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- implicate order 20

the result of the action of the operator d dx on the function x 2 is another function namely 2x Some functions however behave in a peculiar way when subjected to this operator d dx For example the derivative to x of the function e 3x is equal to 3e 3x In this case we see that the original function e 3x is recovered but now multiplied by a certain number which is in the present case the number 3 Functions that after being subjected to a certain operator return in their original form are called eigenfunctions of that operator and the numbers with which those eigenfunctions turn out to be multiplied are called the eigen values of the operator So with each operator is associated a set of numbers a set of numbers because the operator can act on several different functions resulting in different numbers to appear This set forms the spectrum of the operator This spectrum can be discrete when the eigenvalues form a discrete sequence like the set of natural numbers 0 1 2 3 or continuous when those eigenvalues form a continuous sequence like a set of real numbers for instance all those between 0 and 1 So we now have a mathematical formalism that expresses observable quantities and their possible values The possible results of measuring such a quantity observable are just the set of eigenvalues of the corresponding operator which stands for that particular observable If those operators really stand for observables then their eigenvalues must be real numbers which include natural numbers and not say complex numbers which contain the notorious square root of minus one and which play elsewhere in the quantum formalism an important role Well if we restrict ourselves to so called hermitean operators then it is guaranteed that their eigenvalues are real numbers and thus can represent sensible results of measurements which are always expressed as a certain number times some appropriate unit for example 12 344 millimetres where 12 344 is a real number Moreover the operator must be linear which means that it transforms sums into sums and multiples into multiples Formally the operator O is linear if when it turns a state vector as such representing the state of a system or particle V 1 into V 1 and V 2 into V 2 then also V 1 V 2 becomes V 1 V 2 and nV 1 becomes nV 1 where n is some number By no means all operators are linear however The operation of squaring numbers is a counter example It turns 1 into 1 2 into 4 but since it turns 1 2 3 into 9 we see that this is not equal to 1 4 The linearity of operators to be used in quantum theory is a necessary condition because otherwise the phenomenon of superposition could not be expressed In quantum theory an eigenfunction stands for a definite state of say an electron It is expressed as a state vector in some abstract vector space We are now able to understand why our information about the states of motion state vectors of say an electron is so restricted in quantum theory If we could know both where a particle was and also what it was doing it would then have to be in a state which was simultaneously an eigenstate of the position operator x and also an eigenstate of the momentum operator p The mathematicians tell us that this would only be possible if the operators x and p were to commute which means that when they are multiplied together in a given order i e applied one after the other they must give the same result as when they are multiplied in the reverse order just like in our ordinary arithmetics 2x3 3x2 6 However for operators this is not usually the case In quantum theory it turns out that the operators x and p do not commute which is why there cannot be a state in which they both take definite values This is Heisenberg s celebrated principle of uncertainty for the quantities position and momentum which reveals itself strongly only in the quantum mechanical domain of the microworld Now quantum mechanics says that the Schrödinger wave equation describes a smooth and determined development of states of say an electron as it goes about its business as long as it not being interfered especially by an observation While the electron originally was in a state of superposition of possible values as soon as it is interrogated by some observational device it shows one of those superposed values as predicted only statistically i e in terms of chances by quantum theory In the language of operators the electron originally was not in an eigenstate of position but in some superposition of such states After the act of measurement it finds itself in an eigenstate of position i e in an eigenstate of the position operator corresponding to the eigenvalue which is the result of that particular measurement and which result was statistically predicted by quantum theory Every act of measurement has this character of entailing instant change Beforehand our system is not in general in an eigenstate of the observable i e of the observable quantity under discussion we are intended to measure but rather it is a superposition of such states Afterwards the system is in that particular eigenstate selected from the original superposition which corresponds to the eigenvalue actually obtained as the result of that measurement This discontinuous change is called the collapse of the wavefunction The idea is that the probability which was originally spread out in a wavefunction covering the particle being here there and perhaps everywhere is now all concentrated here It has collapsed onto itself Now BOHM maintains that quantum theory all by itself does not describe time and consequently not the phenomenon of process To remedy this can be done by the extension of quantum theory with the concept of implicate

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Open archived version from archive - General Ontology I

and justifies to initially study them in isolation which in fact has occurred in the development and differentiation of the sciences As has been established earlier all Hartmannian categories are to be understood as If Then constants where the Then component is HARTMANN s concretum which is the set of all entities that are so and so determined by the category the If Then constant while the If component indicates the range of relevance of the category The way a category is formulated namely as an If Then constant lets this category be valid everywhere However its range of relevance generally is limited And the degree of generality of the Then component should exactly match that of the If component which means for example that a Hartmannian category applying to the Real Sphere as a whole i e to the whole temporal set of layers that at least includes the macroscopic inorganic the organic the psychic and the super psychic layers has as its If component this whole set of layers ultimately derived and emerging from the non physicalized mathematical layer and consequently its Then component the so and so determined i e the concretum has the same degree of generality as this whole set of layers taken together range of relevance HARTMANN presents his general categories of the Real as a whole as oppositional pairs for example the pair Form Matter This pair now must be represented by the corresponding If Then constant The If component is the Real Sphere i e the mentioned set of layers which together form the range of relevance of the category The Then component is the concretum of the category namely being permeated by Form Matter relations of all beings and processes of the Real Sphere The If Then constant i e the Hartmannian category as analysed will then read If we are considering the Real i e if the Real as such is considered to be there then we find the relation Form Matter permeating all of the Real In the same way we should consider the other general categories And when we discuss more special categories having a more limited range of relevance e g categories obtaining only within the macroscopic inorganic level then the If and Then components have the corresponding degree of lower generality We shall normally indicate categories by their single name as HARTMANN does but nevertheless always understand them as If Then constants As has been said HARTMANN s ontological layers must be re interpreted as complexity levels and the Theory of Levels is presented in the First Part of my Website namely in the document Structural Levels It is extensively discussed in the book GÖDEL ESCHER BACH of HOFSTADTER D 1979 1999 and also in the book The Cosmic Blueprint of DAVIES P 1987 1989 A higher level phenomenon like a certain behavior or structure depends first of all on determinants in that same level But according to our hypothesis the higher level phenomenon same level cause secondly depends wholly on features of the previous level According to HARTMANN this is only partially the case because he assumes that novel features are being added when we ascend from a lower layer to a typical high layer feature that in turn and ultimately depends on features of the lowest level These downward dependencies are however seldom straight forward but often of a very i n d i r e c t nature It can be so indirect that the higher level determinants appear to be self sustained i e independent of lower level entities The higher level is in a sense more or less sealed off from the previous levels The relevant determinings trickle through the intervening levels and sublevels all the way up to that higher level And because of this prolonged trickling up i e traversing a long distance during which much happens the lower level features ultimately responsible for the higher level features become as such unrecognizable to the investigator Especially when we ascend from the physical level to the organic but also from the organic to the psychic the lower level laws are in a way harnessed by the higher level structures in the same way as we harness some natural laws for our purposes with respect to for instance the construction of devices or buildings The phenomenon of being more or less sealed off of a higher level from lower levels is the result of a discrete jump to higher functional complexity as we ascend from a lower to a higher level e g from the physical to the organic And even within one and the same layer for example the organic we find in evolution sudden jumps to new organic types For the time being such jumps remain little understood but they have occurred many times If we would assume that the rules obtaining in each level of the set of dependency levels as described above can in principle be changed then this set of levels always demands the existence of one ultimate inviolable level the rules of which can never be changed Based on the above considerations we can summarize the l a y e r e d w o r l d in the following diagram Figure 3 The Total World and its complexity layers and sublayers Because the Real Sphere as such i e as temporal is just an epiphenomenon emerging in virtue of complexification of mathematical structures this Real Sphere is still ideal i e it consists of certain hypercomplex mathematical structures determining it still to belong to the Ideal Sphere of Existence which is the whole objective world anyway While the non physicalized layer is atemporal the real sphere is temporal which implies that all its sublayers physical inorganic organic psychic super psychic are temporal and in this particular sense physicalized The discrimination and interpretation of complexity levels In order to understand the complexity levels and with it an attempt to grasp how HARTMANN himself understands them in his case as ontological layers we must consider a level X as it is in itself i e we must take it only insofar as it is in itself How do we do this Every level except the lowest one is dependent on the previous level This previous level should be seen as a relatively inviolable level or substrate The lowest level viz the non physicalized mathematical level is the absolutely inviolable level When we now consider one particular level let us call it X we deliberately ignore the next lower level i e the inviolable level on which X is dependent and the inviolableness expresses this dependence and this is equivalent to consider X only insofar as it is X For example the psychic level is surely based on the individual organism having such a psychic level So the psychic is dependent on an o r g a n i c structure which in turn depends on a physical chemical structure etc and in this sense the psychic is spatio temporal But the psychic AS psychic i e only insofar as it is psychic is not spatial while it is still temporal because the phenomenon of experiencing itself and discriminating itself from its not itself in a conscious way is not spatial In this sense and only in this sense the category of s p a c e breaks off when we enter from the organic into the psychic The non spatial aspect is however dependent on the s p a t i o temporal aspect of the substrate of the psychic the organism The non spatial aspect is an epiphenomenon emerging from the s p a t i o temporal by a discrete increase of functional complexity When we say following for the time being an for the sake of argument HARTMANN that the psychic layer is not spatial by ignoring the next lower level which is s p a t i a l temporal it could sound like throwing away the baby together with the bath water when we conceptually remove the spatial then what is left is of course non spatial However we do not actually remove the spatial at all because we have said that the non spatiality of the psychic layer is d e p e n d e n t on the s p a t i a l temporal substrate i e on the next lower level which is s p a t i a l temporal According to HARTMANN each layer is characterized by the appearance in it of what he calls a c a t e g o r i c a l n o v u m i e some quite new category or set of categories new with respect to the categorical content of previous levels Such a novum determines the new face of such a layer and influences the categories that come in from lower layers It places these categories in a new setting and partially transforms them into variants new versions of the old lower layer categories He does however not indicate where such a novum comes from which is of course consistent with its being a genuine novum after all But although the latter is new with respect to the set of categories of the previous level it must somehow emerge from what went before I think it is reasonable especially in the context of our hypothesis of mathematical wholeness to interpret such a categorical novum as we see it for example in the psychic layer as implied by emergent material structures which appear as a result of a discrete jump like increase of functional complexity These emergent structures still in the case of the psychic as material structures as such still belong to the organic level but bring with them the psychic which means that within the psychic itself they are not material structures anymore but immaterial structures dependent however upon the corresponding material structures of the organic layer Indeed the psychic layer cannot also according to HARTMANN exist without the organic layer on which it rests We should realize that this transition from the organic to the psychic is less dramatic than the transition from the mathematical to the physical because in the mathematical even the time category is absent so that complexification of it resulting first of all in the physical is hard to visualize Figure 4 The Mathematical Level Atemporal Level and the four sub levels of the Real Sphere Temporal Level A jump in complexity somewhere within the mathematical level results in the physical level A jump in complexity somewhere within the physical level results in the organic level A jump in complexity somewhere within the organic level results in the psychic level A jump in complexity somewhere within the psychic level results in the super psychic level As the above diagram indicates the complexification responsible for emergent phenomena refers to the jump like increase of complexity within a certain layer representing a complexity level resulting in a new layer on top of it The higher layers are always carried by i e are dependent on their previous layer They enjoy however a certain independence of them in the sense that having acquired its higher level features it continues its business wholly within the context of these features The latter consist of new behavior and structure not present in the previous layer The problem with all this concerns the new In the present context new should mean fundamentally new which in turn should mean that the new feature is not only not derivable from features of the lower layer where not derivable means that it cannot be understood from the lower level features but also not actually derived i e not actually generated from lower level features But then the new feature could not be the result of just a jump like or not complexification of features in the lower level This problem cannot really be solved But we have at least an indication where to look for its solution namely the mentioned j u m p like increase i e discontinuous increase in functional complexity because a g r a d u a l increase certainly cannot create something new in the just defined sense In the mean time we should not confuse such a jump with the jump like phenomena as they occur in certain dynamical systems When a certain parameter of such a system is increased beyond a certain critical value the present state of the system becomes unstable and the system quickly slides off into another state that is stable resulting in new behaviors or patterns However this can easily take place within one and the same complexity layer i e no new layer is formed at all So the jumps we have in mind must be much more dramatical If it were such that new categorical elements appear then the corresponding transition would be ontological On the other hand we insist that any jump of whatever magnitude does not add any features from outside the relevant layer otherwise a higher layer is already presupposed So it seems that we are forced to drop the non derived clause from our definition of new and only retain the non derivable which only concerns our ability to know The non derivable gives us the impression of new and only in this way we must consider the new The underivability stems from the fact that between the old and the new there is a very complex entanglement of events compressed into a very small space or one of its analogues and in a very small interval or one of its analogues which gives us the impression of a jump and which is not or cannot be understood i e it cannot be understood as a succession of states leading from the old to the new I assume that in the present case larger wholes suddenly appear from a multitude of cooperating elements and these new wholes now behave as if they were elements themselves i e they interact according to certain higher level rules All this means as already concluded earlier that HARTMANN s layers are not ontologically new i e they are not fundamentally different from each other which means that the dependence of higher layers on all the previous layers which also HARTMANN admits is t o t a l This in turn implies that the world is categorically homogeneous i e one set of fundamental i e ultimate categories or determinants reigns throughout the world implying that there are no ontological layers at all but only levels of complexity The transformed and new categories prevailing in a particular layer are then derived categories But because the above mentioned entanglement intervening between the old and the new cannot be analysed i e the new categories and corresponding features are underivable we have no choice than to work with these new categories And as long as the intervening entanglement or knot is not understood i e not wholly disentangled the possibility of an ontological status of the layers cannot be excluded Preparation and motivation to study Hartmann s theory of layers In what follows we will try to get as close as we can to HARTMANN s theory of the ontology of the Temporal World i e of the whole system of layers above the mathematical This system of layers is his Real Sphere of Existence In this we will not follow in all rigor the above expositions about the derived and the cognitive non derivability and with it about the definition of the new the novum Our demand of consistency of the whole exposition i e consistency between what follows and what went before will be somewhat mitigated This will give us room for d e v e l o p i n g a theory about the existence and content of the layers instead of just p r e s e n t i n g it fully fledged which is after all not even possible However we will in a stronger fashion emphasize the dependence of the higher upon the lower i e stronger than HARTMANN already does implying that the higher categories If Then constants are not absolutely fundamental but only relatively so resulting in the fact that the whole set of layers including the mathematical one is ontologically homogeneous The whole theory of layers or levels is meant to give some preliminary understanding of the ever increasing wealth of structures and subtleties as we ascend from the purely physical ultimately from the purely mathematical via the organic and psychic subjective spirit to the super psychic The latter is the supra individual layer of human culture and institutions the objective spirit It cannot be expected that the above given inherent problems of such an investigation will be solved certainly not by one writer all by himself The only intention is to trigger further thought and discussion with respect to those problems The NOVUM i e the genuinely new feature appearing in a higher layer and representing this layer consists in a very special complexification The latter results not only in a highly complex structural unit but also complexifies its relevant surroundings in such a way that that unit obtains a specific m e a n i n g like we see in DNA molecules Just for themselves these molecules are just complex chemical structures and as such belong to the physical layer But their having acquired a specific meaning elevates the whole system DNA its chemical environment onto a higher structural and functional level The mentioned meaning consists of certain definite r u l e s determining from now on the whole structure This is the NOVUM and leads to Life provided this kind of meaning generating complexification is such that the whole structure is not only determined by the new rules but only by these new rules i e when it can now go its business all on its own which means for example that it also takes care for the selective import of matter and energy and the selected export of matter and entropy and thus guaranteeing its own continuation as a process The dependency of the higher upon the lower is a consequence of the higher having emerged after and in virtue of the entanglement of jumpy i e more or less abrupt complexification of domains of the lower layer So without that lower layer there will not be any entanglement of complexification and consequently no emergence of a higher layer Thus the presence of the next lower layer is in any case a conditio sine qua non i e a condition that must in all cases be satisfied there could be more such conditions that should be satisfied and only when all such conditions are satisfied we have sufficient ground and the consequent will then necessarily follow for the higher layer to exist at all And what is important the lower layer determines the higher layer ONLY as a conditio sine qua non and certainly we here have not to do with a determination by a category of its concretum but with a concretum concretum determination which itself stands under a certain category The lower layers determining as we now know only as a conditio sine qua non are i n d i f f e r e n t as to what higher layer can emerge after the entanglement of complexification i e whether it be Life Consciousness or whatever A layer all by itself does not have a tendency to complexify i e self organize into a higher layer it only has the ability the potential to do so Something comparable with that what was exemplified for the case of the emerging of the organic layer is the case with respect to the emergence of the psychic and the super psychic layers from their respective lower layers They go their business according to special rules however not contradicting the lower level rules The fundamental and thus most general laws of Nature not necessarily as they are described by science but as they really are have room for such high level special rules The same goes for the case of more special natural laws like crystallization laws or laws of other types of dynamical systems Also they leave room for high level special rules Those low level dynamical systems are then selectively fed with certain ingredients which deflect their course and become moreover included into a larger i e high level dynamical system while those same special low level laws are at the same time constraining those high level rules but only as a conditio sine qua non for them The indifference of a layer with respect to the content of higher layers and also to their appearance at all also implies a certain independence of the higher layer with respect to the lower because the latter does not determine the specific content of the higher layer apart from it being a conditio sine qua non The only dependency of a particularly structured higher layer upon the next lower layer is the p l a y that is allowed by the lower layer The lower elements in the sense of lower level laws or rules cannot be trans formed but only be o v e r f o r m e d HARTMANN discusses also the possiblity as it is supposed to occur at the transition from the organic to the psychic of becoming o v e r b u i l t in which case the lower layer remains only in its function of being the carrier of the higher The latter is then only dependent on the capacity of carrying of the lower layer But because we here entertain just a mitigated version of the NOVUM in general we for the time being do not consider a layer to be over built Life as a definitively n e w phenomenon is affirmed not only by philosophers but also by many natural scientists so for example DAVIES P The Cosmic Blueprint 1987 9 p 101 1989 edition Biology will never be reconciled with physics until it is recognized that each new level in the hierarchical organization of matter brings into existence new qualities that are simply irrelevant at the atomistic level Every lower layer co determines it is true the next higher layer but at most only as its matter i e its substrate matter that becomes in formed The independence of higher layers and ultimately the independence of their corresponding categories their nomos is only a partial independence generally because of the mentioned conditio sine qua non o n l y that the lower features are for the higher and more specifically because of the dependence of their content on the degree of available play o n l y a play that is allowed by the lower categories which in most cases are laws And this only partial independence and thus not complete independence causes the higher categories not to be absolutely fundamental The only categories that are truly fundamental are those of the lowest layer namely the mathematical So insofar as absolute fundamental categories are concerned the whole layered system of the World is ontologically homogeneous as found out earlier But the higher layers can nevertheless not be fully understood when we only consider the lowest and absolutely fundamental categories because the lower layer as it is in itself is always o p e n with respect to the presence at all and to the content of some possible higher layer that can appear on top of it That is because of the i n d i f f e r e n c e of the lower layers and their categories as to the specific content of the higher layers that can come next and which are then being carried by them The higher categories say higher forms of concretum concretum determination cannot in any way act against the lower categories but only with them i e having them for themselves to exploit or harness them they can assume whatever quality that fits into the available play of lower categories But despite this autonomy of their s they remain in a certain dependence upon the lower They are thus free within their dependence HARTMANN Der Aufbau der realen Welt p 548 An instructive example is the o v e r f o r m a t i o n not trans formation of the c a u s a l n e x u s i e the repeatable connection between states of dynamical systems The causal nexus as such is not a whole series of states but just the generation of one state effect out of another cause In itself the causal nexus is not directed to predetermined end states If the cause changes for example when it is perturbed a different effect will ensue while the original effect will not take place So causality does not lack direction but lacks direction to a specific target This indifference of the causal nexus as to the final result makes it susceptible to be harnessed i e to be over formed which means that a definite influence can be applied to the cause resulting in a corresponding effect i e a definite specific target In an organism we have causality and this causality in itself as we saw not being directed to a specific end result is susceptible to noise perturbations coming from outside the organism The latter however blocks this noise and posits its own causal factors resulting in a specific outcome In this way a specific structure or behavior generating law appears which is however fully compatible with causality The latter is just over formed not trans formed Of course here the ability of the organism to discriminate between factors that are relevant to it or irrelevant or even noxious is already presupposed But this ability can initially be in a very primitive state i e just a physical function and not yet requiring an organic function In a later stage of organic constitution this ability will become more subtle and effective It is best to assume that both phenomena over forming of causality and the mentioned ability of discriminating between the self and the not self go hand in hand and represent in fact one and the same event of jumpy complexification In this way we get systems in which the elements seem to cooperate in such a way that causality is in a sense masked and that a specific goal is being realized by the dynamics of such systems Such goal oriented directiveness as if teleology came into play although compatible with all physical laws Again the laws how they really are in themselves not insofar as we suppose to know them is irrelevant to physics but relevant to biology What is needed is a comprehensive theory of organization a theory that investigates and explains emerging structures and laws already apparent in the crystallization process and in some other inorganic dynamical systems and that goes all the way up to investigate the staggering organizational processes that take place in living beings The recent dynamical systems approach has entered this line of research but is still a long way from explaining fully even the crystallization process let alone those of organisms And the investigations of Nicolai HARTMANN in the 1930 s and 1940 s which are conducted at a very fundamental and general level already are despite the mentioned inherent difficulties very providential in this respect and should be studied by at least all philosophically minded biologists and physicists who want to place their subject into a broader context And above all it should be studied by philosophers which has as far as my knowledge goes hardly taken place Let s now return to our main concern with respect to HARTMANN s ontological layers which we will interpret for the time being as complexity levels namely the vexed problem of the NOVUM i e the new We again try to illustrate it but now in some more detail in the case of the o r g a n i c in which we do not present a reconstruction of its actual origin but of its constitution Some aspects of the foregoing discussion will be more or less repeated while others are added In the physical layer the mode of determination is causality where determination does not mean a category concretum determination but a concretum concretum determination in the present case standing under the category of causality In the complexification that will eventually lead to the organic layer to emerge complex molecules will be formed by the above described harnessing over formation of causality and the physical laws based on it These molecules encode blueprints for the synthesis of proteins in definite amounts and temporal order These proteins assist in the realization of biochemical reaction chains etc By means of feedback loops the whole system is then controlled All this results in the fact that the organism grows and maintains itself under import of matter and energy and export of matter and entropy according to certain intrinsic rules that are only relevant i e geared to large organic units that play the role of system elements and that interact according to these rules which are consequently high level rules or laws only as such appearing at high structural levels and not present at lower levels And such a rule is not just the result of a simple summation of lower level rules they are of a different nature all together i e their qualitative content is different High level rules can also be observed in inorganic systems but in organisms they imply dynamical sub systems A rule implies a dynamical system because a rule or law of such a system is in a dispersed fashion immanent in the elements of the system that show a resistance against perturbations i e those systems are not allowed to be fed by alien elements As has been said this can initially be just some purely physical function but should further be sophisticated hand in hand with the origin of the higher level rules themselves into the ability to discriminate between Self and not Self which is by the way the essence of Life And in this way there is just a selected set of causal factors admitted while others are neutralized And only in this way we have a self contained set of rules and with it a self contained dynamical system that connect certain specific high level states to subsequent states and finally to certain special high level end states Organisms are therefore called autopoietic systems i e systems that produce themselves that do themselves in contrast to them being done by something else Such self contained sets of high level rules or laws and the corresponding dynamical systems do not occur in the inorganic world where dynamical systems can and will accept in principle any alien factor coming in its way Under circumstances such factors can be damped by such an inorganic system it is true but when the system is unstable it will alter its course in virtue of such an incoming factor Such inorganic systems are because of this in many cases not precisely repeatable in the wild One of the exceptions are presented by the formation of crystals which however involve relatively simple equilibrium systems So the above discussed strongly self contained high level dynamical laws though not occurring in the inorganic world are completely compatible with causality and with the fundamental physical laws based on it When we examine the e l e m e n t a r y state transitions of which the high level state transitions are composed of we will see that they represent each for themselves purely causal connections The NOVUM thus consists of the fact that these elementary causal connections stem from a highly selective i e choosy environment within the system itself resulting in the organization of particular causal connections in turn resulting in a particular and stable succession of complex high level states leading to some specific end state or cycle With all this we have in a crude and more or less vague way described the o v e r f o r m i n g not trans forming or h a r n e s s i n g of causality The latter is as it were the matter for the former which then is its form And so with the presence of such high level laws which taken all by themselves represent a qualitatively different type of NEXUS different from that of causality pure causality still exists at the lower level while at the higher level it is over formed And as has been said in the same way physical laws presupposing causality can be over formed And because the resulting law is of a different type namely a different type of nexus and moreover self contained and high level we call it a n e w c a t e g o r y i e a new determinant in HARTMANN s sense a c a t e g o r i c a l n o v u m as such responsible for the appearance of the organic layer which has taken up many categories from the inorganic layer but more or less being modified by the novum on top of the inorganic physical layer One must realize that the organic layer does not as such consist of individual organisms In each individual organism including humans we find both layers the inorganic and the organic And in some higher organisms yet another layer appears in addition to these two And as is clear from the above the appearance emergence of the NOVUM and with it of the new layer is accomplished by a leap like increase in functional complexity of physical matter This increase of complexity follows a very long and highly entangled path compressed into a very small reaction container The foregoing complicated and more or less unclear discussion that tries to explain the phenomenon of over forming is no more than an attempt to get rid of the awkward notion of the NOVUM as something not being produced somehow but coming out of the blue About the h a r n e s s i n g or over forming of existing physical laws see also DAVIES 1989 The Cosmic Blueprint p 143 and p 149 So in the foregoing we have explained the appearance of the NOVUM namely as the result of over forming HARTMANN however postulates the appearance of the NOVUM as something totally new without antecedents And this NOVUM is responsible for the modification of many categories of the lower layer reappearing in the next higher layer i e is responsible for their over forming Maybe this is true and we keep it in mind i e we accept it for the time being But it implies the NOVUM coming out of the blue which assumption causes the theory to come close to those that assume some non physical directive that is responsible for the organization of many in themselves physico chemical units into functional high level routines as we find them in organisms But as hinted at already maybe we should try to assimulate with the idea of the appearance of something totally new of a totally new principle or set of principles Turning our attention for a while to the m a t h e m a t i c a l non physicalized l a y e r we see something similar to harnessing For example we begin with considering a most primitive starting point the topological space This is a collection of points endowed with only

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Open archived version from archive - General Ontology II

dimensional geometric space However if we interpret this three dimensional space of this three dimensional object as physical space then we have a physical object a physical pyramid the three parts of which still represent the same three group elements 1 p and p 2 So now the whatness of the symmetric structure of this physical object is indeed the group C 3 as given above And the elements of the whatness category that is represented by this group are now indeed pointing not to concepts or their meanings but to objects or their parts which objects and parts can be either purely mathematical or physical Figure 1 The group C 3 interpreted meaning that its elements are represented by parts of an object In order to let the image represent a p h y s i c a l object with the same symmetry this object must be transformed into a gyroid pyramid See next Figures We can do this by erecting a line in the center of and perpendicular to the plane figure and then connecting the top with all the angle points of that figure and interpreting the 3 D space as physical space The group elements p p 2 and 1 are represented by the parts of the 2 D figure as indicated but at the same time they are represented by the implied irregular pyramids together making up the whole gyroid pyramid The rotations are about an axis represented by the mid point of the figure Figure 1a Construction of the gyroid pyramid from the gyroid polygon of the previous Figure Black lines belong to the pyramid s base while the red lines are going up i e leave the plane of the base The next two Figures give the finished construction of the gyroid pyramid It has the same symmetry C 3 as the two dimensional figure depicted in Figure 1 and it can be interpreted as a physical object Figure 1b Three fold gyroid pyramid constructed from its base Figure 1 See also next Figure Figure 1c Same as previous Figure Visible areas colored The pyramid consists of three parts that can represent the three elements 1 p p 2 of the above discussed 3 fold cyclic group Every coherent object in the Physical Layer possesses a specific identity viz its ESSENCE The latter is the object s genotypical whatness while its phenotypical expression is the phenotypical whatness consisting of the complete set of intrinsic properties as analysed to their content The latter is equivalent to the total set of entitative constants of the given object In fact entitative constants include in addition to the corresponding properties also the whole disjunctive set of their conditions So the mentioned total set of entitative constants do include at least the object by which these properties are implied In this total of entitative constants each corresponding property is presupposed as analysed This being analysed comes about by the whatness category of that particular property So all the intrinsic properties of the object are in the context of the total of the object s entitative constants being analysed in virtue of the corresponding whatness categories i e their whatness is explicated which here means determined To determine the whatness of a property is to explicate it as it is in itself i e what it is in itself But of course a property does not have a self It has only a self in a certain sense and this wholly in virtue of the whatness of the thing of which it is the intrinsic property So the self of the property derives from the self of the thing Indeed in all this it is just presupposed to be sure that such itselves with respect to certain things do really objectively exist This presupposition could however be false because it could also be true that the specific identity as referred to above of any object thing is completely determined by the total of all relations it has to all other objects And that would amount to assume that the identity of a given object involves at least a whole Layer of existence But this in turn would imply that such an object does not have any identity of its own Its identity would be either that of the entire Layer for example the whole physical universe or that of several such Layers as a whole or of all Layers as a whole or it would be nothing at all And moreover the mentioned relations that it has with other objects vary all the time certainly in the temporal Layers and so would not point to a definite identity or self of such an object The most convincing argument that genuine objects do have a definite identity can be taken from the fact that organisms do exist especially human beings Such an argument establishes that at least certain selves do exist And such selves are not absent in other objects but probably grade down as we descend into the inorganic domain Another argument is the fact that materials do have certain constant properties which we discussed as material constants and thus referring to intrinsic properties presupposing a kind of self of the material the latter as an integrated summation of all the individual constituent objects that make up this particular material e g individual crystals or individual molecules the material of which they are intrinsic properties So despite the all permeating r e l a t i o n a l i t y which as real relationality is one of the general categories of the Real Sphere of existence The temporal Layers there must be things genuine objects beings themselves If we do not suppose this i e if we let everything to consist of relations only we end up in an infinite regress of things being related to each other where those things themselves are just relations of other things where these things are again just So

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junior antisymmetry groups In such a group e 1 is itself not an element of the antisymmetry group The latter e 1 is a permutation of two states These two states can represent any bivalent quality whatsoever including structural features So the transformation e 1 represents an alternating change between these two states say from A to B and from B to A As has been said the transformation Se 1 e 1 S is an antisymmetry transformation and this transformation is again a cover operation but of a different type i e different from the cover operations of usual symmetry The e 1 component is however not a cover operation Well if some particular symmetry transformation S of an ordinary symmetry group is replaced by an antisymmetry transformation e 1 S the resulting set of transformations still forms a group an antisymmetry group which is isomorphic to the original symmetry group According to usual symmetry a pattern potentially representing an antisymmetry group has lower symmetry than when it is explicitly interpreted as antisymmetry group Interpreting such a low symmetric pattern as an antisymmetry pattern involves extra features of the structure viz the above mentioned dual qualities i e extra features of the given structure are now taken into consideration and so we dig deeper into that structure than by usual symmetry alone The same applies to colored symmetry which is a generalization of antisymmetry All the Space Groups including their anti or colored symmetry versions are infinite groups which means that the number of elements of such a group is infinite the order of the group is infinite In addition to the space group symmetry we have the intrinsic point symmetry of the crystal its POINT GROUP which is the translation free residue of the corresponding Space Group Organisms which all lack a periodic structure have only point symmetry Here we must discriminate between intrinsic and extrinsic symmetry The external shape of a crystal as such determined by its Essence is in the wild often distorted by external agents or by inhomogeneous growing conditions resulting in a lower extrinsic symmetry By i n t r i n s i c s y m m e t r y we here mean the symmetry solely resulting from the crystal s Essence which symmetry will appear under uniform growing conditions free from biasing agents The potential presence of such biasing elements is typical for the non mathematical Layers So only there the distinction between extrinsic and intrinsic is relevant In organisms this difference between intrinsic and extrinsic symmetry is even stronger pronounced but in a different way than it is in crystals In organisms not only perturbations from outside can distort their symmetry for example in plants but also and more importantly many functional demands involve deviations from a symmetry initially primarily present Promorphology attempts to uncover and retrieve such primary symmetries It assesses the ideal basic form of a given organism and of all other inorganic non periodic structures for example molecules or atomic complexes for that matter or of those that can be reduced to non periodic structures for example crystals This point group symmetry can be of three types of which only the first one appears in crystals Isometric symmetries distances preserved They involve the usual symmetry transformations like reflection and rotation and give rise to isometric groups which are finite with respect to all discrete cases as in all crystals Equiformity and even congruity are preserved Apart from isometric Point Groups the mentioned Space Groups can also be isometric and they even cannot be otherwise Similarity symmetries distances not preserved This type of symmetry is a generalization of isometric symmetry In it congruity is not preserved but equiformity is still preserved Its typical symmetry transformations are reflection followed by dilation or contraction and rotation followed by dilation or contraction They give rise to similarity symmetry groups They are infinite and involve in addition to the usual symmetry transformations extra structural features dilations and contractions that supplement the description of structure done by just ordinary symmetry transformations Similarity symmetry groups occur as an approximation in some organisms Conformal symmetries circles are preserved This is a further generalization It consists of circle preserving transformations of the plane E 2 O i e the 2 dimensional plane without the origin For it the property of equiangularity has been preserved but not that of equiformity and gives rise to conformal symmetry groups We have as the elementary transformation of conformal symmetry in E 2 O the circle inversion isomorphic with a reflection which will be explained later When no similarity transformations are present such a group is finite It involves in addition to the usual symmetry transformations extra structural features that supplement the description of structure done by just ordinary symmetry transformations Conformal symmetry could in principle occur in some organisms or parts of them when they are projected onto the plane or when its 3 dimensional analogue is considered instead But as has been said of these three types of point symmetries only the first one the isometries occur in crystals And from point group isometries we can derive antisymmetry groups in the same way as was the case in the Space Groups Also here certain a symmetries allow to be interpreted as being a result of an antisymmetry desymmetrization and the antisymmetry group is again isomorphic with the higher symmetry the higher symmetry that would be present if the mentioned asymmetries were absent So also here we can with antisymmetry dig deeper into the structure of a given crystal Organisms do not have space group symmetry but do have point group isometries and also similarity and presumably conformal symmetries And all those symmetries potentially permit antisymmetry to be detected allowing us to dig deeper into the given organismic structures Analogous considerations apply for colored symmetry As has been related earlier the point symmetry of a crystal which here is always an isometry is the translation free residue of the corresponding Space Group of the crystal A particular Point Group however admits of several possible Space Groups The Point Group describes and is the symmetry of a chemical motif that remains when all translations present as symmetry transformations in the crystal structure are eliminated But there are several different ways in which such a motif can be periodically arranged into a lattice For the 32 Point Groups this results in 230 possible Space Groups So if we go from the crystal s Point Group to its Space Group we dig deeper into the crystal s structure Further a particular Point Group of a crystal as well as of an organism generally admits of several possible Promorphs depending on the number and configuration of the antimers For crystals knowing the symmetry of the chemical motif the chemical motif that remains after elimination of all translations we can dig still deeper into that motif s structure i e deeper than by symmetry alone by considering its chemical composition and configuration As a result we can assess its Promorph giving the number and arrangement of antimers of the given chemical motif and on the basis of this adjudge this promorph to the crystal itself and assess its chemical type especially the precise distribution of certain atoms ions or atomic complexes within the crystal lattice For an organism we can assess its basic symmetry by considering reflection planes and rotation axes When these data are supplemented by the number and configuration of antimers we can assess the organism s promorph which in many cases supplements the information about the organism s structure as obtained by symmetry alone See next Figures Figure 1 Test of a recent sea urchin sanddollar with natural indications of the five antimers extending also internally which together build up the whole animal Although almost regularly 5 fold the animal is only bilaterally symmetric and possesses a mirror plane as its only symmetry element Its symmetry therefore is that of D 1 This group has two group elements namely the identity element and the reflection in the median plane The general promorph of the animal is half a rhombic pyramid or still more generally half an amphitect pyramid and so belongs to the general Heterostaura Allopola It has however five similar antimers and so does not belong to the Allopola zygopleura The true promorph of the body of this animal is half a 10 fold amphitect pyramid expressing in this way the possession of five antimers and implying the promorph to belong to the Pentamphipleura itself being a species of the Allopola amphipleura See next Figure Figure 2 Oblique top view of the promorph of the bodies of irregular sea urchins It is half a 10 fold amphitect pyramid with the five antimers corresponding to five body parts in the animal indicated by colors It therefore belongs to the Pentamphipleura Allopola amphipleura The brown plane facing the beholder is the bisection face associated with the bisection of the 10 fold amphitect pyramid Figure 3 Half a 10 fold amphitect pyramid representing the promorph of an irregular sea urchin The latter s five antimers are indicated by colors The median plane is a mirror plane and as such the only symmetry element of the sea urchin s symmetry The symmetry group therefore is D 1 This group has two group elements namely the identity element 1 and the reflection in the median plane m In the promorph they are represented by its two halves separated by the median plane These halves are asymmetric units But as can be seen each such a half representing a group element consists of two and a half antimers so each group element is in the present case interpreted as a certain set of antimers more precisely as a set of five half antimers or equivalently by two and a half antimer The antimers bodily counterparts of an object are equal or similar and in organisms functionally equivalent body parts that are arranged around the body s main axis or when no such axis is present around the body s center In bilateral objects they are arranged symmetrically with respect to the body s median plane Antimers are important for the assessment of the promorph of such a body See for this and for what is stated in the subscripts of the above Figures the introductory document in the Series on Group Theory Subpatterns and Subgroups Part XIII To see this document in a separate window click HERE The symmetry of an organism as formally implied by its promorph evidently comes about in spite of its one and definite grouptheoretical what is it by a quite different process as is the case for the same symmetry in crystals In the latter the actual generation which is strictly just a co generation of their symmetry is based on lower level processes that have to do with lowest energy configurations of atoms ions or atomic complexes configurations that are going to make up the crystal structure The higher level processes in organisms responsible for their basic symmetry are the result of harnessing lower level laws as explained in Part I The same applies to the phenomenon of growth In crystals this consists in the apposition deposition of constituents molecules atoms ions or atomic radicals onto the surface of the growing crystal without changing the crystal so far formed This apposition takes place in order to reduce surface energy caused by dangling chemical valences So this way of growing is a low level process In organisms on the other hand growth proceeds by intussusception i e by the intake of nutrient constituents into the interior of the organism accompanied by complex metabolic processes This way of growth often changes the organism so far formed It is a high level process a high level growth The mentioned intussusception is possible because an organismic body is in a semi liquid state in contrast to crystals which are solid objects There do exist so called liquid crystals it is true and there intussusception is possible but in that case it is a simple low level process without being accompanied by genuine metabolic processes All the mentioned high level processes embody a leap of increase in material complexity elevating the inorganic to the organic Functional demands absent in inorganic beings where only energy demands count play a decisive role in constituting the organismic structure They have to do with locomotion suspension in water and even ornaments the latter in connection with sexual reproduction All these are high level features and many of them must be maintained against the tendency to lowest energy states Therefore organisms are not thermodynamically equilibrium structures like crystals are but are dissipative structures held far from thermodynamic equilibrium That s why organisms must feed themselves Functional demands are not the sole generators of organic forms In all organisms we can find certain fundamental dynamic subsystems that represent stable dynamics within the developing organisms and result in some basic morphological structures generic forms as we see it for example in phyllotaxis in plants These basic morphological structures often are expressed in the final promorph of the organism or part of it See for all this GOODWIN B 1994 and 2001 How the Leopard Changed Its Spots When all these analyses are done with respect to a given crystal or organism we are left with two types of asymmetric units units that cannot be further analysed grouptheoretically The first type of asymmetric unit is a symmetry left over i e it represents the ultimate motif representing in turn the ultimate group element and consists in the case of crystals of two subtypes The macroscopic left over asymmetric unit resulting from considering the crystal s Point Group The microscopic left over asymmetric unit resulting from considering the molecular symmetry rosette 3 dimensional obtained after all translations have been eliminated This rosette has the same symmetry as the macroscopic intrinsic point symmetry of the crystal The second type of asymmetric unit involves the ultimate promorphological unit which sometimes coincides with the just described symmetry left over unit but sometimes not In the case of crystals it is an antimer or half an antimer of the molecular rosette Macroscopically single crystals do not have genuine antimers because their structure is microscopically periodic Macroscopically these crystals are structurally homogeneous Organisms on the other hand are macroscopically heterogenous and definitely do have antimers macroscopically Crystals often occur as twinned forms In these cases two or more crystal individuals of the same species have grown together in a definite and regular way Such a twinned condition sometimes allows for macroscopic antimers to be detected These antimers are then represented by the crystal individuals which have grown together Consequently such twinned crystals are at a higher structural level aperiodic So in a formal morphological sense they represent a transition from single crystals which are wholly periodic to organisms which are aperiodic with some exceptions among Bryozoans and maybe some other colony like organic individuals It should be realized that in most cases crystal individuals grow together in a totally irregular way resulting in crystalline aggregates as we find them in most rocks Twinning on the other hand is the result of a regular process according to some definite rules yielding typical and repeatable forms In the case of organisms the ultimate promorphological unit is always a macroscopic antimer or half an antimer If such an antimer is symmetric normally according to the symmetry group D 1 it can grouptheoretically be further analysed as to its parts representing group elements What is left then is normally half an antimer which cannot be further analysed grouptheoretically If the antimer is already all by itself asymmetric it is as such a left over asymmetric promorphological unit and cannot be further analysed by Group Theory The structure of all these left over asymmetric units can only be understood by means of a study of the dynamical systems or subsystems that took part in their generation In organisms this involves consideration of organic functions and their evolution The aim of all this research is not to figure out the structure of crystals or of organisms but to establish general ideas and concepts concerning 1 intrinsic properties as intrinsic properties 2 their relation to the Essence of the given object intrinsic being and 3 most important the expression of that Essence and those properties in terms of Categories or equivalently in terms of If Then constants Of these we have the following types as established earlier Whatness Categories In our introductory document Second Part of Website Series on Group Theory Subpatterns and Subgroups Part XIII we distinguished them as just whatnesses contrasting them with categories but here we consider them as whatness categories We have two subtypes Whatness Category of a given intrinsic object being viz its Essence As phenotypical expression of the genotypical Essence it is a composed whatness consisting of all the object s intrinsic properties The latter are simultaneously or successively expressed or suppressed by the Essence as it acts as the dynamical law according to which the object is generated and maintained Think of a caterpillar developing into a butterfly As Essence it is the genotypical whatness of the object Whatness Category of an intrinsic property As such it determines the what is it of that property resulting in that property as analysed in an ontological sense The properties which will concern us most are as has been said intrinsic Shape intrinsic Symmetry and Promorph Entitative Constant i e a category determining the implication of an intrinsic property of which its whatness is already presupposed by a given intrinsic object Nexus Category causality general and special laws This category connects states of a dynamical system and as such when all possible trajectories are considered is a dynamical law The fundamental conservation laws of Physics are contained within such a law If it is a dynamical law of a totality generating system

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they are taken generally as just Shape Symmetry and Promorph and also when special examples are discussed for the sake of them understanding generally Crystals and Organisms Shape Symmetry and Promorph Growth Rate Vector Rosette A fter having specified and delimited the investigation concerning special If Then constants categories resulting in an investigation in which we compare crystals and organisms with respect to intrinsic Shape intrinsic Symmetry and Promorph as some of their intrinsic properties we now show how we shall deal with them in more detail Crystals and Organisms Often it is attempted to more or less equate crystals and organisms For example HAECKEL E Generelle Morphologie der Organismen 1866 Kristallseelen 1917 and PRZIBRAM H Die anorganischen Grenzgebiete der Biologie 1926 And indeed there are many similarities between the two Organisms as well as crystals are intrinsic beings i e they are in principle repeatable patterned entities originated by some regular process Both occur in the form of individuals which here means that they are not only individual which we could also say of a certain color to which we point with the finger but genuine individuals individua that are ontologically subsistent i e that are ontologically independent They are substances in the sense of our updated Substance Accident Metaphysics First Part of Website Both show the phenomenon of growth Both show healing of wounds damage by regenerative growth at the wound site resulting in restoring the damage Both show a high degree of specifity with respect to chemical composition symmetry shape and many other properties Besides these similarities there are still others having lead to setting up so called c r y s t a l a n a l o g i e s For some researchers they were supposed to be more than that resulting in equating crystals and organisms Of course they admit differences but these were not supposed to be of a fundamental nature However it is generally acknowledged that organic beings differ strongly from inorganic beings of which the most instructive representatives are crystals Crystals are thermodynamically equilibrium structures They represent a lowest energy configuration of their constituent atoms Organisms on the other hand are thermodynamically far from equilibrium structures To maintain themselves they must continually import matter and energy and export matter and entropy Organisms consist of parts organelles or organs that execute special and often highly complex functions All these functions are integrated and geared toward the maintanance of the whole In crystals there are no such parts Organismic structure is ultimately the result of some evolution and consequently is historic in character An organism contains an instruction code for its individual development and maintenance and this code is being transmitted and transformed improved etc during evolution by means of natural selection of certain mutants Crystals lack such an inbuilt code and lack evolution Their structure is not historic The organismic body is in a semi liquid state while that of a crystal is normally in a solid state sterro crystals There are however also some so called liquid crystals rheo crystals but the majority of crystals are solid entities minerals Although some crystal species have very complex bodies indeed for instance many natural silicates the material complexity of organismic bodies by far surpasses that of crystals In the organismic body also when it is already fully developed complex metabolic processes take place all the time while crystals are apart from some internal vibrations static objects A crystal has a periodic microstructure while organisms do as far as we know not possess such a structure they are tectological These differences and many more clearly indicate the high degree of difference between crystals and organisms They cannot however belong to different worlds In Part I we have interpreted this difference as originating wholly from a difference in degree of material complexity However more or less paradoxically not in a gradual way The Organic Layer has come into existence from the Inorganic Layer by a sudden leap of complexity resulting in the basis of Life that can now evolve further and further by improving and diversifying this basis What we now see in the behavior and inner workings of an organism is the execution of harnessed special natural laws or rules i e low level rules as they reign in the Inorganic Layer are over formed but not contradicted resulting in high level rules only as such appearing at higher structural levels of the organismic body So while for example growth is in crystals a low level process consisting in sterro crystals i e solid crystals of apposition of microscopic parts onto the crystal surface in organisms part of the food after it has been pre processed is being introjected into the interior of the organism i e from the gut into the organism s interior where it is being transformed and distributed within the body Although the ultimate result of the feeding activity of the organism is like in crystals its enlargement it is a very complex process resulting in a high level growth Growth Rate Vector Rosette in Crystals We will now discuss a feature viz the Growth Rate Vector Rosette from which we can derive with respect to crystals as well as with respect to organisms intrinsic Shape intrinsic Symmetry and Promorph We will discuss it first for crystals and then for organisms Crystals can originate and grow from solutions vapors melts and from other crystals Generation from solutions is for us the most important one because it is most suitable for a comparison with organismic bodies In the case of crystallization from a solution we have to do with two phases viz the dissolved phase and the crystalline phase Under different temperature pressure conditions the stability of one phase or another can be different Say we have a solution of some chemical substance and say the mentioned thermodynamic and concentration conditions are such that the dissolved phase has become unstable while the crystalline phase is stable then crystallization will occur along the following lines The creation of a crystal must start from a seed i e a very small fragment of the crystalline material belonging to the same substance that is dissolved or it must start from some substitute of such a seed Let us consider the former First we should in order for a crystal to be formed have the accidental formation of an embryo that s how we will call the just mentioned seed in the solution This is in principle possible because in the solution the particles of the dissolved substance constantly bump into each other bounce away etc So among the many temporary configurations arrangements there will from time to time appear a configuration having the structure of the type to be crystallized If such an embryo is of a sufficient size the surface energy of that embryo which embryo can now be called a crystallization nucleus does not cause the total free energy of the nucleus to increase when it grows but to decrease and consequently the nucleus will grow resulting in a crystal The intrinsic Shape of the growing crystal will now be determined by the different atomic aspects as presented to the nutrient environment by the several crystal faces This atomic aspect consists of two factors The chemical nature of the growing face atomic aspect s str The number of lattice nodes that the face encounters depending on the geometry of the lattice Both factors together determine the set of growth vectors of the corresponding set of growing crystal faces resulting in a number of vectors originating from the crystal s central point and directed outward In short we get a Growth Rate Vector Rosette Each individual vector in this rosette indicates the direction of growth of a particular face while its length indicates the growth rate So this rosette indicates anisotropic growth i e different growth rates of a given crystal in different classes of directions resulting not in a spherical shape but in a polyhedric shape of the crystal This shape as intrinsic shape is determined by 1 and 2 and consequently it is determined by the internal structure of the crystal which is represented by its Space Group plus chemical composition i e a particular type of lattice provided with particular chemical motifs of which by implication not only the symmetry is given but also the complete chemical nature and atomic configuration The internal structure describable by the Space Group Chemical Composition is itself directly caused by the relevant fully physically interpreted crystallization law which is the dynamical law of the dynamical system that consists of the growing crystal in its growing environment This dynamical law is metaphysically interpreted as the Essence of the crystals i e the essence of any individual single crystal that is generated according to that particular dynamical law Other crystal species are associated with different dynamical laws In order to explain all this i e about growth rates and Shape in crystals we will a little further down reproduce some material from two documents from the First Part of Website viz from The Morphology of Crystals and from The Internal Structure of Crystals Chemical nature of the atomic aspect determining growth rate Forms If we take a face of a crystal of a particular symmetry Class a face cutting off certain distances of the corresponding system of crystallographic axes and if we subject it to all the symmetry elements rotation axes mirror planes center of symmetry of that Class then this face will produce copies of itself according to the symmetries of that Class If we happened having to do with the Asymmetric Class or if the prevailing symmetry elements of some other Class do have a special orientation to it then the face will not be so multiplied This set of copies we call a Form in the crystallographic sense and the constituent faces of such a Form are crystallographically equivalent More about these Forms see the above mentioned document The Morphology of Crystals Some Forms are closed i e their faces together form a polyhedric figure like a cube or a tetrahedron Others are open like pyramids without their base prisms without their top and bottom faces or are just a set of two parallel faces or just one face While the closed Forms can exist as real crystals the open ones must be combined with other Forms of the same crystal Class such that a closed figure is resulting But also closed Forms can combine with each other one cutting off the corners of the other So we can think of a crystal as consisting of one or more Forms of the corresponding crystal Class to which the crystal belongs This boils down to imagining a crystal as made up of faces How the above concept of basic crystallographic Form has its foundation in the crystal s internal structure One might wonder why we can so easily speak of faces that together build up a Form consisting of the same type of face i e consisting of equivalent faces Surely Crystals are the product of some Lego set of faces that can be attached to each other But no crystals are not hollow structures but filled solids without container walls In order to explain why the above constructions are nevertheless conceptually possible and founded in real state of affairs let us consider a certain fictitious two dimensional crystal consisting of two different sorts of constituents say two different species of ions electrically charged atoms or groups thereof with opposite electrical charge Let us represent those ions by two kinds of disks gray ones and red ones Because like charges repel each other and opposite charges attract each other each negatively charged ion wants to collect as many positive ions as its nearest neighbors as possible and each positively charged ion wants to collect as many negative ions as its nearest neighbors as possibe And every ion avoids a position next to any ion having a charge of the same sign The following crystalline structure will be the result of this in this figure we depict only the internal structure and not its external boundaries so we depict just an arbitrary fragment of the crystal Figure 1 Internal structure of a two dimensional ionic crystal consisting of two different sorts of ions of opposite electrical charge The total symmetry of this structure is according to the Plane Group P3m1 Its Point Group Crystal Class is 3m isomorphic to D 3 To study this Plane Group geometrically click HERE for the document where it is discussed First Part of Website Internal Structure of Crystals Part V This structure allows several kinds of flat faces to be developed Each type of face presents a different atomic aspect to the growing environment the nutrient environment In the next figure we show that the structure under consideration allows three different types of faces A B and C corresponding to three atomic aspects to the environment The blue lines indicate these possible faces and we must interpret the aspects as true aspects in so far as they are facing outwardly i e towards the environment Figure 2 Examples of the three different kinds of faces allowed by the structure according to the three different atomic aspects pictured below Figure 3 Three different atomic aspects of the structure of the figures 1 and 2 When we look to aspect A we see that six faces are possible i e six faces differing in orientatation but showing the same aspect to the growing environment The nutrient material cannot distinguish among these six When conditions in the environment are the same around the entire crystal all six faces will grow at the same rate And if they are the only faces appearing then the ideal crystal will have the shape of a regular hexagon as shown in the next figure and will not reveal the lower symmetry of the structure which has the symmetry of an equilateral triangle not of a regular hexagon Figure 4 Regular hexagon bounded by A faces only This hexagon does not however possess the full symmetry of a regular hexagon One set of mirror lines is suppressed and the rotation axis is three fold not six fold To show this clearly let us look to a two dimensional structure which has the full symmetry of the hexagon Figure 5 A structure showing the full hexagonal two dimensional symmetry This structure is that of a two dimensional crystal consisting of only one atomic species It has full hexagonal two dimensional symmetry namely a six fold rotation axis and two sets of mirror lines each set consisting of three equivalent mirror lines The structure of Figure 4 on the other hand has a three fold rotation axis instead of a six fold one It lacks one set of mirror lines namely the red ones Therefore this structure is analogous to a hemihedric Form from the domain of three dimensional crystals Such a less dimensional Form can only combine with other such Forms belonging to the same symmetry Class So our hexagonal Form as depicted in Figure 4 can only combine with a triangular Form Such a triangular Form could be bounded by three B faces the B faces alone can only form triangles because only three of them having different orientations are possible in our structure as depicted in the Figures 1 and 2 if indeed these three faces grow slowly enough to appear Figure 6 Triangular Form possessing the full symmetry of an equilateral triangle It has the same symmetry as the hexagon of Figure 4 and can therefore be combined with it The fact that faces that grow slowly enough will appear and remain while fast growing faces disappear quickly sounds a little paradoxical but let me explain this before we continue with the possible combinations of Forms The growth rate of a crystal face depends in so far as only the crystal is concerned on the atomic aspect s str the face presents to the nutrient environment It can further depend on external influences for example on the presence of certain chemicals impurities in that environment A second factor determining the growth rate is the geometry of the lattice of the given crystal see below Now the slower the face grows the more likely it will survive while relatively fast growing faces will ultimately disappear as the next Figure illustrates Figure 7 Successive stages in growth of an imaginary two dimensional crystal Growing faster than the x faces the y faces may finally disappear So when the A faces of the structure depicted in the Figures 1 and 2 appear and the B faces grow slowly enough for them also to appear then the hexagon al outline will be mixed with a triangle i e we get a combination of two Forms both having the same symmetry Figure 8 The combination of two Forms in a two dimensional crystal Here three corners of the hexagon are cut off by a triangle and thus revealing the structure s true symmetry But it could also be that the B and C faces both grow very slowly relative to the growth rate of the A faces The latter will disappear In this case we will get a combination of two triangles And because the aspect to the nutrient environment is different in both cases B aspect and C aspect see Figure 3 the faces will almost certainly grow at a different rate resulting in triangles of different sizes and in this way again reflecting the lower symmetry of the structure i e lower than the full symmetry of a regular hexagon See the next Figure Figure 9 Combination of two unequally developed triangles in a two dimensional crystal The triangles cut off each others corners So for two dimensional crystals we have now explained the growth of faces and the combinations of Forms For three dimensional crystals

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internal structure can be seen as being eidetically determined by its lattice prescribing the mode of periodic repetitiveness within the crystal provided with chemical motifs where with respect to the mentioned internal symmetry only their point symmetry matters and not the additional geometric and structural features of these motifs This lattice plus symmetry representatives of the chemical motifs constitute the crystal s Space Group and for two dimensional crystals we have their Plane Groups Such a space or plane group is then a set of symmetry transformations that forms a definite algebraic structure under the operation of combining those symmetry transformations a structure which is called a group When we now conceptually eliminate all translational elements i e eliminate all translations wherever they occur in the crystal from this Space Group or from the Plane Group in the 2 D case we end up with the translation free residue the symmetry of which is then the crystal s Point Group The latter is the eidetical If Then constant whatness determinant whatness category determining what the intrinsic non translational symmetry as it happens to occur in the crystal in itself is This Point Group thus eidetically determines the intrinsic symmetry of the single crystal as a macroscopic non translational object because all translations which are by themselves microscopic are eliminated And this symmetry of the crystal reflects the symmetry of the microscopic chemical motif that remains after all translations are eliminated So we can let represent the crystal s intrinsic point symmetry by a macroscopic motif having this same point symmetry This symmetry very often does not coincide with the s h a p e of the given crystal even not with its intrinsic shape that results from a crystal being fully developed within uniform conditions So in order to express these features conveniently we can insert the appropriate macroscopic motif into a drawing of the intrinsic shape of a given crystal species While the symmetry of this motif expresses the crystal s intrinsic point symmetry the morphology of this motif together with its symmetry determines the promorph i e it determines the number of antimers Recall that in many cases a given definite point symmetry can allow for several different promorphs to be possible So in the diagrams of 2 D crystals next document we will depict the following items Intrinsic Shape Vector Rosette of Actual Growth Intrinsic Point Symmetry Distribution of symmetry elements The crystallographic Forms of which this shape consists The crystal s macroscopically interpreted Antimers The Promorph based on the second and fourth items where point symmetry and promorph will in a number of cases be symbolized by an inserted macroscopic motif See next two Figures reflecting the microscopic motif as translation free residue And the detailed structure of this motif reflects in addition to the point symmetry the number and arrangement of antimers Recall that the Promorph for three dimensional objects is the simplest geometric solid expressing geometrically all the symmmetries and the number and arrangement of the antimers of such an object For two dimensional crystals we will indicate the two dimensional analogue of such a solid Such analogues will manifest themselves mainly as polygons They often can go by the same names as their three dimensional counterparts All this will be done for two dimensional crystals only because they are easy to depict and nevertheless reflect some important features and general principles of 3 D crystal structure It is further clear that we here concentrate on the geometric aspects of crystal structure not on chemical types And while these aspects are causally dynamically dependent on the supposed chemistry of the crystal constituents we presuppose them i e these aspects and consider them eidetically The next two Figures give some examples of inserted macroscopic motifs and their symmetry Figure 11 Possible two dimensional motifs of two dimensional crystals The symmetry of a macroscopical version of such a motif inserted in a drawing of a crystal shape can indicate the crystal s intrinsic point symmetry The morphology of the motif allows for the crystal s promorph to be assessed because it shows the number and arrangement of antimers The next Figure illustrates the symmetry of those motifs which are according to the groups D 2 D 1 and C 1 Figure 12 Symmetry of the two dimensional motifs of the Figure above Mirror lines indicated by red lines Two fold rotation axis indicated by a small solid green circle The C 1 motif does not have any symmetry at all besides the trivial symmetry transformation 0 0 or equivalently 360 0 rotation about any axis Shapes of Two dimensional Crystals and Crystallographic Forms Crystals in our case 2 D crystals are bounded by possible faces Possible crystal faces are either parallel to the edges of the unit mesh which is that microscopic unit that is repeated periodically or follow oblique stacking boundaries such as indicated in the next Figure Figure 13 Possible faces red of a 2 D crystal with rectangular building blocks unit meshes As has been explained earlier the total set of crystal faces present on a given crystal is constituted by one or more Forms in the crystallographic sense Recall that such a Form originates as follows If we suppose one of the possible faces to be present then the presence of copies of this face can automatically be expected depending on the intrinsic point symmetry of the crystal and on the position of this face with respect to the symmetry elements in 2 D crystals reflection lines and or rotation axes of the crystal When this produces a closed figure it could be a crystal If only an open Form is produced for instance two parallel lines and nothing more then such a Form must be combined with one or more other Forms in order to represent a crystal So the shape of a crystal consists of one or more stronger or weaker i e prominent or not so developed crystallographic Forms In this way a rectangular 2 D crystal could consist of four three or two Forms depending on the intrinsic point symmetry of the crystal i e on its Crystal Class In the ensuing series of diagrams we will indicate the crystallographic Forms involved in making up a given intrinsic shape of some 2 D crystal Crystals and Organisms It is especially the promorph that eidetically connects crystals and organisms by determinative threads Apart from symmetry the promorph relates to purely morphological features often not accounted for by symmetry alone The number of antimers is generally not determined by symmetry alone but also by conspicuous morphological features The intention of this and the following documents indeed is to bring crystals and organisms together as far as possible realizing of course that they have only a limited set of features in common Crystals are periodic structures based on certain lattices and although it is perhaps possible to suppose such lattices also to be present in certain organismic types it is far from certain An organic lattice could be present in the form of some chemical system something like a Turing pattern that provides for position information in a developing organism it is true but in many other cases like in spiral organisms it cannot be as such present Often the regular spatial structure of organismic bodies is determined by certain stable and robust dynamical subsystems operating during morphogenesis without the necessity of some pre existing or pre generated organic lattice as for instance in the phyllotaxis of plants where leaf distribution in the growing plant is accomplished by a simple robust dynamic of repulsion of leaf primordia resulting in the overall symmetry and promorph of the plant in the form of discrete spirals leading to a basic form that is somehow intermediate between Spiraxonia continuous spirals and Stauraxonia pyramids and bipyramids See for a System of these promorphological categories our System of Basic Forms in Second Part of Website A general pattern theory will connect crystals and organisms by determinative whatness categories and in this way connect the Mathematical Layer on the one hand with the Physical and Organic Layers on the other which are otherwise separated by a huge complexity gap The latter already exists between the Physical i e Inorganic Layer and the Organic Layer but is especially prominent between the Mathematical Layer and the Non mathematical Layers and therefore Temporal Layers i e between the Ideal Sphere s str of Existence and the Real Sphere s str of Existence See for this Part One Promorphs as whatness categories Here we repeat from Part I a section on the ontology and essence of promorphs or basic forms in order to clearly understand what promorphs in fact are and especially what they are as co determinants of the overall spatial structure of crystals and organisms Intrinsic symmetry is one of the aspects of the structure of a being The promorph is also such an aspect It presupposes intrinsic point symmetry and is in many cases equivalent to it apart from the fact that it involves exclusively geometric and dimensional aspects of structure while in other cases the promorph goes beyond symmetry especially in all cases where the number of antimers counterparts of a body is not fixed by symmetry alone So like symmetry the promorph is an intrinsic property which is implied by the Essence of the given being and is as such part of that being s whatness We can say that an object where we always mean an intrinsic being has or has not symmetry And only when we consider the trivial group C 1 to be a symmetry as well we can say that every object has a certain symmetry or equivalently is symmetric In the same way we can say that an object has or has not a promorph And only if we consider completely irregular bodies namely bodies that do not possess a definite geometric body center nor repeating parts nor similar or homologous parts as having a promorph as well i e if we consider the Anaxonia acentra of our Promorphological System to represent a promorph as well then we can say that every object has a promorph or equivalently is promorphic If on the other hand we do not so consider then all promorphic bodies and only promorphic bodies show a definite pattern of polarities with respect to their axes as well as to all other parts of their overall structure A body axis can be either homopolar which means that both ends of that axis represent or are embedded within equal or similar body parts or heteropolar which means that one end of that axis represents a body part unequal to the body part represented by the other end of that axis Think of head tail as representing a heteropolar axis of the body of many an animal As with axes these polarity features can also be present in other structural elements of a body It is possible to express these polarities in a geometric way and this is what Promorphology is doing it finds the simplest geometric bodies or figures that fully express these polarities geometrically which is equivalent to geometrically express the intrinsic point symmetry of the object and the number and arrangement of its antimers See for Promorphology Second Part of Website Main Section Basic Forms Promorphological System So in line with the Category of Symmetry we can establish the Category of Promorphology and the corresponding more special promorphological categories Recall that categories are If Then constants and are consequently expressed in an ontological fashion i e as determinants and not as logical definitions Category of Promorphology whatness category If an object possesses at least a geometric body center or allows for certain definite body axes then it is promorphic If we consider the Anaxonia acentra to be a promorph as well then all beings are promorphic The concretum of this category is all beings objects that are promorphic but more strictly it is promorphic as unanalysed The corresponding logical definition would run as follows An object is called promorphic if it contains at least a geometric body center or allows for certain definite body axes The logical and ontological expressions are logically equivalent but the logical definition is not a category General Category of Promorph whatness category If a given being object is promorphic then its body possesses a definite pattern of polarities not only of its body axes but of all parts of its overall structure This pattern is its promorph The concretum of this category is all beings objects that have a promorph but more strictly it is promorph as unanalysed Although these two categories viz the Category of Promorphology and the General Category of Promorph are more or less equivalent it is contributing to clarity if we present them separately A promorph or equivalently a stereometric basic form for 3 D space or planimetric basic form for 2 D space implies and contains the corresponding symmetry group while the latter in turn implies and contains the corresponding abstract group See for symmetry group and abstract group The Intermezzo of Part XIII of the Series Subpatterns and Subgroups in Second Part of Website Click H E R E to see this Intermezzo in a separate window To leave it again close the window To actually go there click the just given Second Part of Website link then scroll the left frame through the Series on Group Theory then click on SEQUEL TO GROUP THEORY two dimensional patterns and then finally click on Part XIII of the Series on Subpatterns and Subgroups and scroll down to INTERMEZZO So the structure of the General Category of Promorph can be visualized as follows The following is a statement that is wholly equivalent to that of the General Category of Promorph but is as such not ontologically meant but methodologically A promorph is the simplest geometric solid expressing the intrinsic point symmetry of a given being and the number and arrangement of that being s bodily counterparts antimers This simple geometric solid directly depicts the polarity pattern of that given being In the exposition of the special categories of promorphs we will use this methodological definition while nevertheless interpreting it o n t o l o g i c a l l y i e m e a n i n g it according to the content of the above General Category of Promorph So let us then give a special promorphology category Category of Allopolar Pentamphipleural Promorph or just Category of Allopola pentamphipleura whatness category If we have a promorphic being object of which the pattern of polarities can be expressed as half a ten sided amphitect pyramid and in a 2 D case as half a ten sided polygon then the promorph of this being is that of the Allopola pentamphipleura The mentioned pyramid viz half a ten sided amphitect pyramid must then be explicitly indicated Figure 14 Oblique top view of the promorph of the bodies of say irregular sea urchins Besides sea urchins also many flowers have this promorph It is half a 10 fold amphitect pyramid with the five antimers corresponding to five body parts in the animal indicated by colors It therefore belongs to the Pentamphipleura Allopola amphipleura The brown plane facing the beholder is the bisection face associated with the bisection of the 10 fold amphitect pyramid The concretum of this category is all beings objects that have this promorph but more strictly it is this promorph as partly unanalysed namely as just belonging to the Allopola pentamphipleura or just Pentamphipleura The Category of Allopola pentamphipleura implies and contains the Symmetry Group D 1 and the latter in turn implies the Abstract Two element Group The Abstract Two element Group can be given by the following group table Its elements are not in any way interpreted They are what they are in virtue of their relations as defined in that group table From the table it is clear that the element a is the group s identity element If we now interpret the element b as a reflection in a plane which is possible because this element has like a reflection a period of 2 i e bb identity element and call it correspondingly m while at the same time seeing the identity element as a rotation of 0 0 no turn or 360 0 full turn and thus denoting it by 1 we obtain a symmetry group namely the dihedral group D 1 isomorphic with the Abstract Two element Group and defined by the following group table This group D 1 can itself be a whatness category namely the Category of Bilateral Symmetry as we defined it HERE in Part I So the structure of the Category of Allopola pentamphipleura as such a special promorph can be visualized as follows This promorph viz that of the Allopola pentamphipleura which as such is just a geometric figure involving just 3 D metric space and not physical space and so still being a purely mathematical entity can now be physically interpreted viz inorganically or organically We will often find it interpreted organically as in irregular sea urchins and in flowers of at least many Scrophulariaceae However this organic interpretation of the promorph does not mean just its materialization i e transition of the involved space from mathematical to physical space of the promorph We generally don t find organisms appearing as pyramids or other such polyhedral solids The promorph especially that of an organism is not present in it just like that It is an i d e a l basic form reflecting the pattern of polarities that is as whatness category only directing the organism s overall spatial structure Indeed it is only directing the latter not rigorously determining it because there are many other organic factors that will partially overrule it The actually emerging overall spatial structure of the given organismic individual

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above but now with a compatible motif s str black inserted in order to show the equivalence of all four vectors a b c d of the vector rosette And the next Figure depicts the same but with another also compatible motif i e we still have the case of a rectangular crystal having D 2 symmetry and four antimers Both Figures also represent the first mentioned type of configuration of the four antimers which we here depict again but without the Vector Rosette Figure above Interradial configuration of the four antimers green yellow of the above described rectangular two dimensional crystal with intrinsic D 2 symmetry The second mentioned type of configuration of the four antimers is depicted next also without the Vector Rosette Figure above Radial configuration of the four antimers green yellow of the above described rectangular two dimensional crystal with intrinsic D 2 symmetry To symbolize this second type of configuration we insert a motif black Radial configuration of four antimers emphasized by an inserted motif To see in what way the motif shown by a comparable motif also having D 2 symmetry is actually only microscopically present and then periodically repeated in the rectangular crystal see the relevant Figure in Part V Sometimes such a motif is smeared out as it were because of glide reflections of which only the reflection component is macroscopically visible because the translations are very small When in addition to the simple translations also the translational component of such glide reflections is eliminated turning them into ordinary reflections which is what we do when looking to the crystal macroscopically we are in the present case left with a translation free residue with D 2 symmetry See for this latter case the relevant Figure in Part V The next two Figures depict and name the promorph of the above considered two dimensional crystals It is either the 2 dimensional analogue of a rectangular pyramid in fact it is its base with four antimers yellow green first mentioned type of configuration of the four antimers or second mentioned type of configuration of the four antimers see for further explanation below the 2 dimensional analogue of a rhombic pyramid in fact it is its base also with four antimers The next Figure explains these two promorphological possibilities They depend on how the motif lets itself be interpreted in terms of antimers Figure above The chemical and morphological features of a two dimensional chemical D 2 motif determine which parts of it to be interpreted as antimers In the case of a motif consisting of four antimers the promorph can then be either that of the Autopola Orthostaura Tetraphragma interradialia or that of the Autopola Orthostaura Tetraphragma radialia The spatial configuration of the four antimers green yellow indicated by numerals is different in each case See for this difference also our Promorphological System of Stereometric Basic Forms Promorphs at Autopola in Second Part of Website BASIC FORMS See also next three Figures The delineation of the four antimers can be expressed in several equivalent ways as is shown in the next Figure for the radial configuration Alternative ways to express the delineation of the four antimers in the radial case The next five Figures are about cross axes and directional axes Figure above System of cross axes of a rectangle A rectangle but also all other amphitect laterally compressed polygons possesses a special system of cross axes Generally cross axes are lines that go through the center where each such line connects two definite parts for example corners or centers of sides of a polygon When as in amphitect polygons there is one and only one pair of such axes of which the members are perpendicular to each other we call them directional axes In the Figure the lines dd and d d And if the directional axes are radial then we call the corresponding promorph radial as in Autopola Orthostaura Tetraphragma radialia If on the other hand the directional axes are interradial then we call the corresponding promorph interradial as in Autopola Orthostaura Tetraphragma interradialia The next Figure shows the system of cross axes of a square It consists of two pairs The members however of each pair are perpendicular to each other implying that we cannot distinguish one unique pair as representing the directional axes of the figure which means that this figure doesn t have directional axes at all System of cross axes of a square The next Figure finally shows the system of cross axes of a six fold amphitect polygon Figure above System of cross axes three pairs red blue blue of a 6 fold amphitect polygon The perpendicular ones red are the directional axes of the polygon In the above drawing there is another pair of cross axes of which the members seem to be perpendicular to each other in addition to the pair red mentioned earlier But the perpendicularity of that second pair is only of an accidental nature implied by the special demensions of this particular figure Generally in any amphitect polygon all cross axes involve angles different from 90 0 except for one pair of them See next two Figures Six fold amphitect polygon with its cross axes Six fold amphitect polygon with its cross axes D 2 symmetry and two antimers Chemico morphological features of motifs could be such that there are only two antimers present in the rectangular two dimensional crystal while its symmetry is still according to D 2 We can depict such antimers green yellow as follows Note radial R and interradial IR directions a somewhat longer rectangle is chosen here in order not to let the antimers look like squares Figure above A two dimensional intrinsically rectangular crystal with intrinsic D 2 symmetry and with only two antimers green yellow And with the Vector Rosette of Actual Growth added The next Figure is the same D 2 crystal two antimers but now with an appropriate motif s str inserted black to show more clearly the equivalence of all the four vectors a b c d of the Vector Rosette of Actual Growth blue lines The next Figure gives the promorph and its name of this crystal D 2 two antimers It is a 2 D analogue of a rhombic pyramid with two antimers Because there is only one way of arrangement of the two antimers possible the simplest geometric figure to express this promorph is a rhombus The rectangle of the crystal has two types of sides long and short while the rhombus has only one There can in principle be any even number of antimers dependent on the geometry of the translation free chemical motif The Figures below show the cases of six and of eight antimers D 2 symmetry and six antimers Figure above A two dimensional rectangular D 2 crystal with six antimers green yellow and indicated by numerals And with the Vector Rosette of Actual Growth added Figure above A two dimensional rectangular D 2 crystal with six antimers green yellow blue and indicated by numerals Vector Rosette of Actual Growth added The corresponding promorph and its name is given in the next Figure It is a two dimensional analogue of an amphitect pyramid with six sides The amphitect polygon representing the two dimensional promorph has two types of sides as does the rectangle of the crystal The amphitect polygon is however more appropriate to represent the promorph because each of its six corners represents one of its six antimers while in the rectangle we do not have six corners nor six sides to represent the antimers Figure above Promorph of a D 2 two dimensional crystal with six antimers green yellow blue D 2 symmetry and eight antimers and with the Vector Rosette of Actual Growth added The next Figures give the promorph and its name of the above crystal D 2 eight antimers It is the 2 D analogue of an amphitect pyramid with eight sides Figure above Same as previous Figure Median lines of antimers omitted in order to let the latter stand out more clearly The degree of flattening of the amphitect polygon that depicts this promorph is arbitrary as long as it remains flattened so the next will do also C 2 symmetry Chemico morphological features of motifs could be such that the true symmetry of the rectangular two dimensional crystal is not D 2 but has a true symmetry according to the Cyclic Group C 2 Its only symmetry element is then a 2 fold rotation axis Its symmetry group therefore consists of two group elements viz the identity element and a half turn p where p 2 1 identity element In spite of this the shape of the crystal can still be rectangular Two fold motifs i e motifs with C 2 symmetry arranged according to an oblique lattice where the angles between the translations happen to be 90 0 i e where the meshes of the crystal lattice happen to be rectangles periodic stacking of which results in a rectangularly shaped crystal The shape of our rectangular C 2 crystal consists of a combination of two crystallographic Forms One consisting of two parallel horizontal faces indicated by red coloring the other of two parallel vertical faces indicated by dark blue coloring Crystals with intrinsic C 2 point symmetry can in principle have any even number of antimers We will give some examples C 2 symmetry and four antimers The four antimers yellow green of this case i e the case of a rectangularly shaped 2 D crystal with C 2 symmetry having four antimers can be depicted as follows where the C 2 symmetry of the rectangular crystal is indicated by red hooked lines They indicate the equivalence of the vectors a and d and of the vectors b and c and the non equivalence of the vectors a and b and of the vectors c and d and further the non equivalence of the vectors a and c and finally of the vectors b and d of the Vector Rosette of Actual Growth of the crystal A somewhat better representation of an intrinsically rectangular two dimensional crystal with intrinsic C 2 symmetry and with four antimers is the following and with the Vector Rosette of Actual Growth added The next Figure gives the promorph and its name of the above crystal C 2 four antimers It is a two dimensional analogue of the four fold amphitect gyroid pyramid i e it is its base The four antimers are indicated yellow green Its three dimensional counterpart i e the corresponding pyramid is given in the next Figure slightly oblique top view C 2 symmetry and two antimers Chemico morphological features of motifs could be such that the true symmetry of the rectangular 2 D crystal is again C 2 while it possesses only two antimers These two antimers green yellow can be depicted as follows where the C 2 symmetry of the rectangular crystal is indicated by red hooked lines They indicate the equivalence of the vectors a and d and of the vectors b and c and the non equivalence of the vectors a and b and of the vectors c and d and further the non equivalence of the vectors a and c and finally of the vectors b and d of the Vector Rosette of Actual Growth of the crystal A slightly better representation of our two dimensional rectangular C 2 crystal with two antimers is the following and with the Vector Rosette of Actual Growth added The next Figure gives the promorph and its name of the above depicted rectangular crystal C 2 two antimers It is a two fold amphitect gyroid polygon and as such a two dimensional analogue of the corresponding three dimensional two fold amphitect gyroid pyramid Its three dimensional analogue the pyramid is depicted in the next Figure slightly oblique top view There can in principle be any even number of antimers dependent on the geometry of the translation free chemical motif The next Figures show the case of eight antimers C 2 symmetry and eight antimers Or the same symmetry C 2 and number of antimers differently expressed and with the Vector Rosette of Actual Growth added The next Figure gives the promorph and its name of the above rectangular crystal C 2 eight antimers It is an 8 fold amphitect gyroid polygon and as such the two dimensional analogue of an 8 fold amphitect gyroid pyramid i e it is its base The eight antimers green yellow are indicated The next Figure gives this same promorph but now with the median lines of the antimers omitted in order to let the latter stand out more clearly D 1 symmetry Chemico morphological features of motifs could be such that the true symmetry of the rectangular two dimensional crystal is D 1 which implies that the only symmetry element of such a crystal concerning its point symmetry is a mirror line The point group D 1 is isomorphic with the point group C 2 See next Figure where the red line is a mirror line The rectangular shape of the crystal consists of a combination of three crystallographic Forms One such Form consists of one horizontal crystal face indicated by red coloration in the next Figure The second Form consists of a horizontal crystal face indicated dark blue parallel to the first one The third Form finally consists of two parallel vertical crystal faces indicated green See next Figure Such a D 1 rectangular crystal in principle admits of any number 1 of antimers as long as they are symmetrically arranged with respect to a mirror line We give some examples D 1 symmetry and two antimers Such a case two antimers green yellow is depicted in the next Figure where the D 1 symmetry of the crystal is indicated by an inserted motif black Figure above A two dimensional intrinsically rectangular crystal with intrinsic D 1 symmetry and two antimers green yellow The D 1 symmetry of the crystal is expressed by the insertion of a motif black With another way of expressing the D 1 symmetry of the crystal red hooked lines and with the Vector Rosette of Actual Growth Added Figure above Same crystal as in previous Figure Vector Rosette of Actual Growth added The two antimers green yellow indicated by numerals The next Figure is the same D 1 crystal two antimers but with the former motif s str black inserted within the crystal to more clearly show the equivalence of the vectors a and b the equivalence of the vectors c and d and the non equivalence of the vectors a and c and finally the non equivalence of the vectors b and d of the Vector Rosette of Actual Growth blue lines The next Figure gives the promorph and its name of the above rectangular crystal D 1 two antimers It is an isosceles triangle and as such the two dimensional analogue of half a rhombic pyramid The two antimers green yellow and radial and interradial directions are indicated D 1 symmetry and four antimers Here as comparable with the D 2 case discussed above we have to do with two possible configurations of the four antimers viz two antimers at each side of the mirror line which we call interradial or two on the mirror line and one at each side of it which we call radial Interradial and with the Vector Rosette of Actual Growth added Radial or alternatively but promorphologically equivalently and with the Vector Rosette of Actual Growth added The next two Figures give the corresponding promorphs and their name of the above rectangular crystal D 1 four antimers It is either dependent on the above mentioned configuration of the four antimers an isosceles trapezium and as such the two dimensional analogue of a trapezoid pyramid i e it is its base or a bi isosceles triangle and as such the two dimensional analogue of a bi isosceles pyramid i e it is its base The four antimers green yellow and the radial and interradial directions are indicated Interradial case perpendicular directional axes interradially positioned Radial case perpendicular directional axes radially positioned The next three Figures elaborate a little more on the directional axes of D 1 promorphs Allopola with four antimers Allopola Zygopleura eutetrapleura in fact their two dimensional analogues The two possible promorphs were given just above Figure above Cross axes two pairs blue red of the isosceles trapezium the basic form of the Eutetrapleura interradialia The perpendicular cross axes red are the directional axes while the other axes blue are just cross axes The latter are bent but this is neither typical nor necessary as the next Figure shows Figure above Cross axes of the isosceles trapezium the basic form of the Eutetrapleura interradialia Two sets of straight cross axes red blue The perpendicular ones red are the directional axes of the isosceles trapezium The horizontal directional axis does not connect the centers of opposite sides but this is to be expected because of the absence in the promorph of a horizontal mirror line Also the basic form of the Allpola Zygopleura Eutetrapleura r a d i a l i a can be depicted by a polygon viz a bi isosceles triangle such that it has straight cross axes Figure above Cross axes of a bi isosceles triangle the basic form of the Eutetrapleura radialia All cross axes are straight The perpendicular ones red are the directional axes of the bi isosceles triangle D 1 symmetry and six antimers Here we have six antimers three at each side of the mirror line and with the Vector Rosette of Actual Growth added Indicating the arrangement and number of antimers to express a certain promorph often allows for some freedom The next two Figures are wholly equivalent to the two Figures directly above The six antimers are indicated by

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express these rates in terms of the number of unit rectangles in a unit of time and thus not in terms of units of distance It is then clear that in the present case in order for crystal growth to result in a rectangular crystal Rate lo must be three times as big as Rate sh And indeed generally we can say that if the stacking of rectangular building blocks should result in a square crystal the relation Rate lo lo sh Rate sh measured in terms of numbers of building blocks must be satisfied Although such cases of the formation of square crystals out of rectangular building blocks can be imagined to occur we will not pursue them further Instead we consider the cases where the properties of the chemical motifs are such that with respect to the plane groups P2mm C2mm P2mg P2gg Pm Cm Pg P2 and P1 the lattice meshes become squares See next Figures Figure above Two dimensional square crystal point symmetry D 2 and supported by a primitive rectangular point lattice indicated by connection lines where the two translations happen to be of equal length The above Figure depicts a two dimensional crystal with plane group symmetry P2mm and point group symmetry D 2 Earlier viz in Part V we depicted a crystal with the same intrinsic symmetry but having an intrinsic rectangular shape The difference between the two must lie in the motif But we have used the same motif And we will keep on doing this because our drawn motifs commas or other figures only symbolize the point s y m m e t r y of the actual chemical motifs of the crystal The next two Figures also show a square P2mm crystal as in the Figure directly above but now with its motifs rotated anticlockwise by 45 0 implying that also the mirror lines are so rotated Figure above Same as previous Figure Some symmetry elements indicated 2 fold rotation axes small yellow solid circles and mirror lines m demonstrating that the symmetry of the pattern is indeed according to the plane group P2mm The next Figures illustrate that a symmetry pattern having C2mm symmetry can nevertheless be such that it is consisting of s q u a r e building blocks unit meshes Figure above Construction by means of auxiliary lines of a two dimensional square crystal with C2mm plane group symmetry and consequently D 2 point group symmetry See next Figures Figure above Two dimensional square crystal point symmetry D 2 and supported by a centered rectangular point lattice indicated by connection lines where the two translations happen to be of equal length Compare with its rectangular analogue of Part V The next five Figures analyse the above square crystal Figure above Two dimensional square crystal of previous Figure Some equivalent points indicated Together they form the centered rectangular point lattice of the crystal In the present case the meshes happen to be squares as a special kind of rectangle Compare with its rectangular analogue of Part V Above Figure Some points of an alternative set of equivalent points of the crystal of the previous Figures Like the set depicted above they also form a possible centered rectangular lattice of the crystal The next Figure shows how according to this lattice D 2 motifs are repeated Above Figure Two dimensional square crystal of previous Figures Some motifs highlighted All this in fact shows D 2 motifs being repeated according to a centered rectangular point lattice which demonstrates that the pattern is indeed a C2mm pattern despite its square meshes Above Figure Two dimensional square crystal of previous Figures The repetition of the motifs can also be described with a rhombic point lattice instead of with a centered rectangular lattice as was done above In our present case the rhombi of this lattice happen to be squares which are a particular species of rhombus Compare this square crystal with its rectangular analogue of Part V Inspecting the present Figure one would be tempted to see a D 2 motif which is repeated by two independent translations perpendicular to each other indicating that the plane group symmetry of the pattern is P2mm instead of C2mm But this is not so because the reflection lines of the motif are not aligned with the edges of the square meshes which should be so when the pattern s symmetry was according to P2mm as can be seen in the earlier Figure depicting a P2mm crystal where we see a D 2 motif repeated according to a primitive rectangular point lattice where in the present case the shape of the meshes has become a square The next Figure shows this more clearly Figure above Same as previous Figure viz 2 D square crystal with plane group symmetry C2mm and point group symmetry D 2 but rotated 45 0 to clearly see the non alignment of the motif s mirror lines with the edges of the square meshes Compare with the earlier Figure depicting a P2mm crystal The motif s mirror lines only aligns with the edges of the meshes when we have chosen to describe the repetition of the motifs by means of a centered rectangular lattice See above where such a repetition of the motifs viz a repetition according to the centered rectangular lattice the meshes of which have in the present case become squares is depicted and where one can see that the mirror lines of the motifs do align with the edges of the square meshes To obtain the above pattern i e a C2mm pattern allowing for s q u a r e lattice meshes in particular that of the above Figure from the corresponding pattern having rectangular lattice meshes as was depicted earlier in Part V it is not enough simply to change the rectangle that outlines the rectangular unit mesh into a square If we do this we get the following pattern next Figure and when we inspect

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