE W Introduction to Mineralogy 2000 Figure above Hydrogen bonding in ice Image a A polar H 2 O molecule is formed by covalent ionic bonds between H and O using 2p orbitals on O See for the covalent bond and the ionic bond our essay The Chemical Bond in First Part of Website The different types of chemical bond can be mixed up in a single such type The O has greater claim on the electrons so the H consists of little more than its positively charged nucleus a single proton Charge is tetrahedrally distributed positive at the H atoms and negative at nodes on the opposite side of the O Image b Each negative node on the molecule attracts a positive node an H atom on an adjacent H 2 O molecule to form hydrogen bonds that hold the water molecules together partially and changeably in liquid water totally and constant in ice Two crystallographic unit cells are outlined in the ice structure with dashed lines The top face and also the bottom face of the unit cell has the shape not of a square but of a rhombus with angles of 60 0 and 120 0 The crystallographic c axis runs parallel to the vertical edges of the unit cell After NESSE W Introduction to Mineralogy 2000 Above we have spoken about snow crystals that possess six virtually equal outgrowths Although sometimes these outgrowths are a bit dissimilar their strong similarities in most other cases stand out We have indicated that it could be hard to provide a local explanation for this phenomenon i e an explanation solely based on local interactions of crystal constituents Instead we probably have to admit a global factor at work ensuring the symmetry of the whole crystal especially its morphological symmetry which is its crystallographic symmetry but now also expressed in all macroscopic details of the crystal We have spoken about the fact that in snow we encounter an almost unlimited number of different morphologies of these outgrowths on different crystals but almost always the same morphology within a single crystal The next Figure shows some of these different morphologies actually found in snow Figure above A selection from the huge set of actually occurring different outgrowth morphologies in snowflakes Each image shows only one of the six outgrowths For a larger version of this Figure showing more details click HERE Taken from microphotographs of whole snow crystals in W A BENTLEY and W J HUMPHREYS Snow Crystals 1931 1962 We have discussed the high degree of similarity of the morphology of the six outgrowths of a snow crystal as something not fully understood by natural science I can however imagine that the reader cannot believe this because it looks so much to be just a trivial problem while moreover I am not a professional chemist or physicist So let me quote the natural scientist Philip BALL in his book H 2 O A biography of water of the year 1999 p 177 178 when discussing branched snowflakes comments between square brackets A continuing mystery about dendritic snowflakes is why all six of their branches seem to be more or less identical The theory of dendritic growth explains why the side branches will develop at certain angles but it contains no guarantee that they will all appear at equivalent places on different branches i e at corresponding locations on the different outgrowths of the crystal causing their s h a p e their morphology to be the same or the theory of dendritic growth contains no guarantee that the side branches will grow to the same dimensions The sameness of these dimensions of at least the main branches is more or less guaranteed by the equal growth rates implied by equal atomic aspects Indeed these branching events are expected to happen at random Yet snowflakes can present astonishing examples of coordination as if each branch knows what the other is doing One hypothesis is that vibrations of the crystal lattice bounce back and forth through the crystal like standing waves in an organ pipe providing a degree of coordination and communication in the growth process Another is that the apparent similarity of the arms is illusory a result of the spatial constraints imposed because all the branches grow close together at more or less the same rate But for the present the secret of the snowflakes endures We can elaborate a little further on the appearance of outgrowths on a snow crystal i e as to their general structure that they all in all star shaped snow crystals have in common One such general feature is the fact that the side branches that shoot off from the main axis of an outgrowth do this by making an angle of 60 0 with this main axis The next two Figures try to give more insight into this phenomenon Figure above Growth of a hexagonal crystalline plate blue only a part of it shown by increments yellow green Figure above Formation of an outgrowth blue at a corner of a hexagonal crystalline plate blue in the context of crystal growth by increments like in the previous Figure but smaller These increments as representing possible crystal faces are indicated by x y z etc and a b c etc The Figure is not meant to present a full explanation of the appearance and general morphology of an outgrowth but is only meant to give some indications that could stimulate further pondering It probably gives some hint why the angles that the side branches make with the outgrowth s main axis are all 60 degrees Liquid H 2 O Snow crystals are made of water The water molecules are held together by hydrogen bonds The resulting crystalline structure is rather open When ice melts the structure is allowed to telescope inwards because the net of hydrogen bonds becomes partly broken up becomes more or less random and dynamic And because of this telescoping liquid water is more dense than ice The precise situation in liquid water is only partially understood Computer models indicate that most water molecules in liquid water have two or three hydrogen bonds and a relatively small proportion has four or none Clearly there is a significant rearrangement of hydrogen bonds when ice melts In particular whereas hydrogen bonding in ice links each water molecule into rings of six the most common ring structure in liquid water contains five molecules not six And while each vertex of the ice network is tetrahedral the confluence of two hydrogen bonds departing from two hydrogens themselves covalently ionically bonded to an oxygen atom and two hydrogen bonds in which the lone pairs of electrons of that same oxygen atom participate in liquid water

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  • General Ontology XXIXd
    Implicate Order i e it having more dimensions than the Explicate Order are shown in the above diagram Other but related effects of multidimensionality are shown in the next Figure above Fish tank analogy of the Implicate Explicate Order with respect to the multidimensionality of the Implicate Order with respect to the Explicate Order As given by David BOHM 1980 The Implicate Order is here represented by three dimensional space while the Explicate Order is represented by two dimensional space One and only one fish i e a three dimensional fish is brought into a fish tank Two TV cameras A and B are pointed to the interior of the fish tank under an angle of 90 degrees To each camera is connected a TV screen Both TV screens are placed side by side in another room i e another room than the one in which the fish tank was placed When looking at the two TV screens we see two similar fishes moving about independently However on closer inspection we see that their movements are highly correlated i e the movements of the one fish in some way correlate to the movements of the other So there must be some connection between the two fishes And in fact this is the case because our two fishes are just two dimensional projections of one single three dimensional fish We can let the two fishes stand for two p a r t s of some larger whole These parts then seem to be somehow correlated with each other like the six outgrowths of a snow crystal And the reason that they correlate whether behaviorally or morphologically is that they are enfolded into each other in some higher dimensional space This mutual enfoldment of the two 2 dimensional fishes results in one single 3 dimensional fish And this fish represents a holon We can use this analogy when thinking how parts of some whole can be mutually connected in such a way that when one is changed the other changes correspondingly and immediately But after such thinking the space aspect must be subtracted again because we assume that the Implicate Order has no extension i e no spatial aspect All the logic of extension including dimensions is however present in the Implicate Order but only in a noëtical form like the fact that all geometric features can be described algebraically The Implicate Order and the Virtual Existence of Parts or Elements of a Holon Because of its apparent irrationality the concept of virtual existence of parts or elements of an actually existing holon should be clarified still further So before we pick up the general discussion again we first try to make clear what this virtual existence precisely means and how precisely the Implicate Order is involved in this In First Part of Website we have argued that any individual true substance in its metaphysical sense or equivalently any intrinsic being is the result of interaction of certain elements i e certain initially given material entities an interaction according to a dynamical law immanent in this collection of elements and triggered into action by certain specific circumstances Such an intrinsic being say a crystal or an organism can be considered to be a unity in virtue of that one dynamical law which now constitutes its genotypical domain But with respect to its i e the intrinsic being s phenotypical domain i e the visible product of the corresponding dynamical system of interacting elements it is not a strict unity but merely a collection of entities which are the mentioned elements now organized into a certain pattern This means that this intrinsic being is reducible to its elements i e to the particular set of entities elements harboring the dynamical law As such this intrinsic being is not a strict holon Within the general theory in the mentioned First Part of Website this was no problem But in the present discussion it is because here we have assumed on the basis of some more or less convincing arguments or indications that at least molecules single solid crystals and organisms are true holons i e entities that are not reducible to lower i e less complex entities or to their elements i e those entities that were the elements of the dynamical system that generated the molecule crystal or organism We then investigated how such a holon as holon must look like So now the alleged fact that any intrinsic being as the product of a dynamical system is reducible to the set of elements the interaction of which according to a dynamical law immanent in them lead to that being poses a problem In a given organism we see organs bones cells etc which are in all probability actual and discrete entities Alternatively however we could say that we see actual and discrete qualities or generally properties not of these entities organs bones etc but qualities directly of that organism as a whole causing the mentioned entities to be only virtually present in that given organism And it is this brand of virtuality which we above called an apparent irrationality In the present discussion we try to circumvent such a brand of virtuality by involving the Implicate Order The second reason to invoke the Implicate Order is the fact that even when we accept this apparent irrationality of virtual organs bones cells etc we still have to do with genuine distances between qualitatively determined sites within one and the same organism or crystal for that matter which in a way destroys its assumed strict unity And although organs bones cells etc seen as actual discrete entities are not identical to the very elements of the dynamical system that generated that given organism they can be seen as complex elements or derived elements or elements after the fact of the organism because it is constituted by these organs bones cells etc And if we now demand a strict unity of the given intrinsic being this demand based on certain features like the arms of snowflakes or holistic features in organisms then we must declare all directly or indirectly visible discrete parts of it not as being actually there but existing only virtually in that intrinsic being which means that some p r o p e r t i e s of those parts or elements are virtually preserved i e preserved in a virtual way because these parts or elements are themselves now virtual So while these properties are virtually preserved they are now as properties of the whole intrinsic being itself actually present and thus now appear not as genuine entities anymore but as actual q u a l i t i e s of that intrinsic being whether it be an organism or a crystal All this results in the strict oneness or unity of that intrinsic being But what precisely does this virtually existing of parts and elements mean And why we still see in the case of organisms at least discrete parts Well as already explained above we assume these parts or elements are enfolded into each other within the Implicate Order The discrete parts we see are nothing more than a projection from the multi dimensional Implicate Order onto the three dimensional Explicate Order One should realize in all this that the parts or elements as they are conceived of in the Implicate Order namely as enfolded into each other are not some sort of second version or duplication or copy of those same parts or elements as they are seen in the Explicate Order They are one and the same as the next diagram illustrates Diagram above A structure depicted as a zig zag line in the Implicate Order here depicted as 2 dimensional space is projected onto the Explicate Order depicted as a one dimensional space a straight line red The result of the projection is a collection of discrete points each of which stands for a discrete part of that collection This collection is as it is seen in the Explicate Order itself a certain being or thing because we assume that these points parts appear together not successively in the Explicate Order or in different words we assume that these parts have approximately the same implication parameter which means that their degree of enfoldment when still residing within the Implicate Order is approximately the same and also not much different from the other parts of the whole structure zig zag line but different from that of other structures The discrete parts as they present themselves in the Explicate Order aren t discrete as indeed the full picture reveals So these discrete parts aren t parts or elements afterall We call such parts or elements virtually existing parts or elements All these parts are not in any way separated from each other Only by cutting the whole the zig zag line into chunks we get discrete parts But then these are not parts of that whole because that whole doesn t exist anymore they are all by themselves now and thus no parts or elements at all Remark We can assume that a given structure or part of it when it is unfolded and thus then present in the Explicate order is also still present in an enfolded condition in the Implicate Order in such a way that we have to do with a constant unfolding and enfolding or equivalently with a constant projection and injection where and because of it the mentioned structure is constantly being projected and injected and thus as seen within the Explicate Order constantly being created de created and re created So such a structure is then intrinsically dynamic on a fundamental level See Third Part of Website MODEL OF ENFOLDING AND UNFOLDING The Ink in Glycerine Model One can see by the way how enormous the consequences are of the simple statement that certain complex things say crystals or organisms are ontologically not only epistemologically irreducible to their elements or to other lower i e less complex entities And such a statement we encounter in many many writings An intrinsic being or equivalently a mixtum perfectum or shortly mixtum which is the product of the interaction of elements of a dynamical system like what is a billion times taking place in a bulk chemical reaction is now ex hypothesi one entity and as such a c o n t i n u u m Nevertheless such a continuum is not necessarily homogeneous it can be heterogeneous and thus a heterogeneous continuum because of the following reasons Generally such an intrinsic being is generated from a set of different kinds of elements These elements end up at different locations within the resulting intrinsic being and become virtual Some properties of these elements are preserved after the intrinsic being has been generated resulting in a spatial mozaic of properties across that intrinsic being These properties are actual not virtual and as such they only can derive from actual natures or essences But this would imply that the intrinsic being consists of actual elements representing these natures denying it to enjoy strict unity So these properties are not preserved properties of the elements but just properties of the intrinsic being And these properties are very similar to the corresponding properties of the elements because the intrinsic being mixtum perfectum is of course very closely related to its elements So the elements of a mixtum perfectum or shortly mixtum are only virtually present in the mixtum because they are folded within the Implicate Order as explained above while those properties of the elements that are preserved after the mixtum has been generated are now properties not of these elements but of the mixtum resulting in a heterogeneous continuum which as such is a true holon Continuation of the general discussion about the existence of genuine Holons Also for molecules we must expect that they are such holons i e such heterogeneous continua Although they consist of atoms or ions they until this day do not seem to be reducible to those atoms i e they cannot again until this day be calculated from the properties of constituent atoms Even the simplest of all molecular systems the hydrogen molecule ion containing two protons and one electron presents insuperable problems SHELDRAKE 1988 p 122 Its properties can be calculated only by making a series of simplifying assumptions For complex molecules and crystals even more drastic approximations and simplifying assumptions have to be made in order to apply a mathematical analysis These calculations have given an increased understanding of some of the properties of molecules and crystals but this is a very different matter from predicting their forms and properties from first principles SHELDRAKE 1988 p 122 So it is not yet demonstrated that the structures of molecules or crystals are implicitly contained or potentially present in the set of constituent atoms And consequently all this could be an indication that molecules and crystals are non reducible entities i e holons Moreover for molecules and crystals chemical bonds are crucially involved which means that quantum conditions are imposed on them implying that molecules and crystals are in no sense mechanical entities Also for the atom we must conclude that it is a totality or holon not reducible to its constituents The hydrogen atom to be precise its lightest isotope for example consists of one positively charged proton and one negatively charged electron that is somehow associated with that proton If we have just a mixture i e an aggregate of individual and independent protons and electrons we can expect it to emit a continuous spectrum perhaps caused by emission of electromagnetic energy wherever the trajectories of these particles happen to be accelerated or curved in whatever rate or intensity If on the other hand we have a collection of hydrogen atoms we see its spectrum to be a discontinuous one i e a spectrum of discrete colored lines These spectral lines of the collection of hydrogen atoms are precisely predicted by a theory that is supplemented by BOHR s quantum conditions Several possible different energy states are divided i e distributed over the many individual atoms of the mentioned collection which explaines all the different lines in the spectrum The mentioned quantum conditions cannot be applied to an aggregation i e just a collection of protons and electrons because in the case of such an aggregate it is assumed that while calculating the energy Coulomb s electrostatic law and the mechanical laws are valid while at the same time this is denied by the quantum conditions So it is clear that when formally changing from an aggregate of protons and electrons to hydrogen atoms or from a proton and an electron to a hydrogen atom for that matter we must necessarily invoke BOHR s quantum conditions which transforms the atom into a non mechanical system And it is these quantum conditions that constitute the NOVUM when we change from proton electron aggregate to hydrogen atom And this means that the hydrogen atom is a non reducible entity a genuine holon or totality And such a totality is therefore a unity in the present case a heterogeneous continuum See HOENEN P in Dutch Philosophie der Anorganische Natuur Philosophy of the Inorganic Nature 1947 The quantum conditions are the folowing When the energy of the system that constitutes the hydrogen atom does change it does so not continuously as assumed in classical physics but discontinuously i e the energy is quantisized and the system shows discrete energy levels in which Planck s constant h plays an essential role The system that constitutes the hydrogen atom only emits light when the electron by whatever reason jumps from a higher energy level to a lower The electron not actually jumps in the sense of traversing intermediate states It does not traverse intermediate states It is either in this state or in that state where the latter discretely differs from the former In virtue of such a quantum jump energy is emitted that is equal to the difference between that of the two energy levels To invoke these quantum conditions was necessary because a classically conceived hydrogen atom cannot be stable Molecules consist of atoms that are connected to each other by chemical bonds See First Part of Website The Chemical Bond Many molecules involve so called covalent bonds described by the theory of orbitals and they surely entail quantum conditions So such molecules are like free atoms totalities or holons Crystals can be considered as giant molecules but now with a reticular or equivalently periodic structure where often also covalent bonds are present as in diamonds In other crystals however ionic bonds prevail which just consist of electrostatic attraction between ions as we see it in salt crystals NaCl Again in other crystals whole molecules are elements of a crystal This is the case in ice crystals H 2 O Here these water molecules are held together in the crystal by hydrogen bonds explained in the previous document Also here we have a case of electrostatic attraction Whether in ionic and hydrogen bonds quantum conditions are involved I am not sure but we must realize that generally these bonds at least covalent ionic and metallic bonds occur in mixtures i e occur in mixed forms in which one type of bond dominates in the given chemical bond I think it is not unreasonable to assume that in all cases of chemical bonds quantum conditions are involved implying that every chemical compound and every single solid crystal is a genuine totality or holon Organisms consist of a host of chemical compounds and ions Most of these are taken up into higher i e more complex structures and although chemical bonds can be involved in such higher structures this is generally not so And this means that if we consider organisms as holons they are so by other reasons capacity to regeneration to regulate to reproduce etc And so we could theorize that already within the Inorganic Physical Layer of Being there are genuine holons And each such holon involves a NOVUM as we change from their constituents their parts or their elements to that holon itself This NOVUM is a new principle a new category or If Then constant not to be found among those of the constituents And indeed the appearance of a new principle causes the corresponding concretum to be an i r r e d u c i b l e entity and thus a holon But it does not define a new ontological layer as is the case with the NOVUM connected with the living state It is a physical If Then constant which at the most co defines some sub layer within the Inorganic Layer And something comparable occurs within the Organic layer based on the presence of many irreducible living entities i e certain living entities that cannot be reduced to other simpler living entities As the many failed attempts of the historical derivation of one animal group from another show In the Implicate Order such a NOVUM those that are responsible for the emergence of a whole new Layer of Being as well as those that are not is represented by the corresponding holistic simplification as explained earlier And again within the Inorganic Layer we see protein molecules and bimolecular complexes of them that show a morphological structure that is not derivable from the given amino acid sequence of such a molecule SHELDRAKE 1988 p 123 127 So also these are true holons i e wholes not reducible to their parts or elements This non reducibility must be specified as follows If we go from free atoms to the corresponding chemical compound we can distinguish two general main kinds of properties of the compound with respect to its elements Aggregation resultants Totality resultants There are further two kinds of aggregation resultants Additive resultants Constitutive resultants An additive resultant is a property of the compound that results from adding up the corresponding properties of the elements An example is mass and if we want to press it mass energy which does not change before and after a chemical reaction and here we consider a reaction which results in a chemical compound from interacting atoms So the mass of the compound is equal to the sum of the masses of the atoms that constitute the compound A constitutive resultant is a property of the compound that is absent in its elements An analogous case can serve as an explanation what this means Pressure of a gas is a property that is absent from any individual gas molecule because this notion does not make sense for a molecule So pressure of a gas is a constitutive resultant If all properties of a compound are either additive resultants or constitutive resultants and if the constitutive resultants can in principle be calculated from the corresponding properties of the elements The additive resultants can of course always be calculated all these properties can then together be called aggregation resultants and the compound is then just an aggregate i e it is an entity that can be fully reduced to its elements If on the other hand at least one constitutive property cannot even in principle be calculated from the corresponding properties of the elements then this property should be called a totality resultant and the compound is then a totality or equivalently a holon which in virtue of its being a holon involves as seen in the Explicate Order a NOVUM because it is not fully derivable from its elements But although such a holon or totality generally does involve a new but special principle or equivalently a new category or If Then constant i e a special categorical NOVUM it does not entail a new main Layer of Being like we see when going from the Inorganic to the Organic which involves a more general NOVUM Like in the other cases we referred to the NOVUM that is associated with such a holon does not entail the modification of one or more other categories And as has been found out earlier although the structures of many kinds of crystals have been described in detail so that one exactly knows of what parts the crystal consists the ways in which such a crystal takes up its structure as it crystallizes is very obscure In the first place just as in the case of protein structures it is not possible to predict from first principles i e from the constituents as they are in themselves the way in which the molecules atoms or ions will pack themselves together in the crystal lattice Even with quite simple chemical compounds there are many possible lattice conformations that are expected to be equally stable thermodynamically and there is no clear reason why one rather than any of the others is actually taken up during crystallization And like in protein folding there is no way of empirically testing the assumption that the actual lattice structure is uniquely stable from an energetic point of view The molecules in the sense of the given chemical compound simply will not crystallize into the other theoretically possible lattice structure within the same range of thermodynamic conditions and therefore their energies cannot be measured and compared SHELDRAKE 1988 p 129 The second difficulty arises as has been discussed at length earlier in trying to understand the way in which the crystal grows as a whole Somehow as molecules in solution or in a vapor come close to the growing surface of the crystal they snap into place in the growing bulk crystal But the way in which they do this cannot be directly observed and attempts to model the process mathematically are still very crude and have not been very successful so far SHELDRAKE 1988 p 129 where with respect to this he cites MADDOX 1985 No pattern yet for snowflakes Nature 313 93 Such models take into account only local effects on the molecules atoms or ions joining the growing crystal But crystals each as a whole show patterns of symmetry and here we mean not only crystallographic point symmetry but also morphological point symmetry which cannot possibly arise from a sum of local effects Such symmetry we see very clearly in star shaped crystals especially in snow crystals as discussed earlier Snow crystals generally have a six fold symmetry but each is unique Within a snowflake which is a snow crystal the intricate structure of the six arms is very similar if not identical and these arms are themselves symmetrical according to the group D 2 one mirror plane coinciding with the equatorial plane of the whole crystal and a second mirror plane coinciding with one of the mirror planes of that same crystal which itself has a point symmetry according to the product group D 6 x C 2 while this symmetry is denoted crystallographically as 6 m2 m2 m Although the differences among snowflakes may be explained in terms of random variations the symmetrical development within each snowflake i e the six fold repetition of some complex morphology namely that of the arms cannot be explained in this way Lattice vibrations could be significant in this respect but whether they are specific enough with respect to the special morphologies as we see them in the arms is according to me not demonstrated Maybe these lattice vibrations resonate with certain enfolded structures in the Implicate Order and so establish a non local connection between corresponding sites of the growing snow crystal And such direct connections then guarantee the symmetrical repetition of morphologies within the crystal A Speculative Note on the Implicate Order It is possible that in the whole of Reality which here means not only in the Explicate Order but especially also in the Implicate Order there is a historical aspect present and therefore a contingent aspect If this is correct then it would seem that the Implicate Order is embedded within Time But that contradicts our initial assumption It is the Explicate Order where things appear as separated in time and or space because of the unfoldings that take place during projections from the Implicate order So the Implicate Order is timeless And changes in it in fact of it expressing its historical aspect will only be evident in the Explicate Oder If there have taken place certain projections followed by corresponding injections then the Implicate Order is not precisely the same anymore which can become visible in subsequent as seen in the Explicate Order projections If all this is correct then the Implicate Order is not a Platonic static transcendent domain of all possible one might say ideal structures or Forms which are only formally being derived from each other but is immanent in all explicate structures The structures in the Implicate Order are potentially present in the Explicate Order In given particular explicate structures some specific implicate structures are present in near potency And these implicate structures are timeless despite the fact that they are immanent in temporal structures The Explicate Order expresses i e explicates or unfolds the Implicate Order And only from within the Explicate Order we can see how things are in the Implicate Order But if this were all then the Implicate Order is with respect to its content what it is and remains what it is and so does not have a contingent aspect But this aspect is necessary to preserve the unity of the Implicate Explicate Order i e the unity of the world So there also must be events that originate in the Explicate Order like the application of a series of ink drops into the ink in glycerine model device as discussed in Third Part of Website And injection of these applied ink drops changes the Implicate Order A patterned series of ink drops applied to the model device is injected re projected re injected etc into and from the glycerine and could as such stand for say a crystallization event And if a crystal of some totally new substance is formed for the first time comparable to a certain definite and new pattern of applied ink drops the Implicate Order is changed because of the ensuing injections If we express all this in terms of morphic fields we can say that the newly appeared crystal i e the first crystal of a totally new substance creates a field A solution of the new substance becomes supersaturated and thus becomes unstable and finally finds a lower energy state in virtue of the packing of the corresponding particles into an array resulting in some stability of the system Injection of such an array changes the Implicate Order correspondingly which means the creation of a morphic field Subsequent projections which we can identify with resonances triggered by subsequent cases of supersaturated solutions of this new substance now result in the formation of this particular kind of array of constituents of the mentioned new substance again and again and thus the establishment of h a b i t i e a habit of that substance to always crystallize in that particular array within a given range of thermodynamic conditions And from now on crystallization of that new substance will get started easier because of resonance with the newly evolved morphic field If we adopt these speculations the theory of the Implicate Order is brought closer to the hypothesis of formative causation of Rupert SHELDRAKE first proposed by him in 1981 Let s elaborate a little further on these additions to the theory A certain array of packing of particles of the new substance representing a state of lower energy can be arrived at by the system and then in the sequel becomes a habit in virtue of the stabilization of the newly evolved implicate structure or field associated with this array of packing From now on the substance always easily and quickly finds this particular array of packing i e it crystallizes more readily However it must be emphasized that a true holon cannot emerge spontaneously from the system of its constituents for the first time when no morphic field has yet been formed because by definition a holon cannot be reduced to that system It is more than that system alone and this more is not yet present in the Implicate Order so no holistic simplification in this Order and the corresponding appearance of the NOVUM in the Explicate Order can yet take place But it can emerge for the first time by pure accident Such an accident is a one off event and so does not mean a reducibility of the result to the pre existent system of elements because this result did not emerge by intrinsic necessity It could have been co generated by additional extrinsic causes As such i e according to its history of generation this result is not a holon but its actual structure is the same as those entities which will come after it i e which will later be generated anew and which we will interpret as holons As soon as the habit of formation is firmly established by morphic resonance this particular kind of result now emerges with necessity from the initial not meaning the very first system of elements and represents from then onwards in every such case a holon because the necessity which would otherwise entail it not to be a holon but something that is fully reducible to its elements is a necessity via the Implicate Order where holistic simplification is taking place It is now a holon because its generation involves wholeness Of course a pseudo holon as it comes about for the first time never emerges completely by accident First of all a sliding off to lower energy states is for free and thus spontaneously and repeatabe Every system that is unstable will do that because it is always being perturbed and thus never remains in such an unstable state Often there is some small energy barrier to overcome before the system can plunge off into the lower energy state The barrier can be taken when the strength of the perturbation is sufficient If it is not then the system is metastable Further the system of interacting elements ultimately leading to a new pseudo holon will have many parts or substructures that will resonate with similar structures already present in the Implicate order implying that many aspects of the new holon have not emerged by accident Lastly we must realize that the accidental aspect comes wholly from without i e from fluctuations of the environment that are extrinsic and random with respect to the developing holon In a broader context these fluctuations are when they are not quantum fluctuations not random at all but completely determined Continuum or Tight Contiguum While according to our earlier considerations free atoms free molecules and single solid crystals seem to be heterogeneous continua and snow crystals and then with them all solid crystals seem to point to we could say an even stronger sense of continuum namely where the parts and elements are non locally connected organisms seem to be different although they can reasonably be interpreted as continua even in the just mentioned stronger sense Because to see all these entities atoms molecules crystals and organisms as continua of some sort is intuitively difficult but nevertheless very intriguing it is perhaps instructive to summarize and further elaborate on the notion of holon as heterogeneous continuum In his interesting book Philosophie der Anorganische Natuur 1947 referred to earlier HOENEN shows that the results of classical physics do not decide whether any given intrinsic being i e a being which has the specific causes of its generation all within itself is a continuum or a contiguum i e whether it is 1 a strict continuum and thus a holon in its strongest but one sense The stongest sense is that of a non local continuum in which its elements exist only virtually the continuum is as continuum not divided but only divisible and are in a physical continuum just a spatial distribution of qualities of the holon or 2 whether such a being is at most a tight contiguum i e an entity consisting of well defined tightly packed material parts and elements that exist actually in it and are not qualities of something but beings i e things themselves and so resulting in a unity it is true because it is organized and patterned but not a unity in the strict sense All results of classical physics and chemistry about molecules and crystals which are the supra atomic intrinsic beings of the Inorganic World are according to HOENEN for which he advances strong arguments not in fact aimed to answer this question and consequently do not decide on it However as we have seen above quantum mechanics seems to point to continua and not to contigua Further as we saw above dendritic snow crystals point in the same direction With organisms however things are a bit different because they show it seems actually existing parts like organs bones cells etc Nevertheless organisms and especially organisms are considered to be true holons by many authors We ourselves also consider it very likely that organisms it is true have been naturally evolved from inorganic precursors but that they are nevertheless not derivable from these precursors which means that during the emergence of organisms from the mentioned inorganic precursors a NOVUM has appeared And this NOVUM we explained by means of holistic simplification within Implicate Order So when we accept that an organism is a true holon it seems to be so in a different way as it is in inorganic entities And as has been already found out it is this way by which an organism is a continuum that forces us to interpret the continuous nature of all intrinsic beings

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  • General Ontology XXIXe
    can vary implying the possibility of motion But before we jump to the conclusion that the contact theory is in trouble we should analyse this matter still further What in fact is our intuition our imagination of space Well we see ourselves within a space surrounding us And this space we experience as real not as empty It has to be traversed in order to reach out to something successfully We cannot ignore it An additional reason why we experience space as something real is the fact that we cannot imagine pure nothingness Our intuition of nothingness always refers to things that are there i e we refer to nothingness only with respect to the absence of certain things among other things that are present So it is clear that we do not intuit empty space Indeed our intuition of real space has much in common with the theoretical ether of Lorentz The position theory From the viewpoint of the contact theory a world of particles flying about in empty space is absurd because each such a particle would be a world and these worlds would not be in any sense interrelated This is true when departing from the contact theory as the only intelligible way of seeing things especially meaning that the position theory is not Let s investigate this position theory While the contact theory says first contact then position which here means that position cannot exist without existentially presupposing contact and with minimum distance i e neighborhood resulting from immediate contact and distance resulting from mediate i e indirect contact the position theory says first position then either contact resulting from neighborhood or distance Place which in the position theory is called position and motion can never be such that they exclusively involve only one body i e involve one body in an empty space while the whereabouts of all other bodies are totally irrelevant for the place and motion of that one body As we have seen and will show further down a certain place and a certain motion can belong to just one particular body A and not equivalently to all non A but this still means that a place always demands two entities and so does motion Given all this we can now formulate the position theory as follows It assigns to every body a position which is an intrinsic determination accident or modality of such a body resulting in a relation of the latter to other bodies or entities that possess their own intrinsic position This relation can then be either contact resulting from neighborhood or a certain distance If the possibility of distance between bodies embedded into an absolute vacuum without mediate contact by contact with other real extensa really exists then indeed one should in order to explain this possibility resort to the position theory But does such possibility exist As we saw above with respect to intuition or imagination we cannot construct this possibility a priori We do not and cannot intuit or imagine empty space We could however try a different construction Suppose two bodies that touch a third one on both sides They then will be at a certain distance from each other measured by the dimension of that third body Now suppose that this third body is annihilated while nothing else happens Then so one could argue there is no reason why this particular distance should vanish But this argument is false First of all in the beginning the contact theory was presupposed from the double contact distance resulted Then the third body was annihilated whereby it was supposed that nothing else was changed i e it was supposed that this annihilation did not entail other changes But this cannot be because when the third body is annihilated then the two contacts are also annihilated which means that then the conditions as presupposed by the construction for the distance between the two bodies also disappear implying the disappearance of this distance itself Ans so no local relation between the two bodies exist anymore So this construction devised for the demonstration of the possibility of empty space and a distance not based on contact fails While the implication of the possibility of place or position and of motion by the ens extensum i e by the presupposed presence of spatial entities by means of contact is totally intelligible and as such expressed by the contact theory the position theory must assume bodies with intrinsic positions whose nature eludes us totally But in spite of this such a theory could still be true Or does it contain a contradiction It does The contact theory wins Recall that we have found the following From the content of the concept of ens extensum the possibility of contact between two extensa necessarily results And this is completely intelligible It is not just concluded from a broad experience it can be concluded a priori If the position theory were right then the possibility of contact would not directly result from the nature of the extensa as extensa From the latter would first follow intrinsic position with respect to other bodies and only then the possibility of contact or distance Well this could be so but something else not Possessing insight in the ultimate emergence of possible contact namely via the link position entails that we then also should possess insight of that link itself But this insight is totally absent So if this link really would exist we only conclude as to its existence on the basis of a broad experience But then also our conviction of the possibility of contact were only based on broad experience and nothing more But as we have seen this is not so The possibility of contact is a priori clear as soon as the ens extensum is presupposed It follows therefore that the link i e the intrinsic position does not exist which in turn means that the position theory is impossible Consequently we accept the contact theory The contact theory was first proposed by St Thomas Aquinas in the 13th century It is further worked out by HOENEN 1947 the result of which is reproduced here But remarkably HOENEN found it more or less expressed by EINSTEIN Forum Philosophicum I 1930 p 173 He cites Ihm dem Raume geht die Bildung der objektiven Körperwelt voran Ich kann Körper durch sinnliche Merkmale wiedererkennnen ohne sie bereits räumlich zu erfassen Ist in solchem Sinne der Körperbegriff gebildet so zwingt uns die sinnliche Erfahrung dazu Lagen Beziehungen zwischen den Körpern festzustellen d h Relationen der gegenseitigen Berührung Was wir als räumliche Beziehungen zwischen Körpern deuten is nichts anderes Also ohne Körperbegriff kein Begriff räumlicher Relationen zwischen Körpern und ohne den Begriff der räumlichen Relationen kein Raumbegriff The italics are from HOENEN Let us translate this important passage into English and provide it with some comments placed between square brackets The objective world of bodies which we can represent by the ens extensum precedes it space I can recognize bodies by means of sensible features without grasping them spatially here meaning without presupposing space When in this way the concept of body is formed then sensible experience forces us to determine local relations between the bodies i e Relations of mutual contacts And what we signify as spatial relations between bodies is nothing else So without the concept of body there is no concept of spatial relations between bodies and without the concept of the spatial relations there is no concept of space Some might argue that this is all about concepts but the reference to sense experience says otherwise The Ether of Lorentz So it is safe to fully accept the contact theory concerning place position distance and motion And this theory demands that the place of a body comes about by a region of contact of this body with another real entity that is unchangeable and immobile And if we now ask ourselves what this immobile entity would be we can say that the ether of Lorentz is a good candidate It fulfils perfectly well all five conditions for a place of a body 1 It i e the ether is supposed to be real 2 it is supposed to be an ens extensum 3 it is different from the body 4 it makes contact with the body the area contact is three dimensional because the ether is supposed to be penetrable by the body the principal possibility of which was established above and finally 5 it is unchangeable and immobile This ether was although for other reasons established and proposed by LORENTZ HOENEN cites him from his The Theory of Electrons 1909 p 10 One of the most important of our fundamental assumptions must be that the ether not only occupies all space between molecules atoms or electrons but that it pervades all these particles We shall add the hypothesis that though the particles may move the ether always remains at rest From this statement of LORENTZ it is clear that his ether neatly fulfils the demands of place And this we consider to be a fairly good indication of its actual existence So although it cannot it seems make motion with respect to it observable its existence as a medium of localization is now fairly well established HOENEN pp 150 sets out to demonstrate that physical motion is not necessarily symmetrically relative which means that although mathematically it makes no difference when we say A moves with respect to B or B moves with respect to A physically it makes a difference However he does this by invoking the ether of Lorentz i e he presupposes the existence of this ether We cannot accept this as a demonstration that motion is not symmetrically relative Fortunately there is a metaphysical indication that points to this asymmetry when we consider uniform motion When a particle moves uniformly i e with the same velocity no force is acting upon it otherwise the motion becomes non uniform But motion is change namely change of place And moreover the body moving uniformly is something passive and it should therefore to express this be called body being moved uniformly So there must be an active cause But this cause cannot be a force And it must reside either 1 in the body s surroundings and then moving these surroundings with respect to the body or 2 in the body itself and then moving the body with respect to its surroundings or 3 in both body and its surroundings I think it is fair enough to rule out the first and third possibilities because they imply a large extension or realized expansion of this cause affecting many many bodies or particles So when a body is moving uniformly it does so in virtue of an active cause residing in the body not in the surroundings this active cause will later be identified with the impetus All this is at least some indication of motion not being symmetrically relative without invoking an ether like that of Lorentz The contact theory which we have found to be correct demands an immobile and unchangeable medium as a medium of localization And if we look for such a medium we see that the ether of Lorentz fits these demands It is pervading all non ethereal bodies and so is indeed a medium for them implying three dimensional contact with every such body It is permanently at rest according to LORENTZ s hypothesis which means that the place of any non ethereal object immersed in this ether is immobile It is the non ethereal object that moves and therefore changes place However this ether in its function of universal localization medium must be specified further for its own sake as well as in order to assess the plausibility of its existence Ontological analysis of the Ether as universal Medium of Localization Every real full fledged being can be considered to consist of Substance and Accident or in other words of properties accidents plus that substance of which they are properties If it were not so constituted it would be difficult to consider it as real The ether if it is to be a medium of localization must be a real being as established earlier So it must ontologically be constituted by substance and accident But as such the ether can change and would not be an appropriate localization medium A way out of this dilemma could be the following Precisely the naked substance i e its substance without accidents i e that which remains the same during accidental change is the medium of localization This medium would not be real if it did not have accidents properties But it does have accidents It is only that these accidents do not participate in the ether being a universal medium of localization Only its substantial aspect does That this not seen as a being but as an aspect of a being is real enough can be illustrated by the substantial aspect of any ordinary physical body that is an intrinsic being Let s take snowflakes They all differ in appearance even when grown in uniform conditions Especially they differ in morphology But their chemical composition is the same and also their crystallographic symmetry expressed by their Space Group is the same As was established in First Part of Website Crystals and Metaphysics the Essence of a single crystal is its Chemical Composition plus its Space Group symmetry And this Essence represents the substance and was called the genotypical domain of the given being which here is a crystal From this Essence flows part of the crystal s phenotypical domain consisting of intrinsic properties like the crystal s point symmetry its electrical and other physical properties and also its chemical properties In addition to these properties the crystal also displays extrinsic features constituting the other part of its phenotypical domain such as resulting from irregular growth caused by fortuitous irregularities of the growing environment This description should however be a little amended In fact the mentioned crystal s chemical composition plus its space group symmetry because as such this is already a full fledged physical structure already belongs to the crystal s phenotypical domain the domain of the observable consequences of the crystal s Essence its genotypical domain and the latter can be identified with the particular relevant crystallization law as dynamical law of a dynamical system which is the growing crystal in its nutrient environment It is this crystallization law that as Essence of this particular crystal cannot change because in that case we would obtain a specifically different crystal Realize that the Essence insofar as it is this particular Essence cannot change Only the crystal can change and when this change is a substantial change and thus not an accidental change we have to do with a specifically new Essence of the crystal This could happen for instance when a given crystal metamorphoses its intrinsic structure as a result of change of certain external conditions like temperature and pressure In the same way the direct phenotypical consequences of the Essence of the crystal for example the space group symmetry and point group symmetry as intrinsic properties and only as intrinsic properties remain the same while other features like the particular shape and morphology that the crystal has actually taken up changes when say ice crystals are concerned from snowflake to snowflake Of course the intrinsic properties could change but only as properties not as intrinsic properties of the initially given crystal and then they become intrinsic properties with respect to another species of crystal for instance with respect to chemical composition when Oxygen is replaced by Sulfur at low temperatures or when its space group symmetry turns into another space group symmetry These are substantial changes about which we spoke just above But in a snowfall this does not happen And it is this case we consider illustrative for something remaining the same within something that changes constantly And the ether is now considered to be something that remains the same and can as such serve as a medium of localization within an ontologically broader entity that can and does change all the time This ontologically broader entity which as such must be a full fledged being is composed of a particular substance plus its accidents intrinsic or extrinsic and could or even should be because a universal medium of localization not only demands just extension but extension in and around all objects that obtain place some physical field Maybe several physical fields are involved that however ultimately unite to be one field after all and the ether would then be that aspect of this field that remains the same And this aspect will then consist of the unified field s Essence plus that what necessarily flows from it i e that which is immediately implied by it which is the set of intrinsic properties What we observe however is neither this Essence nor the intrinsic properties insofar as they are intrinsic properties but only actually existing features The Essence plus intrinsic features does not change as long as the substance does not change i e is not annihilated This means that our unchangeable entity demanded by the contact theory is a theoretical construct not a physical entity And this goes someway to explain why the Michelson Morley experiment failed to demonstrate it Motion can only be measured when it is motion of a body i e of an ontologically complete entity with respect to another such body This interpretation here presented of the nature of the localization medium differs from that of HOENEN because he maintains that this medium is a full fledged substance i e a complete being its substance plus its accidents We say that the medium of localization is only an ontological part of such a complete being It is the theoretical construct or equivalently the ideal entity as outlined just above And its reality wholly derives from the reality of the complete being of which it is an ontological part If our medium of localization by means of which a body obtains its place were uniform and homogeneous then every place would be the same which means that motion with respect to this medium would be impossible But our medium is not necessarily homogeneous We have said that this medium of localization insofar as it is a universal medium of localization is the constant structure that is immediately implied by the Essence or intrinsic nature of the complete entity of which the medium is just an ontological part And if we now assume that this constant and permanent structure is non periodic otherwise many places separated by certain distances are exactly the same with respect to local structure as well as to orientation then our medium is a true universal medium of localization It is an abstract heterogeneous continuum If there really exists such a universal medium of localization then the place of a body is a unique place when it is related to this medium But can a place really be unique i e can we demonstrate this without assuming this medium Yes we can We will derive it from absolute simultaneity So the first thing we must do is to derive the latter The Theory of Relativity presupposes the possibility of Absolute Simultaneity What the theory of relativity demonstrates is that absolute simultaneity when it occurs cannot even in principle be m e a s u r e d Observers which are observing two events as to their simultaneity or non simultaneity will not agree about this when their frames of reference each represented by a particular coordinate system move with respect to each other with a high speed approaching the speed of light This however does not mean that according to the theory of relativity as a theory of natural science absolute simultaneity does not e x i s t because then the following philosophical position and thus not a result of natural science has been smuggled in That what cannot in principle be measured and thus cannot be observed does not exist Of course that still doesn t demonstrate the possibility of absolute simultaneity but it also doesn t disprove it Before we continue we must elaborate a little more on the just given philosophical position And to begin with there is another philosophical position to which we like to adhere the principle of intelligibility If something is not intelligible then it cannot exist We will now show when the first mentioned philosophical position applies and when it does not Suppose there is something of which we definitely i e in the sense of direct intellectual insight know that it has quantity And suppose we can establish that this quantity cannot even in principle be measured while this impossibility is neither caused by imperfect measuring devices nor by fundamental physical constraints but comes about by a cause within that entity itself then such a something i e this entity is unintelligible And according to the just given principle of intelligibility it then cannot exist So in this case the first philosophical position not observable in principle cannot exist is valid Indeed this position is only valid if the in principle non observability relates to the object itself i e to the object in question It is not valid when this in principle non observability relates either exclusively to technical imperfections of the instruments but this is aggreed upon by everybody or exclusively to absolute i e in principle physical impossibility of observing and measuring because here we can imagine cases where we know beforehand the presence of say a property of something by evident intellectual intuition while this property is not observable because of fundamental physical reasons And in such a case the philosophical position not observable cannot exist does not hold A simple example could be the following Suppose that it is impossible even in principle by physical reasons to position measuring devices such that they can inspect the rear side of the moon Then we cannot conclude that the moon doesn t have a rear side because we intuitively know that the moon being an extensum i e a spatial thing must necessarily have a rear side Let s continue with the problem of the existence of absolute simultaneity We should realize that we here are trying to determine something of the nature of time not how it can be measured What is then time Well we could with ARISTOTLE define time as follows Time is the numerable aspect of change for instance motion with respect to the before and after Let us supplement and elaborate on this definition Every change unequivocally determines an order of before and after This temperal order does not exist apart from of the different changes it is within these changes but is not a privilege of some particular change or motion When a given moment of one or another change motion or generally process corresponds i e is simultaneous with a certain moment of another change then all that precedes that moment of the first change also precedes that same moment of the second change And if the end of a given change coincides with the beginning of another change then the whole order of before and after resulting from the second change comes after the order of before and after of the first change In this way all changes contribute to one and the same order of before and after in the whole universe It is because of this that we are entitled to speak of one universal time one universal order of before and after This does not mean that time proceeds independently from change What exists is nothing else than the order of before and after in concrete moving or changing matter The above given definition of time and also the just given elaboration on it seems to be contradicted by the theory of relativity especially the notion of simultaneity But there is a caveat here While the philosophical definition of time admits of the possibility of the existence of objective and absolute simultaneity and therefore holds that there is just one universal time the theory of relativity says and rightly so that absolute simultaneity cannot even in principle be measured And as has been said it is now important not to jump to the conclusion that it then doesn t exist On the contrary absolute simultaneity is although not measurable presupposed by the theory of relativity This is because in the theory of relativity the possibility of transformation is assumed Transformation here means that the location place and moment in time of a given event i e one and the same event in the universe can be determined in more than one system of reference coordinate system And this means that observers in different systems of reference that move with respect to each other all can observe and record this event be it that they will find different values of place and time at which this event takes place Transformation formulae admit to express time and place as measured within one particular reference system to be expressed in terms of another reference system that moves with respect to the first one The ensuing argument demonstrating that absolute simultaneity is because transformation is by the theory of relativity supposed to be possible presupposed in the described situation was first given by Van MELSEN A 1955 Natuurfilosofie written in Dutch There is an English edition of this work not a translation preceding Natuurfilosofie Van MELSEN A The Philosophy of Nature 1954 If we for the sake of convenience limit ourselves to two systems of reference I and II that move with respect to each other with a uniform motion i e with constant speed then the possibility of transformation presupposes that every event recorded in I can also be recorded in II The transformation equations then exactly determine what the values of registration in II will be when given in I or vice versa One can now express the presupposition based on every transformation possibility also as follows The two systems which move with respect to each other should be constantly in contact with each other i e penetrate each other otherwise not every event could be recorded in I as well as in II This being constantly in contact however means that at whatever chosen moment of time all points of II coincide with certain though unknown points of I After all there must be a constant correspondence transformation possibility between the two reference systems Precisely the existence of such correspondence presupposes something with respect to time still apart from its measurement It is namely presupposed that at the same moment when say point P 1 in system I coincides with point P 1 in system II another point P 2 in system I necessarily coincides with one or another point in system II Otherwise the two systems would not be in contact with each other at that particular moment implying that there does not exist any sensible application of the transformation equations See next Figures Figure above Two systems of reference I and II moving with respect to each other Both systems must be imagined to be extended indefinitely See also next Figure The systems of reference with respect to the assessment of place should be understood as coordinate systems For three dimensional reality they must be three dimensional Figure above Two systems of reference I black and II red moving with respect to each other Both systems must be imagined to be extended indefinitely They are coordinate systems each provided with an origin indicated by a green point From such an orgin the location of some object can be indicated with two coordinates i e it can be measured So the application of the transformation equations presupposes that at every given moment let us say the moment that is marked by a certain event at point P 1 in system I all points of I whatever their distance is from P 1 coincide with certain points of system II And in this fundamental sense and only in this sense one can legitimately speak of absolute simultaneity This simultaneity does not mean that clocks indicate a same point in time It means that a given moment in time is not limited to one place one point in a system Or expressed in other words Absolute simultaneity according to its fundamental philosophical meaning expresses nothing more than the fact of the coexistence of the parts of the universe Van MELSEN 1955 p 236 In fact in virtue of the discovery of the presupposition a simultaneity is discovered that is objective partly due to the fact that the simultaneity was not found by means of signals such as light The simultaneity is in this case found in a direct way and therefore it is absolute REMARK In the above argument we considered two events that occurred simultaneously viz the coincidence of point P 1 of system I with point P 1 of system II on the one hand and of point P 2 of system I with point P 2 of system II on the other But these are not really events in a physical way So one could conclude that the implied simultaneity is not of a physical nature either and therefore not real And thus the argument which was supposed to be about real simultaneity seems to be invalid However I don t think that things are that bad The systems of reference I and II of the argument m o v e with respect to each other making them enough physical for the argument to be relevant Moreover we could tentatively add the following Suppose that at the location and moment of the coincidence of point P 1 of system I with point P 1 of system II there happens to take place some physical event and that at the location of our point P 2 of system I while coinciding with point P 2 of system II there also happens to take place a physical event Then from the fact of the simultaneity here in the sense of mathematical coexistence of the two coincidences P 1 with P 1 and P 2 with P 2 it follows that also the two physical events took place simultaneously It is of great importance to realize that the above described presupposition present in the theory of relativity and based upon the supposed possibility of transformation of the simultaneous coincidence of points of one reference system with those of another that moves with respect to the first does not mean that this coincidence can be objectively registered To see this let us analyse the following case Suppose we have two systems of reference I an II as described above which uniformly move with respect to each other If we take into account the possibility that the constant speed involved in this motion can be very high i e can approach the speed of light then the transformations that relate these two reference systems must be according to the theory of relativity the so called Lorentz transformations for place and for time Suppose further see this Figure above that at a certain moment point P 1 of system I coincides with point P 1 of system II while this moment of coincidence is recorded as t 1 If we further for the sake of convenience suppose that system II moves in the direction of the x axis of system I then with respect to the position of P 1 in system I only the x coordinate counts We suppose that by measurement in system I it is assessed that the involved coordinate is x 1 And now according to what has been said above at the same moment when t 1 is measured another point P 2 of system I will necessarily coincide with one or another point of system II Let us call this point P 2 a little later we say something about its position The difficulty now at hand is that observers in I and II will not agree among each other as to which points P 2 and P 2 of both systems will coincide at the time t 1 The position of point P 2 of system I and with respect to I at the time t 1 is measured to be x 2 This point will coincide with a point P 2 of system II This point thus becomes identical to the point with location x 2 but the location of this point as assessed within system II is different It is different according to the Lorentz transformation of a coordinate The new coordinate x 2 then is What we until now have is that the observer in system I maintains that point P 2 with the coordinate x 2 coincides at time t 1 with a point P 2 of system II However the observer in system II sees the coordinate of the point P 2 as being not x 2 but x 2 the value of which is given by the above formula So they disagree about the position of the point P 2 and they explain this by maintaining that the measurement of the coinciding of the points P 2 and P 2 was not done at the same time Observers in system II that find themselves at the points corresponding with P 1 and P 2 of system I will maintain that their colleagues in system I at P 1 and P 2 did not assess the point in II that coincides with P 2 in system I at the same moment as that moment in which P 1 and P 1 coincide According to the observers in system II the time t 1 that the observer in system I at point P 1 uses has according to the Lorentz transformation for time the value while the time used at P 2 by observers in system I according to them i e the observers in system II has the value So according to the observers in II these moments differ and they conclude therefore that the observers in system I were wrong about the simultaneity of the two events viz the coincidence of P 1 with P 1 on the one hand and the coincidence of P 2 with P 2 on the other The observers in I will however maintain that they have measured at the same moment So according to them the coincidence of P 1 with P 1 took simultaneously place with the coincidence of P 2 and P 2 Van MELSEN 1955 p 236 237 So indeed we see that the necessary presupposition of the existence of objective simultaneity in the sense of being implied by every accepted possibility of transformation between systems of reference namely that the places of one given coordinate system are continually in contact with those of another coordinate system doesn t say anything about the possibility to actually and objectively measure this simultaneity There is no possibility to measure this simultaneity And this is a true result of the theory of relativity Therefore the concept of simultaneity does not make sense in natural science but it does make sense as one of its presuppositions and as such it is a truly philosophical concept So in our metaphysical and thus philosophical analysis of reality which is a consideration of the way and status of Being and beings we can accept the existence of absolute simultaneity And such a simultaneity implies the existence of absolute place absolute position because of the following reason If at location x 1 an event g 1 takes place then a simultaneous event g 2 cannot take place at location x 1 It can however in principle take

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  • General Ontology XXIXf
    a biography of water 1999 BALL P writes p 177 8 of the 2000 Phoenix edition A continuing mystery about dendritic snowflakes is why all six of their branches seem to be more or less identical The theory of dendritic growth explains why the side branches will develop at certain angles but it contains no guarantee that they will all appear at equivalent places on different branches or will grow to the same dimensions Indeed these branching events are expected to happen at random Yet snowflakes can present astonishing examples of coordination as if each branch knows what the other is doing One hypothesis is that vibrations of the crystal lattice bounce back and forth through the crystal like standing waves in an organ pipe providing a degree of coordination and communication in the growth process Another is that the apparent similarity of the arms is illusory a result of the spatial constraints imposed because all the branches grow close together at more or less the same rate But for the present the secret of the snowflakes endures That the latter explanation involving spatial constraints is in correct is proven by the two snow crystals that we already depicted earlier in Part XXIX Sequel 4 the right image in one Figure and the left image in the next Click HERE to see them Several explanations have been offered for this unanimity in the detailed behavior of a snowflake in the six directions but none has yet been generally accepted wrote HOLDEN MORRISON as we saw above Such an explanation could involve lattice vibrations or focus more on external conditions of snow crystal growth The latter kind of explanation is offered by I PETERSON Islands of Truth A Mathematical Mystery Cruise 1990 p 131 Given the tremendous variations among different snowflakes why do the six arms of a single snowflake look so much alike Experimental studies reveal that environmental conditions at the six tips are a lot more similar than the environmental conditions between one snowflake and another The six arms of a snowflake each less than a millimeter long tend to see the same temperature distribution and vapor density Two neighboring but separately drifting snowflakes are far enough apart to encounter significant differences However the conditions surrounding even a single tiny snowflake aren t completely uniform A close look at any real snowflake reveals that its six arms aren t identical There are always some imperfections This kind of explanation was also given by LIBBRECHT 2003 p 47 where he discusses the genesis of a stellar snow crystal while it is blown to and fro by the wind As it reached snow crystal adolescence the crystal blew suddenly into a region of the cloud with high humidity The increased water supply made the crystal grow faster which in turn caused the corners of the plate to sprout small arms Because the humidity increased suddenly each of the six corners sprouted an arm at the same time The arms sprouted independently of one another yet their growth was coordinated because of the motion of the crystal through the cloud The crystal subsequently blew to and fro in the cloud while it grew following the will of the wind As it traveled the crystal was exposed to different conditions Since a snow crystal s growth depends strongly on its local environment each change of the wind caused a change in the way the crystal grew Again each change was felt by all six arms at the same time so the arms grew synchronously while the crystal danced through the clouds As the crystal grew larger and ever more ornate it eventually became so heavy that it floated gently downward out of the clouds to land on your mitten The exact shape of each of the six arms reflects the history of the crystal s growth The arms are nearly identical because they share the same history However when discussing branching as an instability phenomenon LIBBRECHT writes the following p 52 comments between square brackets emphasizing bold is mine Consider a simple hexagonal plate crystal as it floats through a cloud Because in case of the commencing of rapid growth that as such is only limited by the rate of diffusion of water molecules to the surface of the growing crystal the hexagon s six points stick out a tiny bit while their direction is that of the growth direction of a fast growing face type that had grown itself out of existence water molecules are a bit more likely to diffuse to the points than to anywhere else on the crystal The points then tend to grow a bit faster and before long they stick out farther than they did before Thus the points grow faster still The growth becomes an unstable cycle the points stick out a bit they grow faster they stick out more they grow faster still This kind of positive feedback produces what is called a branching instability even the tiniest protruding points will grow faster than their surroundings and thus protrude even more Small corners grow into branches random bumps on the branches grow into side branches Complexity is born Instabilities like this are the heart of of pattern formation and nature is one unstable system heaped on top of another Instabilities are responsible for many of the patterns you see in nature including snowflakes This latter statement about the random bumps is more or less badly compatible with the assumption as stated by LIBBRECHT and given above that the morphological near identity of the six arms of a single snow crystal is accounted for by near identical conditions of temperature and humidity If the structure of an arm is determined by the occurrence of random bumps which then amplify during further growth then the six arms would not be near identical because as r a n d o m l y occurring bumps these bumps cannot be expected to exhibit the same pattern of distribution of these bumps along each branch The distribution pattern must turn out to be different on each of the six arms resulting in arms that are significantly different with respect to their detailed morphology because the bumps will amplify during growth We will return to this matter shortly The explanation of the near identity of the six arms of a single stellar snow crystal which is often actually the case if we exclude all instances of distortion by local external agents resulting in asymmetric crystals in terms of their near equal falling history and thus in terms of their being subjected to near equal conditions of temperature and humidity sounds pretty convincing And indeed as it were not for the just mentioned discrepancy that could be what actually happens If true the near equality of the six arms is caused by l o c a l factors and not by the crystal as a whole implying that then there is no evidence of any holistic factor involved in the formation of snow crystals But this would not lead to a diminishing of philosophical interest that snow crystals offer because our statements as they are put down in Part XXIX Sequel 3 and Sequel 4 show by means of studying those snow crystals what the ontological consequences which are huge are if holistic factors were indeed at work be it in crystals organisms or whatever These consequences are so radical that they in a way represent a pro for the reductionistic character of all of Reality So even when we advocate in our relevant documents a holistic interpretation of Reality we do so critically And if this Reality turns out to be wholly reductionistic that s fine Only the truth matters Theory may not transform into ideology The genesis of the different morphology of the branches of stellar snow crystals and their near identity in a single crystal Let us now investigate these branches or outgrowths of stellar snow crystals more closely The morphology of the branches are according to me determined by four main factors Crystal lattice Temperature of the immediate surroundings of a developing snow crystal Humidity of the immediate surroundings of a developing snow crystal The occurring random bumps on the branches and side branches as a result of the fact that the formation of stellar snow crystals is a non equilibrium crystallization process involving runaway growth at certain locations in the crystal Let s investigate these four factors more closely Crystal lattice The arrangement of chemical units in an ice crystal is according to a three dimensional hexagonal lattice The Space Group of Ice which is the lattice filled in with motifs which here are water molecules implies a Point Group expressing the Crystal Class which is the Dihexagonal Dipyramidal Class of the Hexagonal Crystal System It is the most symmetric Class of this System The lattice provided with the water molecules induces certain preferred directions of the arms as they sprout from an initial hexagonal plate These directions correspond with vanished fast growing faces One of them indicated as c in the next Figure So the particular type of lattice provided with water molecules is responsible for the fact that generally all stellar snowflakes have six arms going out from the six corners of an initial hexagonal plate Continued branching results in side arms sprouting at 60 0 angles from the previous higher level arm The direction of all the arms and side arms correspond to i e are parallel to the growth direction of fast growing and vanishing prism faces Figure above The face c grows fast with respect to the faces a and b ultimate prism faces of the snow crystal And because the latter faces taper the face c will eventually grow itself out of existence The lattice structure as a co determinant of the form of snow crystals is not a variable but a constant factor Temperature of the immediate surroundings of a developing snow crystal As was found out above temperature controls the habit change of snow crystals Temperature change effects a change in the relative growth rates of the two types of faces viz the top and bottom faces of the hexagonal prism on the one hand and the prism sides on the other This results in crystals of different habits viz plate like or columnar crystals The range of this variation has been experimentally investigated by MASON B 1963 The results are also reported in TILLER W The Science of Crystallization 1991 1995 Variation of ice crystal habit with temperature If we only consider the crystals as viewed along the c axis i e if we only consider a projection of the crystal onto the plane parallel to the basal faces of the prism this habit change is not visible Temperature is a macroscopic variable It cannot apply to a single molecule or a smal cluster of individual molecules So it doesn t act in a strict local fashion Humidity of the immediate surroundings of a developing snow crystal Higher humidity effects higher absolute growth rates These rates can be so high that non equilibrium crystallization takes place resulting in the formation of branches or a distinctive hollowing of the basal faces of the prism All this is the result of an instability in faceted growth which causes the protruding edges to grow faster than the centers of the faces Generally we can say that with increasing humidity the wispiness of the ice crystals increases The changes of broad morphological snow crystal types with changes of humidity supersaturation and temperature are outlined in the morphology diagram as given in LIBBRECHT s book and website and which was adapted from a diagram by Y FURUKAWA The water saturation line gives the humidity for air containing water droplets as might be found within a dense cloud Click HERE to see this diagram and close the window to return Humidity is a macroscopic variable It cannot apply to a single molecule or a smal cluster of individual molecules So it doesn t act in a strict local fashion If in the falling history of a single snow crystal the humidity changes we can expect to see some sort of alternation between branching and faceting determining the overall shape of a branch The occurring r a n d o m b u m p s on the branches and sidebranches as a result of the fact that the formation of stellar snow crystals is a non equilibrium crystallization process involving runaway growth at certain locations in the crystal The appearance of random initial bumps at one or another location on the growing crystal is a local variable and is mainly responsible for the d e t a i l e d morphology of the arms of a stellar snow crystal This is implied by what LIBBRECHT says on p 52 of his book where he discusses the branching instability LIBBRECHT generated snow crystals in a convection chamber He created small snowfalls at several temperatures The most dramatic snow crystal growth was obtained at minus 15 0 Celcius and at high supersaturation resulting in the growth of plates and stellar dendrites snow crystals with ramified arms About them he writes on his website These crystals were all grown under quite similar conditions although some grew for a longer period of time before falling onto the observation window The longest growth time was about two minutes Even the small variations in temperature and supersaturation within the growth chamber resulted in the great variety of forms seen This demonstrates that the final snow crystal shape is very sensitive to growth parameters especially at minus 15 0 C when the supersaturation is high The temperature of the chamber was held at minus 15 degrees Celcius and generally the conditions were quite similar as LIBBRECHT informed us And of course small variations in temperature and supersaturations within the growth chamber occur but I assume that these are not actually measured On his website LIBBRECHT shows the following crystals as a result of this experiment at minus 15 degrees Celcius The first image of this series shows a crystal with a tip to tip diameter of 140 microns 0 140 mm and all the other images of this same series as results of the minus 15 degree experiment are shown at the same scale The next image shows a mosaic of snow crystals grown at minus 15 degrees Celcius I think it is reasonable to surmise as a possibility that it is not the above mentioned small variations in temperature and supersaturation that are mainly responsible for the difference in the detailed morphology of the resulting snow crystals I propose to consider the possibility that the occurrence of random bumps on the growing crystal especially when it grows rapidly and therefore sprouting arms is far more important in determining the detailed morphology of the branches But naturally that would mean that these random bumps and their ensuing amplification are then copied onto equivalent sites on the growing crystal which in turn would mean that the crystal ultimately grows as a whole and not entirely by local factors Of course the morphological effect of the occurrence of local random bumps on the growing crystal is dependent on the degree of humidity because unstable growth necessary for these bumps to become morphologically effective only occurs at high levels of humidity The degree of humidity indeed determines the overall wispiness of the crystal and the six branches probably experience the same degree of humidity We might then think that the randomness of the distribution pattern of the random bumps in one single arm is also expressed by the fact that any single arm is not perfectly symmetric and in the fact that the six arms of a single crystal are not exactly identical And this would mean that the crystal does not grow as a whole afterall but grows as a result of local factors only However I think that the six fold symmetry or more precisely the D 6 symmetry of snow crystals as seen along their c axis as they are usually depicted is far to good for growth by local factors alone to be true With the same experimental devices LIBBRECHT also grew snow crystals at minus 2 degrees Celcius They are again plate like although the growth rates are not as high as at minus 15 degrees Celcius Thus the plates are smaller but they also show significant differences in their detailed morphology The following images as results of this experiment were given at LIBBRECHT s website As LIBBRECHT notes the larger crystals at the far right which apparently grew at the highest supersaturation levels show distinctive rounded arms These reflect the microscopic roughening transition that occurs near the melting point the ice surface becomes microscopically rough so the crystal boundaries are no longer faceted The random nature of the morphology of the arms of stellar snow crystals It is time to have a close look at the arms of some stellar snow crystal We then see when we go from the crystal center to the end of one branch that this arm records a series of more or less random events that the crystal has experienced during its fall See next Figures Figure above Arm of a stellar snow crystal Adapted from LIBBRECHT The Snowflake 2003 Photography by Patricia Rasmussen Figure above Same as previous Figure Only about one half of the arm depicted to emphasize the mentioned sequence of freezing events Adapted from LIBBRECHT The Snowflake 2003 Photography by Patricia Rasmussen Figure above Arm of a nother stellar snow crystal Adapted from LIBBRECHT The Snowflake 2003 Photography by Patricia Rasmussen Figure above Same as previous Figure Only about one half of the arm depicted to emphasize the mentioned sequence of freezing events Adapted from LIBBRECHT The Snowflake 2003 Photography by Patricia Rasmussen Figure above Arm of yet another stellar snow crystal Adapted from LIBBRECHT The Snowflake 2003

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  • General Ontology XXIXg
    projection of the first mentioned planes Figure above The structure seen along the crystallographic c axis of the above mentioned hexagonal layers as part of the lattice structure of ice As such this two dimensional lattice admits of hexagonal unit meshes but the smallest possible unit mesh is a rhombus with 60 and 120 degree angles as indicated Alternative and equivalent rhombic unit meshes are also indicated Red dots represent atoms or molecules Figure above The structure seen along the crystallographic c axis of the above mentioned two types of layers as part of the lattice structure of ice Red or blue dots represent water molecules Alternative triangular sectors are associated with water molecules blue See also next Figure Figure above Same as previous Figure The layer with the blue dots is identical to the layer with the red dots The only difference between the two is that they are shifted with respect to each other Figure above Same as previous Figure If seen not as a projection but as a three dimensional structure consisting of hexagonal layers two types of molecules one type shifted with respect to the other then we can discern the following The center of each empty triangle is the location of a six fold screw axis perpendicular to the plane of the drawing namely a 6 3 axis which is a combined symmetry transformation rotation by 60 0 followed by a translation along the rotation axis The length of the translation is half the unit translation along the crystallographic c axis Three of such screw axes indicated Each molecule itself indicated as a red or blue dot is the location of a 3 fold rotation axis On the basis of the above we have indicated the position of one 6 3 axis in FURUKAWA s drawing of the lattice structure of ice Top part of Figure Crystallographic structure of ice I h Open and solid spheres indicate the Oxygen and Hydrogen atoms respectively Each Oxygen atom is connected with four neighboring Oxygen atoms by hydrogen bonds thin solid lines which makes a tetrapod arrangement Bottom part of Figure Illustration for the corresponding crystallographic axes and planes Basal and prism faces correspond to the 0001 and 1010 faces respectively Position of one 6 3 axis indicated It runs parallel to the c axis Figure and subscript adapted from FURUKAWA website Fig 4 Figure above Same as previous Figure Top face of alternative unit cell indicated black The corners of this latter unit cell are the locations of 6 3 screw axes Such a unit cell can together with three other such unit cells form a hexagonal prism the central axis of which is a 6 3 axis When all translations are eliminated i e if we derive the Point Group symmetry indicating the Crystal Class from the Space Group symmetry this axis will become a six fold rotation axis We continue with our above series of ice lattice drawings depicting hexagonal layers stacked along the c axis layers that are horizontally shifted with respect to each other The view direction is down along the c axis Figure above The structure seen along the crystallographic c axis of the above mentioned two types of layers as part of the lattice structure of ice Red or blue dots represent water molecules Alternative triangular sectors are associated with water molecules blue Possible hexagonal unit meshes six triangles and rhombic unit meshes two triangles are indicated Figure above Same as previous Figure Each water molecule in an ice lattice is connected to four others by hydrogen bonds For each water molecule three such bonds are shown the fourth bond is perpendicular to the plane of the drawing and connects the depicted structure with a copy of it See next Figure where the construction of the pattern of hydrogen bonds is completed Figure above Same as previous Figure Each water molecule in an ice lattice is connected to four others by hydrogen bonds For each water molecule three such bonds are shown the fourth bond is perpendicular to the plane of the drawing and connects the depicted structure with a copy of it In the present Figure the construction of the pattern of hydrogen bonds is completed It turns out to be a hexagonal array consisting of open hexagons Hydrogen bonds perpendicular to the plane of the drawing not shown See also next Figure Figure above Same as previous Figure Open hexagons each consisting of six water molecules indicated As such they represent the structure of ice If we now remove all black connection lines we re left with an array of open hexagons Figure above Structure of ice seen along the crystallographic c axis Each dot red or blue represents a water molecule Each such water molecule connects with four others by hydrogen bonds Three such bonds shown for each water molecule The fourth bond is perpendicular to the plane of the drawing Figure above Same as previous Figure Possible rhombic unit cell indicated The the center of each hexagon is the location of a 6 3 screw axis four of them indicated if the Figure is interpreted as representing a 3 dimensional view down the c axis If the Figure is 2 dimensionally interpreted then we see a rhombic unit mesh containing two 3 fold rotation axes coinciding with the two molecules of that mesh and a 6 fold rotation axis at each of its corners The last obtained picture of the lattice structure of ice can now be compared with some other Figures from the literature that also depict the ice lattice as seen along the c axis Figure above Ice structure viewed down the c axis i e down the 6 fold rotation axis of a single crystal Water molecules consisting of one oxygen large sphere and two hydrogens small spheres are arranged in a hexagonal array Hydrogen bonds between water molecules are shown with thinner lines The positions of the H hydrogen and O oxygen are

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