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  • wings 15g note 937
    surface adsorption always sets in automatically the living substance doesn t have to bother itself steeringly and stimulatingly with the appearance at the right moment of this important element Here we have an example that sophisticated and effective phenomena are

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  • wings 15g note 938
    may also be viewed as consecutive exchange as we see it for instance in the truncated migration of H ions in water through consecutive water molecules Also in this case

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  • wings 15g note 939
    939 What in it happens to aggressive substances also many poisons is as to fundamental knowledge not important because it then is about abnormal unphysiological states of affairs Back to

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  • wings 15g note 940
    need to be optically visible We believe that genuine simple i e mono layered separating membranes thus not external boundary membranes do not exist anywhere because basically they cannot occur and we see in it an experimentally testable indication for or against our view namely the bilayer nature of all denaturation membranes Still to be investigated is the possibility that a bilayer membrane in its generation turns into a seemingly

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  • inhoud SITE 2a
    Website Evolution of all insects paleo entomology Evolution of flight and thus of wings in insects Insect evolution in terms of aut ecology and in terms of the Implicate and Explicate Orders Further elaborating on the noëtic theory of organic evolution Setting up Natural Philosophy space place motion time qualities and quantities The theory of Being qua Being Continuation of the Special Series e mail Please write in Subject entry METAPHYSICS in order for me to be able to distinguish your mail from spam Philosophy of Being HOMEPAGE Sequel to Group Theory Symmetry of Two dimensional patterns Subpatterns and Subgroups in 2 dimensional space Part I Introduction Subpatterns and Subgroups Part II Antisymmetry Groups Color Symmetry Groups Subpatterns and Subgroups Part III Friezes p11 p1g pm1 p12 Subpatterns and Subgroups Part IV Friezes continued pmg p1m pmm Subpatterns and Subgroups Part V Ontology of symmetry groups Antisymmetry in Friezes p11 p11 p12 p12 p12 p11 p1m p1m p1m p11 p1m p1g pm1 pm1 pm1 p11 pmg pm1 Subpatterns and Subgroups Part VI Antisymmetry in Friezes pmg p1g pmg p12 pmm pmm pmm pm1 pmm pmg Subpatterns and Subgroups Part VII Motifs s str in Ornaments Introduction Difference between P31m and P3m1 Subpatterns and Subgroups Part VIII Motifs s str in Ornaments Sequel P1 P2 Pm Pg P2mm P2mg P2gg Subpatterns and Subgroups Part IX Motifs s str in Ornaments Sequel Cm C2mm P4 P4mm P4gm Subpatterns and Subgroups Part X Motifs s str in Ornaments Sequel P3 P31m P3m1 P6 P6mm Subpatterns and Subgroups Part XI Antisymmetry in Ornaments P1 P2 Subpatterns and Subgroups Part XII Antisymmetry in Ornaments Pm Subpatterns and Subgroups Part XIII Antisymmetry in Ornaments Pm Ontology of symmetry groups Subpatterns and Subgroups Part XIV Antisymmetry in Ornaments Pg P2mm Subpatterns and Subgroups Part XV Antisymmetry in

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  • 2-D patterns I
    pattern by just a half turn and a quarter turn situated as follows Figure 2 By using a half turn indicated as a small ellipse and a quarter turn indicated as a small square at the positions indicated the P4mm pattern can be generated from the initial motif unit purple Larger motifs such as or might also have been selected to generate the P4mm pattern Now the selection of a larger motif will always correspond to a subgroup of the full group of symmetries of the pattern provided the symmetries of such a basic unit i e the selected larger motif consisting of more than one motif units are symmetries of the pattern as a whole For example in Figure 1 each motif of the form has D 4 symmetry and the pattern is the same in effect as a pattern of square motifs i e motifs of the form arranged with their centers forming a square lattice as we see such a lattice in Figure 1 Now if these motifs were all given a rotation through 45 0 we should still have P4mm but if they were rotated through say 30 0 we should merely have P4 Each motif would still possess its individual D 4 symmetry but of these symmetries only the half turns quarter turns and three quarter turns would belong to the symmetries of the whole pattern so obtained And because the other symmetries namely the mirror lines of the D 4 symmetries do not belong to the full group so obtained these D 4 symmetries cannot form a subgroup of the full group which is now the group P4 The next Figure shows such a square lattice provided with 30 0 rotated square motifs Figure 3 A square point lattice indicated by auxiliary lines such that the lattice points coincide with the intersection points of those lines Each point of the lattice is provided with a square motif that is rotated 30 0 with respect to the lattice lines Because of the insertion of these rotated square motifs in the square net the latter looses all its mirror lines Only the rotations 90 0 180 0 and 270 0 of the square motifs belong to the whole pattern Consequently those square motifs do not represent D 4 subgroups or more exactly a D 4 subgroup and its cosets of the whole pattern In virtue of all this the latter is a P4 pattern So provided the symmetries of the chosen larger motif i e a motif consisting of more than one motif unit with which we generate the P4mm pattern are symmetries of the pattern as a whole such a larger motif corresponds to a subgroup of the full group of symmetries of the pattern For example the subgroup D 1 C 2 1 at Figure 4 Generating the P4mm pattern Addition of yet another horizontal mirror line and a glide line at 45 0 will generate the whole pattern with the table corresponds to

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  • 2-D patterns XIII
    is it of that thing I have written about emanation in the Third Part of this Website which considers a holistic approach to the constitution of Reality Because the abstract group is wholly indifferent with respect to the material world it surely must belong to a different domain of existence which could perhaps be the Ideal Sphere of Being as proposed by Nicolai HARTMANN We expect that all mathematical self contained structures reside in this Ideal Sphere This Sphere distinguishes itself from the Sphere of Real beings by the different way of being of its inhabitants It is a way of being that does not involve time and matter and thus is devoid of genuine processes We could of course suppose that all mathematical structures like abstract groups turn out to exclusively belong to the real sphere of existence and so could although discovered by purely conceptual means be abstracted from the things of the real world while no ideal sphere of existence is present at all This supposition can however be countered by the fact that there as far as I can see surely are some self contained mathematical structures that contradict each other This could be the case for instance with some disparate topological spaces And such each other contradicting structures could because of their high degree of generality not exist side by side in the Real Sphere of existence as abstractable structures Hence there must be some different sphere of existence which then would be Hartmann s Ideal Sphere The precise relationships if any obtaining between these Spheres of existence the Real Sphere and the Ideal Sphere are hard to assess According to HARTMANN 1940 Der Aufbau der realen Welt pp 41 it is not so that the entities or some particular ones among them of the Ideal Sphere represent in the sense of are the essence or principles in the sense of his categories of certain entities residing in the Real Sphere As the things in the Real Sphere have their own principles within themselves or at least within the Real Sphere itself so also have the beings of the Ideal Sphere their principles within that Sphere itself This could be so but should not imply two completely separate worlds And indeed when doing Mathematics we as real beings are in one way or another in contact with the Ideal Sphere For the time being I have characterized this contact such that it consists of a re creation of some objects of the Ideal Sphere when doing Mathematics But all this is not wholly satisfying It s probably better to interpret the Ideal Sphere as a series or stack of metaphysical levels the highest of which becomes downwards along the stack locally further determined specified by the less abstract and thing and process oriented levels which in turn can be received by a formless substrate and so constituting not generating material things And this would imply that the Ideal Sphere is in some way enclosed within the Real Sphere and so is not a disparate world after all Within this all encompassing world the constitution of things consists in the above mentioned emanation from the highest to the lowest metaphysical ontological levels Indeed in the next work to mention HARTMANN p 281 insists upon the interpenetration of the Ideal and Real Spheres of Being resulting in a partly overlap HARTMANN 1935 Zur Grundlegung der Ontologie pp 262 sees as one of the reasons that prove the ontological self containedness in the sense of existential independence or at least that prove their character to be that of beings of mathematical structures the possibility as it turned out of a mathematical treatment of in principle all natural material phenomena of the real world So at least many concepts representing mathematical structures truly point to objectively existing entities because if those structures were just the result of some sort of thinking compulsion their ruling of the material world would be inexplicable But according to me this argument has no bearing on the question at issue because ANY relationship whatsoever between entities and processes in the real world could in principle be treated mathematically In the old days one thought the planets to move in circles and circles are mathematical objects Later however they turned out to move in ellipses but also these are mathematical objects accessible to mathematical analysis And if those planets later turn out to move in egg shaped orbits or wobbling orbits these too can be mathematically analysed because they too are mathematical objects And where in Physics some phenomena seem not to yield to mathematical analysis one immediately sees this as due to the still insufficient state of contemporary mathematics and one sets out to develop new mathematical theories that can deal with these phenomena however weird they and the corresponding mathematical theories are So the whole of mathematics including future theories has no special nature and then happens to be applicable to the real world HARTMANN seems to say Well you have these mathematical structures which are in many cases created in an a priori fashion by mathematicians like the theory of matrices But they must at the same time be different from pure fantasies mind games or a priori constructions because they turn out to rule certain phenomena in the real world like the role matrices turn out to play in Quantum Theory as matrix mechanics which deals with the sub microscopical domain of the real world And being thus not only a priori constructions of the mathematician s mind means HARTMANN that they possess an objective existence Ansichsein But as I ve said mathematical structures are not at all anything special they could be any conceptual structure provided they are each for themselves logically consistent So although it is true it is meaningless to say that Nature is mathematical It is equivalent to the saying that Nature is natural HARTMANN admits something like this on page 283 of the above mentioned work And all this does not bring us any further than just our finding that in mathematics we ve found a very efficient and exact language to argue about natural phenomena But in principle of course we could have taken any appropriate language also a verbal one to argue about Nature And whatever qualities relations and laws turn out to be the ones of Nature we describe it in that language And this including the use of mathematical language implies concepts These concepts are then supposed to stand for certain phenomena encountered in Nature they are not those phenomena themselves and thus do not enjoy an independent existence Ansichsein which automatically means that the relevant mathematical entities which thus are equivalent to concepts and coherent sets of concepts also do not enjoy an independent existence So the argument that Nature turns out to be mathematical does not necessarily lead to the independent existence of mathematical structures nor to their character of being beings at all One could counter this by indicating that there nevertheless are structural features in the Real Sphere that are as they are by purely mathematical reasons for example the fact that no real genuine single i e non twinned crystal does contain 5 fold rotation axes in its repertoire of symmetry elements Indeed one can mathematically prove that only 1 2 3 4 and 6 fold rotation axes can exist as symmetry elements of p e r i o d i c structures where the internal structure of crystals is by definition periodic So it seems that here a certain mathematical feature determines corresponding real features But a presence of 5 fold axes in spite of the mathematics in real crystals only would mean a contradiction In the present case the contradiction should read There is a real material structure that is periodic and at the same time not periodic So already for the Real Sphere a thing cannot be say A and at the same time non A Consequently it is not a determination from and by the mathematical In the mathematical i e in the Ideal Sphere we have just the same state of affairs with respect to any mathematical object And in virtue of this we can derive the non possibility of 5 fold rotation axes in periodic mathematical structures i e we can derive a contradiction if we assume such an axis to be present And we know that the material counterpart of contradiction cannot occur in the Real that s why we then know that 5 fold rotation axes cannot occur in real material periodic structures One has nevertheless found real crystals that possess 5 fold rotation axes but such crystals are not strictly periodic anymore with respect to their internal structure so they are not crystals afterall as long as we keep including periodicity of internal structure in the definition of a crystal HARTMANN s second set of arguments however does point to an independent existence of mathematical structures He gives three such arguments The a priori nature of mathematical structures The relation between pure mathematics and applied mathematics The indifference of mathematical structures with respect to the material world Let me discuss these three arguments Mathematical structures and theories can be erected in the absence of observation Sometimes observation is introduced in the case of checking complex results by means of a computer and new mathematical phenomena could be discovered by observing the computer s path through the jungle of a mathematical experiment but all this is not genuine observation it remains in the a priori domain A very instructive example of finding mathematical entities without actual observation are groups In a complete a priori manner one can investigate as to what and how many types of so called simple groups that play the same role in group theory as atoms in chemistry there ARE And indeed one has after a long search found the answer laid down in the Classification Theorem This clearly points to an independent existence of such structures which are what they are independent of our thinking about them and of course independent of any empirical observation or empirical experiment HARTMANN 1935 p 277 points to other mathematical structures which in physics play a role of a law Such a law is an example of a priori knowledge of real phenomena i e phenomena of and in the Real Sphere of being in the sense of covering the totality of all possible individual cases and he says This totality is something intrinsic residing in the essence of the relevant phenomena and as such truly ideal The generality of the law as mathematical structure thus makes it ideal with respect to its way of being But as a physical law its elements are already interpreted as to what physical variables and constants they point to So such a law is ontologically comparable to the above discussed symmetry groups Figure 5 while its purely mathematical content or structure is ontologically comparable with the abstract group Figure 4 which is the mathematical content or structure of those symmetry groups Recall that those symmetry groups consisted of already interpreted group elements And only the purely mathematical content i e its logical structure of the law is ontologically independent of relevant phenomena of the real world and thus can exist all by itself albeit in a different mode of existence A physically interpreted law presupposes the material world just like the symmetry groups presuppose an object to which their transformations are applicable The fact that we have pure mathematics in addition to applied mathematics again emphasizes the independence of mathematics as a science which points to an independence of existence of the entities it investigates because there is no essential difference between the two Any piece of pure mathematics can become applied mathematics HARTMANN 1935 p 280 points out that There is a thoroughgoing containment of ideal being within the real The real world is formed through informed and reigned through predominated by ideal core relationships One can also say Ideal Being functions in the Real Sphere as a sort of basic structure And consequently the real world is intrinsically dependent upon it As has been argued above this latter statement is not correct without proper qualification The real world turns out to be such and such according to observation and we then describe it with the appropriate mathematical structures which evidently are not efficient causes Their causal nature is ontological The independence of purely mathematical structures like laws is evident from the fact that we cannot obtain them from relevant real cases by exclusively empirical means the latter only suggest one or several such structures We can only obtain them by contemplating the simple basic relationships themselves i e one can only get them in an exact way from the sphere of ideal being That s what pure mathematics is doing And when we want to apply them these purely mathematical structures should then after their elements have been physically interpreted be tested by observation or experiment and then if necessary be amended or replaced by other such mathematical structures Finally the third argument of HARTMANN involves the indifference of mathematical structures with respect to their possible materializations in the Real Sphere of existence He does however not distinguish between physically non interpreted and physically interpreted mathematical structures In what follows I will so distinguish With respect to this it should be emphasized that in the present document something being interpreted always is meant in an ontological sense i e in the sense of something finding itself as being connected to something else Purely i e not physically or otherwise interpreted mathematical structures like the abstract group given in Figure 4 are indifferent as to their realization in the material world And there will be many mathematical structures that are not were not and will not be realized in that world A purely mathematical structure as such has nothing in it that links it up with the material world it doesn t presuppose this world it is totally indifferent with respect to it not dependent on it It exists on its own behalf Category and Whatness HARTMANN tells us that this is not so the other way around The material world is fully dependent on certain mathematical structures But we have seen that this is not so without qualification The material world IS as it is as we find out as a result of observations and we then describe these results with the language of mathematics and generalize as far as seems possible All this boils down to setting up mathematical models to describe and explain the relevant phenomena So the purely mathematical structures even when their corresponding models which bear a physical interpretation of those mathematical structures turn out to be wholly adequate cannot be seen as efficient causes of the relevant phenomena of the natural material world but as answers to what is it questions They constitute the it is so not the why it is so So an actual dependence of the phenomena of the material world upon purely mathematical structures is out of the question The relevant mathematical structures are ontological causes i e causes only in the sense of constituting not generating the what is it of those phenomena The efficient causes i e the causes of generation must be sought for in physical dynamical systems And only as physically interpreted mathematical laws the dynamical laws of such systems involve true i e efficient or active causality So when HARTMANN claims that the material world which belongs to the Real Sphere of existence is causally dependent on mathematical structures or at least is determined by them as heteronymic entities See below he should add that these structures must have the form of laws and moreover should be in a physically interpreted state But the physically interpreted dynamical law of a dynamical system that generates a coherent physical structure be it an organism a crystal or some other object is however also and at the same time an ontological cause in the sense defined above and as such is the Essence of such an object residing in its genotypic domain while the products of the dynamical law plus initial conditions are among many other features the structure and symmetry of that one object residing in its phenotypic domain i e at its ontological periphery This is fully explained in the First Part of the Website The fact of the necessary co presence of initial conditions and thus of the physical interpretation of the law determines the physical nature of the dynamical system Such a physically interpreted dynamical law in its role of Essence constitutes the what is it of the generated object which here is a real individual independently existing object like a crystal or an organism As such it is comparable with say a symmetry group which also constitutes a what is it But there are two main differences between these two types of physically interpreted mathematical structures First such a dynamical law also and at the same time physically generates that object by organizing the elements of the corresponding dynamical system into one coherent whole a totality while a symmetry group does not physically generate the symmetry of which it is the what is it Second while the symmetry group is the what is it of a single pure feature in the present case the symmetry of a given object the dynamical law as the Essence of the generated object constitutes the what is it of the whole object at once i e of all its features and their mutual relationships generated by that dynamical law So this dynamical law is the what is it not only of the object s symmetry but at the same time of all other aspects as taken together of the object s structure composition and behavior So this composed what is it of the object as a whole its Essence is condensed into the one physically interpreted dynamical law And this further means that this composed what is it is dynamic in character while a symmetry group constituting one pure feature only and moreover not generating it is static The mathematical dynamical law all by itself on the other hand i e uninterpreted is not of a physical nature It does not generate but only co determines the what is it of the generated object as a whole And because purely mathematical structures are not physical structures even in cases where they are moreover realized somewhere in the material world their way of being differs from that of physical entities insofar as they are residing in the Ideal Sphere They have what HARTMANN calls an ideal way of being however not according to me residing in a separate world which also HARTMANN denies or in a different world that nevertheless partly overlaps with the Real World but present as metaphysical levels and so constituting the Ideal Sphere of existence within one and the same world As physically interpreted and thus as present in the Real Sphere the way of being of mathematical structures is realiter HARTMANN says that the Real and the Ideal Spheres of existence both have their own categories although many of them are common between them A category is a determining principle and as such it is an entity that is different from that what it determines or in other words a category is different from its concretum If we look to the Real Sphere we can say that the fundamental conservation laws are categories in this sense They impose certain definite restrictions onto natural processes they so determine But of course these laws which are necessarily physically interpreted as to what physical quantities should be conserved constitute a what is it They do this however only partially i e they form only a constituent of the what is it of the given process a constituent that all relevant processes have in common So a category in the Real Sphere is different from its concretum but is nevertheless part of the what is it of the latter It is a reality category in HARTMANN s sense In contrast to conservation laws a physically interpreted dynamical law that thus relates consecutive states of a dynamical system but implicitly contains the relevant conservation laws constitutes the full what is it of the object as a whole that it generates and so is identical to that what is it But because the object is always also extrinsically determined i e perturbed by agents outside the generating system there is no true identity between the what is it of the object and the object itself and even already so when only the latter s qualitative content is concerned because the what is it is meant to be the intrinsic qualitative content only In the same way we can say that the symmetry group IS identical to the symmetry of the given object Also here we mean the object s intrinsic symmetry Extrinsic agents agents outside the generating system perturb this symmetry resulting in the fact that the what is it of the intrinsic symmetry of the object is not identical to the actual symmetry of that object The latter symmetry can of course also be described by a symmetry group which is however in most cases the symmetry group C 1 which in fact means no symmetry at all It is precisely in virtue of these perturbations as they constantly take place in the Real Sphere that the concept of intrinsic what is it makes sense It is the very point of departure of the metaphysics of PLATO as well as that of ARISTOTLE In all this it is presupposed that there are objects that have something of a self or identity not in an anthropomorphic sense but in the sense that they have some sort of identity core so that we can after some considerable effort it is true distinguish between a qualitative outer and inner with respect to a thing or being resulting first of all in the possibility to objectively distinguish between the constant intrinsic and repeatable features of it and the extrinsically variable features falling onto it and secondly to objectively distinguish between a genuine single unified thing or being and just an extrinsic aggregate of such beings This inner and outer is a central paradigm of the Substance Accident Metaphysics of ARISTOTLE and St THOMAS AQUINAS The First Part of my website is almost wholly devoted to verifying this supposition It has its point of departure wholly from the individual material thing whether it be a human person or a crystal individuum insofar as it sees itself from within And this is a quite different approach from that of the ontology of Nicolai HARTMANN The latter approach is however not because of that in error It is complementary to my approach and should be studied alongside the latter And it is in virtue of the absence of perturbations that in the Ideal Sphere the what is it of an ideal object is always totally identical with that object implying by the way that in such an object the self is not conspicuously expressed The actual symmetry of a hexagon as it i e the hexagon exists in the Ideal Sphere is equal to the symmetry group D 6 i e this D 6 symmetry group is not underlying the actual symmetry but IS that very symmetry Perturbations do not occur in the Ideal Sphere because they imply processes So to extract the what is it from an ideal entity is trivial it is just that entity itself What is not trivial is to analyse the what is it of an ideal object We can then find ideal entities that each for themselves partially constitute that what is it For example the Group Axioms See below partly constitute the what is it of the Abstract Two element Group Figure 4 The other ideal entity making up and completing the what is it of that Group is its order of two i e its two elementness In such cases i e concerning ideal objects we d better not speak in terms of analysing the what is it of an ideal object but in terms of analysing the ideal object itself resulting in its explicit what is it which is wholly identical with that ideal object Entities within the Ideal Sphere partially determining the what is it of other ideal entities are ideality categories in HARTMANN s sense In expounding ideas concerning the ontological status of mathematical structures we stay for the sake of clarity and simplicity within the confines of our special example namely SYMMETRY and still more specificly the symmetries according to the groups D 1 C 2 and C i On top of our scheme of whatnesses with respect to symmetry we can add the group axioms that define a group in general i e which constitute the what is it of a i e any group There are several definitions of a group but they all are equivalent Here is one of them A Group is a set of elements forming an algebraic structure in virtue of one operation let us call it that satisfies the following conditions which are the Group Axioms The operation satisfies the associative property If a b c are any three elements of the group then ab c a bc holds Every equation x a b and every equation a x b possesses one and not more than one solution So our complete sequence of whatnesses is now Group Axioms The partial what is it of any Group the general algebraic whatness Abstract Two element Group Special algebraic what is it It contains the structure of 1 but it is further determined as to the number of its elements This number is 2 which automatically implies that the non identity element must be involute i e its period must be 2 Symmetry Groups D 1 C 2 C i and indeed also the corresponding antisymmetry groups D 1 C 1 C 2 C 1 and C i C 1 briefly discussed earlier Elements interpreted Presuppose object abstract or concrete The Symmetry Group is the what is it of the given symmetry It contains the structure of 1 and if it is one of the three mentioned groups 2 The interpretation of the group elements can to begin with be purely geometrical and if so the group is still a purely mathematical structure with its elements now being geometrical transformations related to each other algebraically Such an interpretation can involve two dimensional space in which C i and C 2 coincide or three dimensional geometric space The interpretation can however also be non geometrical and involve symmetries of an abstract nature In all these cases the symmetry group is still a purely mathematical structure If on the other hand the interpretation of the group elements somehow involves physical three dimensional space then the interpretation is physical and the corresponding symmetry group points to the real material world and is ontologically intermediate between purely mathematical structures and the symmetry of real material objects Promorphs Zygopleura eudipleura Heterogyrostaura Anaxonia centrostigma and indeed all the promorphs mentioned earlier Click HERE for overview Geometrically morphological interpretation of group elements Dimensionality implied If a physical or chemical interpretation of the symmetry transformation can be reduced to a morphological state of affairs microscopically or macroscopically then the interpretation is already considered to be morphological The Promorph is expressed as the simplest possible geometric solid It is the geometric what is it of morphological symmetry It implicitly contains the algebraic structure of 1 and in the present cases that of 2 and 3 It moreover shows the number and nature of the antimers even where this number does not co determine the given symmetry as we could specify the Heterogyrostaura as Heterogyrostaura tetramera or as Heterogyrostaura dimera both having C 2 symmetry See below The group elements so interpreted resulting in the corresponding promorph yield a mathematical structure of which the interpretation is such that it also directly points to certain beings in the real material world and as such also is ontologically intermediate between purely mathematical structures and the symmetry and some other aspects of the structure of real material objects Considering the list of precisely and only those promorphs that can be thought of as further structural specifications of the three symmetry groups D 1 C 2 and C i we can add the following But before we do so we repeat the definition of an antimer Antimers or counterparts are identical or more or less similar parts of a natural body say an organism or a molecular chemical complex that are more or less regularly arranged around that body s main axis for example the two body halves of a human body the five arms of a common starfish the five rays radii of a regular or irregular sea urchin etc As such these antimers are geometrically indicated in the geometric solid expressing the corresponding promorph of that given natural body First these promorphs of the just mentioned list are special interpretations of those symmetry groups They are specified as to be three dimensional geometric closed figures pointing to material objects In this way we get within the overall layer of the relevant promorphs themselves a first geometric what is it layer namely the three general promorphs respectively associated with our three symmetry groups Heterostaura allopola bilateral forms in the broadest sense half amphitect pyramids D 1 Heterogyrostaura amphitect gyroid pyramids C 2 Anaxonia centrostigma triclinic bipyramids C i With respect to the symmetry group D 1 the Heterostaura allopola can be further determined specified to Allopola amphipleura half amphitect pyramids with 4 2n sides n 1 3 4 5 and Allopola zygopleura half a rhombic pyramid or equivalently an isosceles pyramid Together they form a second geometric what is it layer The Allopola amphipleura can be further and finally determined to Triamphipleura half six fold amphitect pyramids Pentamphipleura half ten fold amphitect pyramids Hexamphipleura half 12 fold amphitect pyramids etc They form a third and final i e ultimate geometric what is it layer The Allopola zygopleura can be further determined to Zygopleura eutetrapleura half a rhombic pyramid isosceles pyramid with four antimers or as second possibility a trapezoid pyramid and as final determination to Zygopleura eudipleura half a rhombic pyramid with two antimers These form a third geometric what is it layer The Zygopleura eutetrapleura can be further and finally determined to the Eutetrapleura radialia bi isosceles pyramid with two symmetrically equal lateral halves and Eutetrapleura interradialia trapezoid pyramid with two symmetrically equal lateral halves They form a fourth and final geometric what is it layer The Zygopleura eudipleura cannot be further determined specified With respect to the symmetry group C 2 the Heterogyrostaura can be further and finally determined to Heterogyrostaura dimera amphitect gyroid pyramid with two antimers Heterogyrostaura tetramera amphitect gyroid pyramid with four antimers Heterogyrostaura hexamera amphitect gyroid pyramid with six antimers They form a second and final geometric what is it layer Finally with respect to the symmetry group C i we have the Anaxonia centrostigma They cannot be further determined and consequently represent the first and final geometric what is it layer So by analogy with the relation between Group Axioms Abstract Groups and Symmetry Groups we have the above given relations between General Special and Final Promorphs While the Group Axioms and Abstract Groups do not in anyway point to the material world and thus reside wholly within the Ideal Sphere of existence the Symmetry Groups can in principle do so i e they can when interpreted in a certain way point to the material world and the Promorphs as promorphs do that anyway This means that the physically interpreted Symmetry groups and the Promorphs are ontologically intermediate between the Ideal ande Real Spheres of existence And this further means that the above assessed relations between the promorphs associated with one or another of the three symmetry groups under discussion are of a logical nature and evidently not of a causal nature But within the domains of existence lying beyond the material world the intermediate domain and the Ideal Sphere the logical connections are the special and exclusive mode of determination within these domains They are in addition to be purely conceptually also ontological by nature because those domains have an independent existence First of all independent of thought while the Ideal Sphere is moreover independent of the material world i e of the Real Sphere of existence While an important aspect of STRUCTURE of a material object is SYMMETRY which constitutes an algebraic what is it the promorph partially already transcends symmetry and steps over into other aspects of structure namely some geometrical aspects because it considers the number arrangement and symmetry of the existing antimers of the given material object What remains to be covered of the total structural constitution of that object is the ultimate asymmetric unit of the object the repetition of which constitutes the symmetry of the whole object i e the set formed by all the copies of this unit copied by symmetry transformations forms a group And when promorphs are being considered this ultimate unit is in many cases still a penultimate unit The ultimate asymmetric unit is then an antimer or in the case of the presence of a reflectional symmetric antimer in addition to asymmetric ones half an antimer See Figure 27 Any asymmetric unit cannot be further analysed group theoretically but must be explained as originating from a dynamical subsystem of the overall dynamical system that generated the whole object So a penultimate asymmetric unit representing a group element can still contain several similar antimers of the object Figure 27 but also these do have a certain structure implying sometimes a symmetry of their own like the median antimer in Figure 27 of which in the present case however only one half enters the domain of a group element The other antimers could like this median antimer be symmetric however often only approximately so which in turn implies an ultimate asymmetric unit The ultimate asymmetric unit of the object as the latter is promorphologically represented by the geometric solid of Figure 27 is only approximately a unit because it doesn t build up the object s half by a precise repetition This half which itself is duplicated by the reflection m consists of three asymmetric substructures units namely two more or less different lateral asymmetric antimers and half a mirror symmetric median antimer all lying at the same side of the central median plane of the object But because each of the two lateral antimers is approximately mirror symmetric See for instance Figure 23 we could enumerate five more or less similar half antimers lying at the same side of the median plane figuring as basic asymmetric substructures See for all this below Such an ultimate asymmetric substructure or unit is then a dynamical aspect of the what is it of the object Promorphology It is perhaps instructive to elaborate a little more on p r o m o r p h s because they probably are not familiar to any reader except when he or she has read our exposition of the system of promorphs Promorphological System of Basic Forms on this website and also because of their importance for a general theory of structure of material things The symmetry according to the symmetry group C 2 can promorphologically be expressed as an amphitect gyroid pyramid The only symmetry element that such a pyramid possesses is a 2 fold rotation axis Recall that in cases of geometric symmetry a symmetry element is a point line or plane with respect to which a symmetry transformation is performed But while this symmetry C 2 staying the

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  • subendospheric polyhedra
    here like in all other Polyaxonia and in Homaxonia Although no Organisms are to be expected to have realized Polyaxonia subendospherica several crystals from the Isometric System represent this basic form in several promorphological species These species are represented by the following promorphological categories The Subendosphaerica gyroidea based on the Gyroid or Pentagonal Icositetrahedron The Subendosphaerica pyritoedra based on the Pyritohedron or Pentagonal Dodecahedron The Subendosphaerica tetartoidea based on the Tetartoid or Tetrahedral Pentagonal Dodecahedron First Species of the Polyaxonia subendosphaerica Subendosphaerica gyroidea Gyroids or Pentagonal Icositetrahedra The Subendosphaerica gyroidea are based on geometric bodies called Gyroids or Pentagonal Icositetrahedra The symmetry content of such bodies is 4 3 2 which corresponds to the Pentagonicositetrahedric Crystal Class of the Isometric Crystal System See for details and Figures HERE When you came from the document treating of the Promorphology of Crystals and when you want to go back to it click HERE or click the BACK button of your browser Second Species of the Polyaxonia subendosphaerica Subendosphaerica pyritoidea Pyritohedron or Pentagonal Dodecahedron The Subendosphaerica pyritoidea are based on geometric bodies called Pyritohedra or Pentagonal Dodecahedra The symmetry content of such bodies is 2 m 3 which corresponds to the Duakisdodecahedric Crystal Class of the Isometric Crystal System See for details and Figures HERE When you came from the document treating of the Promorphology of Crystals and when you want to go back to it click HERE or click the BACK button of your browser Third Species of the Polyaxonia subendosphaerica Subendosphaerica tetartoidea Tetartoid or Tetrahedral Pentagonal Dodecahedron The Subendosphaerica tetartoidea are based on geometric bodies called Tetartoids or Tetrahedral Pentagonal Dodecahedra The symmetry content of such bodies is 2 3 which corresponds to the Tetrahedric pentagondodecahedric Crystal Class of the Isometric Crystal System See for details and Figures HERE When you

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