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REMARK : When we, in the course of the ensuing discussions, quote a certain promorph (= stereometric basic form or promorphological category), we will set it as a LINK. This link will bring the reader to the relevant place within the Promorphological System, so that he or she can orient him- or herself as to where that category is situated within the System.

REMARK : The following Theses, or, if we wish to call them, Conclusions, are mainly about the stereometric basic forms of ORGANISMS, to which the Promorphological System was originally (HAECKEL, 1866) geared. But, as we have argued in the previous documents, on the Promorphology of Crystals, we now have included inorganic genuine beings. These are mainly Crystals (single non-twinned crystals, twinned crystals, liquid crystals). So in the following theses we will treat of the promorphs of organisms as well as of crystals, and in this way harmonize both worlds (organic and inorganic world). Everywhere we will  c o m p a r e  organic with inorganic forms. But of course the burdon will lie on the organic forms, because in crystals -- single non-twinned crystals -- there are only 32 promorphs, while in twinned crystals and liquid crystals we will find only a few promorphs, i.e. in them only a few promorphological categories are materialized.

1.  The outer form (shape) of every organism is, as is also its structure (inner form), the expression of the positional relationships of the atoms and molecules that constitute its material body. In crystals these atoms (or ions or molecules) are in a state of thermodynamic equilibrium (lowest energy), while in organisms they are in an energy state conforming to its thermodynamic state, which is never a thermodynamic equilibrium, but a state of far-from-thermodynamic-equilibrium. Organisms are dissipative dynamical systems, they constantly import and export matter and energy. The latter they export (dissipate) as entropy.
Generally organismic individuals as well as crystal individuals are genuine  b e i n g s  (Totalities, intrinsic wholes), generated by certain specific  d y n a m i c a l  s y s t e m s.  So such a dynamical system produces and maintains such a being (organism, crystal). The dynamical law, inherent in the system, is interpreted as the  E s s e n c e  of the given intrinsic product (organism, crystal). This Essence resides in the product's "genotypical domain", while all the special and intrinsic features, generated by this Essence (dynamical law), reside in the product's "phenotypical domain" (See for al this the first Part of this website, accessible via back to homepage). Of course there are many such intrinsic features generated by the dynamical law. We, here on this website, have chosen to emphasize on STRUCTURE, SYMMETRY and FORM, and with respect to them we only concentrate on some of their basic features. We studied  s t r u c t u r e  in Organisms ( Tectology ) and in Crystals (See the last series of documents of the first Part of this website and its continuation into the present -- second -- Part, and especially the previous Series of documents on The Basic Forms of Crystals, that investigate the tectological features in Crystals as a preparation for their promorphological assessment). And here, in the present document, we are summarizing our studies with respect to the stereometric basic forms in Organisms ( Promorphology ), as well as in Crystals. However, with less emphasis on the latter.

2.  The outer forms of organisms are determined by their inner structure. Because the changing nature of orgamisms (i.e. their being process structures) such an outer form is observable only at a particular moment in time. In growing crystals this is also the case, but less dramatic.

3.  The atomic constituents and the molecules that are composed of them, which together dynamically determine the intrinsic shape of the organism, are no different from those that statically determine the intrinsic shape of the anorganism, especially the crystal.

4.  The outer form of every organismic individual is accordingly always just as regular (i.e. according to certain laws) as any inorganic individual, and consequently allows for a mathematical treatment. But in organisms as well as in anorganisms we can distinguish two main groups of forms, namely individual forms with or without a determined stereometric basic form, i.e. a constant basic form.

5.  If a material body, while always possessing a center of gravity, does not possess a definite  g e o m e t r i c  CENTER, then it does not have a determined stereometric basic form. This can be considered equivalent to its having no constant form. When such a center is definitely present, it can either be just a center of symmetry as the only symmetry content of the given body, or it is the mid-point of one or another axial system that can be recognized in the given body. In the latter case the body has one or more AXES, and we can (promorphologically) call such bodies Axonia. They contrast with bodies that do not allow for definite axes to be distinguished. Such bodies we can call Anaxonia, they comprise the totally asymmetric forms, the Anaxonia acentra, and forms that have a center of symmetry as their only symmetry element, the Anaxonia centrostigma. Although we could draw axes through the center of symmetry, which then connect equidistant and equal body parts, those axes are not definite, because we can draw an infinity of them. When one such axis connects special, distinguishable body parts, then we have a form (still) possessing a center of symmetry, but in addition to it allowing for at least one definite axis to be discerned. Such a body then belongs to the Axonia. The Axonia allow for still further promorphological determination on the basis of the nature of these axes and their poles, i.e. on the basis of the regular relationships of the single body parts to one or more definite axes and to their poles.

6.  The axial anorganisms, i.e. the inorganic individuals that allow for one or more body axes to be recognized, occur partly as spheriods ( (Monaxonia) Haplopola anepipeda ), as we see them in liquid crystals, partly as bodies displaying definite faces, crystals. The axial organismic bodies are partly symmetric partly radiate. But these expressions are not definite and can cause confusions.

7.  The individuals, organic as wel as inorganic, which do not have a definite body center can be denoted as  a m o r p h  or irregular.

8.  In  O r g a n i s m s  the promorph or stereometric basic form of the Anaxonia centrostigma and also of all the Axonia is only rarely precisely realized. Usually it is hidden beneath accidental form features. It must be abstracted from them, and as such appears as  i d e a l  basic forms. We represent such a ideal basic form by the simplest geometric body that directly displays the complete symmetry content of the given organismic individual. In non-amorphous organisms we always can discern  a n t i m e r s  (counterparts). These are parts that are placed around the orgnismic individual's main axis in a regular and constant way. Their number, arrangement and shapes are important determinants of the individual's promorph (they determine the nature of its cross axes). The constitution of an organismic individual from an aggregation of antimers is called its  t e c t o l o g i c a l  s t r u c t u r e,  in contradistinction to a periodic structure as we find them in crystals.

9.  In  C r y s t a l s  there are also many accidental form features that mask the basic intrinsic form. The crystals should be considered as having grown undisturbedly in a uniform nutrient medium. But even then the stereometric basic form is hidden, but in this case in a theoretical sense. Crystals have a  p e r i o d i c  structure, which means that they consist of a repetition of a certain microscopical unit, and this repetition is such that the orientation of that unit is everywhere the same in a given crystal (provided this crystal is a single non-twinned crystal, and provided it has not developed defects). All this means that such crystals do not possess genuine  a n t i m e r s  as we see them so clearly in organisms (think of starfishes for example). So strictly speaking we cannot determine their stereometric basic form as we do in organisms. But, deeply hidden in the crystal is a tectological aspect :   it is the "complex motif" that remains if we eliminate (conceptually) all translations. When we do so a certain chemical motif remains, which is equivalent to one formula unit or to a multiple thereof. This complex motif is fully tectological in its make up. Indeed the point symmetry of a crystal -- its Point Group -- representing the Crystal Class to which the given crystal belongs, is the translation-free residue of a family of possible Space Groups (These Space Groups describe all symmetries, point symmetries as well as translational symmetries, present or implied in the given crystal). So the (point) symmetry of the complex motif is the same symmetry as described by the crystal's Point Group (point symmetry). So the symmetry of the promorph of a crystal is already fully described by its Point Group, i.e. by its allocation to one of the 32 Crystal Classes. The promorph itself should then be further (and finally) determined on the basis of the number of antimers (homotypic basic number), insofar as these are indeed discernible (in the Complex Motif) :  The number of antimers not only determines the symmetry, but in some cases also decides between several possible promorphs possessing the same symmetry. For instance, while the regular octahedron has the same symmetry as the cube, they represent promorphs with a different number of antimers :   eight antimers (Octaedra regularia), and six antimers (Hexaedra regularia).
The point symmetry of crystals, as it is determined from a concrete whole crystal individual, is however not solely based on its external geometry, but also on other features like the behavior of crystal faces when they are etched, illuminated etc. This means that sometimes the external geometry of a crystal suggests a higher symmetry than the crystal actually possesses. The geometric form as such then does not represent the crystal's promoph.
In estimating and illustrating the promorph of crystals as well as of organisms, we want, in all cases, to come up with a geometric figure (a geometric body or solid) that fully represents geometrically the intrinsic symmetry of the given crystal or organism. This means that, with respect to crystals, we have to represent the symmetry of a given crystal -- and thus its promorph -- by a geometric solid that sometimes does not as such occur in crystals at all, it just expresses their intrinsic symmetry purely  g e o m e t r i c a l l y.  We've done so in the documents on the Promorphology of Crystals.

1.  The axial forms, Axonia, but also the Anaxonia centrostigma, of  o r g a n i c  individuals (be they cells, organs, antimers, metamers, persons or colonies, and whether these occur as actual (or virtual) bionts -- as such independently existing -- or subordinated to higher-order form individuals), are, just like the  i n o r g a n i c  individuals, the necessary result of the regular (i.e. according to rules) positioning of corresponding body parts around a definite  c e n t e r  (centrum), through which -- in the case of all Axonia -- runs one or several axes (These axes are of course of a conceptual nature, although sometimes materialized).

2.  The number of those determining axes as well as their differentiation and that of their poles, is in  o r g a n i c  individuals (morphonts) much more diverse than in  i n o r g a n i c  individuals (crystals), which implies that the number of the different basic forms (promorphs) is substantially larger in the former than in the latter. This is so because the periodic structure of crystals sets limits as to their possible forms.

3.  Most (but not all!)  o r g a n i c  individuals show their stereometric basic form not so immediately clearly as most (but not all!)  c r y s t a l s  do (many crystals show distortions of all kind, or geometrically display a higher symmetry than they actually possess). The fact that especially in organisms the stereometric form is often masked or unclear is caused by three main factors :

• The semiliquid condition of organisms, in contrast with crystals (sterro-crystals).
• The strongly expressed variability and the abundance of accidental influences.
• The composed tectological nature of organismic structure :   Many organisms consist of a series of metamers (sequential parts). But often the metamers of one and the same person (fifth-order form individual composed of metamers) have different basic forms, and in this way obscuring the basic form of the whole person.

5.  Because of its semiliquid state the  o r g a n i s m  is very adaptible and consequently variable as to its form. Already in virtue of this inconstancy an absolute rigorous stereometric knowledge of specific organic forms is impossible in organisms. In  s t e r r o - c r y s t a l s  this stereometric form enjoys a certain constancy because of the solid state, and is therefore in principle susceptible to a rigorous mathemetical treatment as to their basic forms.

6.  Because most  o r g a n i s m s  distinguish themselves from  i n o r g a n i c  individuals as to their composed individuality (organisms have a tectological structure, while  c r y s t a l s  have a periodic structure with a microscopic repeating unit), resulting in the fact that organisms are clearly heterogeneous while crystals are more or less homogeneous, the determination of the stereometric basic form will meet much more problems than it is the case in crystals. In organisms we sometimes see two interpenetrating basic forms in one and the same individual, these forms can be more or less equivalent to each other, or the one basic form should bee seen as the background of the other.

7.  Because most  o r g a n i s m s  pass through an individual development in which great morphological (and physiological) changes can occur, as in metamorphosis, an absolute stereometric knowledge of their forms is difficult to obtain, because initial transformations are often already under way without yet affecting the external form, which means that already two basic forms are present in one and the same individual. In  c r y s t a l s  we also see an individual development during growth accompanied with a change in external form, but this does not pose serious difficulties for the assessment of their promorph.

8.  Although -- on the basis of the foregoing, and especially because of the semiliquid state of all organisms, but also because their tectological composition involves different symmetries in different metamers of one and the same individual, and, further, because of the organism's almost unlimited ability to adapt to different conditions, and finally because of the changes that occur during individual development -- an absolute stereometric knowledge of organic forms is not in all cases possible, like -- as we have shown -- is possible in crystals, a similar knowledge (as in crystals) is nevertheless possible on the basis of the organism's  i d e a l  basic form that lies at the base of its actual form. If we exclude extrinsic factors, then in crystals we end up with an exact state of affairs with respect to their structure and symmetries, while in organisms we still have to idealize, because even intrinsic factors often mask their underlying symmetrical make up (Without this idealization all organisms -- even when all extrinsic influences are taken out of consideration -- would involve just a very few promorphological categories :   most of them would either belong to the Dysdipleura or to the Anaxonia acentra).

9.  In organic individuals, as in crystals, this basic form expresses itself in a mathematically definite way by the mutual relationships of the axes and their two poles that can be distinguished in every axis.
In crystals the difference of the two poles of such an axis (its heteropolarity) is often not geometrically evident, but must be determined by the behavior of the crystal faces corresponding to these poles (for example by means of etch figures). Theoretically the whole promorph of a crystal is present in the so-called  C o m p l e x  M o t i f  (See the document on the Promorphology of Crystals), which is the chemical motif that emerges after conceptually eliminating all translational symmetry. In this Complex Motif all the axes and their poles, including their differences, are evidently and explicitly present, which means that normally no idealization is necessary to find these axes and their poles. In organisms on the other hand, idealization is often necessary to obtain them.

10.  On the basis of the number of these ideal axes (but in some organisms present as real bodily axes) and the equality or unequality of their poles, certain definite simple stereometric basic forms can be determined on which the actual forms of organisms, but also of crystals (which grow very often more or less irregularly in virtue of external (non-intrinsic) causes), can be based.

11.  The stereometric basic form or promorph of every organismic individual expresses all essential positional relationships of its constituents that determine its shape. For crystals this is valid with respect to the positional relationships of the constituents of the Complex Motif.

12.  Every scientific account of an individual organic form must start with the determination of its stereometric basic form, onto which then the detailed description of its actual form can follow. The latter consists in the determination of axial ratio's and the relative distances of body parts from those axes. In this way we can distinguish different forms in organisms having the same promorph. For example coral fishes, rays and snakes have the same basic form, namely that of the Eudipleura, but their actual forms are different :   many coral fishes are flattened laterally, which means that their lateral axis is very short with respect to the other two axes (main axis and dorsoventral axis). Rays on the other hand, are flattened dorsoventrally, which means that their dorsoventral axis is very short in relation to the other axes. Finally, snakes are strongly elongated along their main axis. Also in crystals we have to do with an axial system. In them we determine the relative intersection distances of a crystal face with repect to the crystallographic axial system and the axial center (i.e. intersection point of the crystallographic axes). In this way the relative orientations of crystal faces are determined. But, as has been said, with these orientations not all symmetries are yet determined :   some properties of the faces need to be investigated in order to obtain the true symmetry (point symmetry) of the given crystal (Because the mentioned Complex Motif -- from which the crystal's point symmetry is immediately evident -- is just a theoretical construct, one determines the crystal's symmetry in the just described way).

13.  On this firm promorphological basis a mathematical knowledge of the organic forms is possible as it is in crystals.

1.  The promorph or stereometric basic form which lies at the basis of every organic form in which either a center is present or, in addition to it a set of axes (or at least one axis), is directly implied with mathematical necessity by the number and size, positioning and connection, the equality or inequality (differentiation) of the constituent form elements.

2.  In simple organisms, i.e. those that are just a single individual of first order, a single Cell, the promorph is consequently implied by the number and size, the positioning and connection, the equality or inequality (differentiation) of the constituent molecules -- especially the macromolecules -- and their complexes, which are composed of all atoms within the organic body. { The promorph of a cell that is a constituent of a tissue is also partly determined by its immediate surroundings }.

3.  In composed organisms on the other hand, i.e. those which are represented by an intrinsic pattern of two or more first-order form individuals, and this means in all organic individuals of second or higher order, the promorph is directly implied (i.e. formally caused) by the number, positioning and connection of the constituent form individuals of the next lower individuality order. { If a certain form individual is not a biont (i.e. not an independently living unit) but a constituent of a higher-order form individual, then its promrph is co-determined by its immediate surroundings }.

1.  All stereometric assessible basic forms of organic individuals as well as inorganic ones divide with respect to their intrinsic body center in three main groups that we call the Centrostigma, Centraxonia and the Centrepipeda (Centrostigma occur in addition to some Axonia also in some Anaxonia, namely those that have a center of symmetry that is then their body center : Anaxonia centrostigma).

2.  In the Centrostigma, the stereometric forms with an intrinsic center, the natural center of the form, i.e. the planimetric body part with respect to which all other parts of the organic or inorganic body have a definite positional (distance and direction) relation, is a  p o i n t.  This is the case in the mentioned Anaxonia centrostigma (in which that center is their center of symmetry), the Homaxonia (spheres) and in the Polyaxonia (endospheric polyhedra).

3.  In the Centraxonia, the stereometric forms with a mid line (axis), the natural center of the form is a  l i n e  (main axis or longitudinal axis). This is the case in all Monaxonia (Spheroid, Bicone, Ellipsoid, Cylinder, Egg, Cone, Hemispheroid, truncated Cone), further in the Bipyramids, Regular Pyramids and the Amphitect Pyramids, i.e. in all Stauraxonia except the Heterostaura allopola. It also is the case in those Spiraxonia (spiral forms) of which the transverse section is a circle, an ellipse, a regular polygon or an amphitect polygon.

4.  In the Centrepipeda, the stereometric forms with a mid plane, the natural center of the form is a  p l a n e  (median plane or sagittal plane). This is the case in the Heterostaura allopola or Zeugita, of which the general basic form is half an amphitect pyramid.

5.  The Centrostigma are the lowest and least perfect forms, while the Centrepipeda are the highest and most perfect. The Centraxonia occupy a middle position in this respect.

6.  All the different basic forms, that appear as subordinated form species of these main groups, allow to be ordered according to their stepwise increasing differentiation of their axes and poles. This sequence of increasing differentiation reflects at the same time the increasing perfection of the form.

7.  So there exists a degree of promorphological perfection of every organism that is implied just by the degree of differentiation of its promorph, and which is first of all independent of its degree of tectological perfection (i.e. it is independent whether the biont is a first-, second-, third-, fourth-, fifth- or sixth-order form individual).
For inorganic beings (crystals) a degree of promorphological perfection is not relevant because crystals are not  f u n c t i o n a l  entities.

1.  With respect to the general presence, condition and positioning of the axes all basic forms divide in two large groups :  Forms with cross axes,  S t a u r o t a  and forms that lack cross axes,  L i p o s t a u r a.
The  S t a u r o t a  include all Stauraxonia, i.e. all pyramidal forms :  bipyramids ( Stauraxonia homopola ), regular pyramids ( Heteropola homostaura ), amphitect pyramids ( Heterostaura autopola ), and half amphitect pyramids ( Heterostaura allopola ). They (i.e. the Staurota) also include those Spiraxonia (spiral forms) that have a cross section that allows for axes to be distinguished.
The  L i p o s t a u r a  on the other hand, i.e. forms lacking cross axes, include all Anaxonia, i.e. forms having no axes at all, further, the Homaxonia (spheres), then all Polyaxonia (endospheric polyhedra), and finally all Monaxonia, i.e. all forms having only one axis.

2.  The Lipostaura or basic forms without cross axes, are generally much lower on the differentiation ladder than the Staurota or basic forms with cross axes. The former are mainly present in lower organic (form) individuals, while the latter are predominantly found in higher organic (form) individuals.

3.  The Lipostauric basic forms either possess no axes at all (Anaxonia), or all equal axes (Homaxonia), or a definite and finite number of constant axes that are however all equal (Polyaxonia), or, finally, possessing only one constant axis (Monaxonia). With respect to all these forms neither the expression radiate, nor the expression bilateral or symmetric is applicable.

4.  All Lipostauric forms are characterized by a lack of a definite number of meridian planes, that intersect in a single main axis, and by which the body is divided into a definite number of equal or similar parts.

5.  All Lipostaura consequently lack definite  a n t i m e r s  (or paramers), if one understands by antimers and paramers only those body parts that lie around the main axis (antimers or paramers) or around some other axis (paramers). In the Polyaxonia we can, in a way, speak of antimers. They are the pyramidal parts that come together with their tips in the center of the body, i.e. they meet in a point (instead of in a line (axis)).

6.  In the Polyaxonia we could interpret the above mentioned pyramidal parts that meet in the center, as metamers or epimers (sequential parts) instead of antimers or paramers (counter parts) :  along every axis of such a polyaxonic form then lie two metamers (or epimers). But because this is only the case where the axes of every two oppositely positioned pyramidal parts are each other's extension, we do not favor this interpretation, and keep interpreting those pyramidal pars of the Polyaxonia as antimers (or one could call them by the neutral term "perimers").

1.  All Stauraxonia or basic forms with cross axes, are higher and more perfect basic forms than all Lipostaurotic basic forms ( = basic forms without cross axes), because in virtue of the presence of certain cross axes, that intersect in the main axis, a greater diversity and possibility of differentiation is given, than in any one lipostaurotic basic form.

2.  The general stereometric basic form of all Stauraxonia is the pyramid, namely either the bipyramid (Stauraxonia homopola) or the single pyramid (Stauraxonia heteropola).

3.  The expression "regular or radiate forms", if one wants to retain these expressions, should be limited to the form categories of the Homopola isostaura (regular bipyramids) and Heteropola homostaura (regular pyramids).

4.  The expression "symmetric or bilateral forms" should, if one wants to retain those espressions, be limited to the Zeugita or Centrepipeda (Heterostaura allopola).

5.  All Stauraxonia are characterized (and as such essentially distinguished from the Lipostaura) by the possession of a definite number of meridian planes, which intersect in a single main axis, and by which the body is divided into a definite number of identical or similar parts.

6.  The corresponding constituent parts of the stauraxonic body, which, by their number, positioning and differentiation (equality or unequality), further determine the basic form of the staurotic individual, either are  p a r a m e r s  (in the first- to third-order form individuals) (For paramers, see HERE and also HERE in Part II of Tectology), or  a n t i m e r s  (in metamers and in linear persons), or  m e t a m e r s  (in planar persons)(The metamers of planar persons are not sequentially ordered but in a branched way), or  p e r s o n s  (in colonies). Promorphologically the most significant are generally the antimers, then the paramers. Their basic form is always pyramidal.

7.  All Stauraxonia divide into two main groups, according to whether the body center is one of the meridian planes (Zeugita, i.e. Heterostaura allopola), or the main axis (Stauraxonia centraxonia, i.e. all Stauraxonia except the Heterostaura allopola).

8. The centraxonic stauraxonians, in which the body center is a line, are either (I) the regular bipyramids (Homopola isostaura), or (II) the regular pyramids (Heteropola homostaura), or (III) the amphitect bipyramids (Homopola allostaura), or (IV) amphitect pyramids (Heteropola heterostaura autopola). In all these forms the two poles of every cross axis, or at least of both directional axes (which are perpendicular to each other) are equal, which means the right body side never is different from the left body side, and also the back side (dorsal half) is never different from the belly side (ventral half). Dorsal half and ventral half are congruent, as well as right half and left half.

9.   The centrepipedal stauraxonians or zeugites on the other hand, in which the body center is a plane (the median plane), either are (I) half amphitect pyramids (homopleural zeugites), or (II) irregular pyramids (heteropleural zeugites) (The zeugites are identical to the Heterostaura allopola). Here at least one cross axis is unequipolar. So in all cases the dorsal side is different from the ventral side, and in the heteropleural zeugites also the right side from the left. They never are congruent. Where the left side equals the right side, as in the homopleural zeugites, they are symmetric, not congruent (i.e. there is no mechanical procedure, say a rotation or translation, that maps them onto each other).

1.  The form group of the Zeugites or Centrepipeda (Heterostaura allopola) forms, as half an amphitect pyramid, the highest and strongest differentiated basic form of organisms.

1.  The spiral forms, or Spiraxonia, are, promorphologically very special forms :   Their main axis is coiled into a spiral. And this can be accomplished in several ways, leading to some subgroups of this category. It is hard to say what exactly the biological significance is of this form, which we encounter in so many animals. Very often this form is the consequence of incremental growth of solid material, like in horns, teeth and shells. The spiral is certainly not so advanced a form, like that of the Zeugites considered above.

1.  The basic form of the organic individuals is the more perfect, the more unequal its constant axes are.

2.  The basic form is the more perfect, the greater the number of unequal axes, and the smaller the number of similar axes is.

3.  The basic form is the more perfect, the more unequal both poles of its axes are.

4.  The basic form is the more perfect, the greater the number of unequal poles, and the smaller the number of similar poles of its axes.

5.  The degree of perfection of the basic form (and consequently that of the whole external form) or the degree of promorphological perfection, generally (but not always) is connected with the degree of perfection of the  s t r u c t u r e,  i.e. the internal form, or tectological order (The highest tectological order is reached by the sixth-order form individuals).

6.  While the tectological perfection (differentiation of the internal structure) generally (but not in all cases) increases during the individual development of the organism, the latter generally (but not in all cases) exhibits an increase in the degree of promorphological perfection (differentiation of the external form and its stereometric basic form).

1.  In the ascending sequence of basic forms many higher or more perfect forms are halfs of the nearest lower or imperfect forms, and consequently mimic the case of hemihedric crystals with respect to the corresponding holohedric crystals (Holohedric crystals are those that belong to the most symmetric Crystal Class of the given Crystal System. In crystals belonging to a hemihedric Class of such a System one or more mirror planes have disappeared, resulting in a lower symmetry).

2.  The perfection process, by which the hemihedric organic basic forms originate from holohedric ones, essentially is a differentiation of the two poles of an axis, effecting the disappearance of a mirror plane.

3.  The diplopolar form (cone, hemispheroid) is the hemihedric differentiation of the haplopolar form (bicone, spheroid).

4.  The heteropolar form (pyramid) is the hemihedric differentiation of the homopolar form (bipyramid).

5.  The homostaurotic form (regular pyramid) is the hemihedric differentiation of the isostaurotic form (regular bipyramid).

6.  The autopolar form (amphitect pyramid, i.e. compressed pyramid) is the hemihedric differentiation of the allostaurotic form (amphitect bipyramid).

7.  The tetractinote form (regular four-fold pyramid or quadratic pyramid) is the hemihedric differentiation of the octopleural isostaurotic form (quadratic octahedron).

8.  The orthostaurotic form (amphitect four-fold pyramid or rhombic pyramid) is the hemihedric differentiation of the octopleural allostaurotic form (rhombic octahedron).

9.  The allopolar form or zeugitic form (half amphitect pyramid) is the hemihedric differentiation of the autopolar form (amphitect pyramid).

10.  The amphipleural form (half amphitect pyramid, as such with (4+2n)/2 sides (n = 1, 3, 4, 5, 6, etc.) is the hemihedric differentiation of the oxystaurotic form (amphitect pyramid of 4+2n sides (n = 1, 3, 4, 5, 6, etc.).

11.  The eudipleural form (isosceles pyramid or half a rhombic pyramid) is the hemihedric differentiation of the orthostaurotic form (rhombic pyramid).

12.  The dysdipleural form (irregular 3-fold pyramid, which can also be represented by a quarter of a rhombic pyramid) is the hemihedric differentiation of the eudipleural form (half a rhombic pyramid).

13.  The Spiraxonia or spiral forms, i.e. forms with a spirally curved main axis, cannot be compared with the forms of crystals. Although many crystals have screw axes, the latter are  h e l i c a l  instead of spiral. They are a combination of a rotation and a translation, causing the screw to fit within the (infinitely thin) wall of a cylinder. Moreover, even this helical screw is not present in the promorph of any crystal, because the promorph is derived from the Complex Motif which itself is derived from the crystal structure by eliminating all translations.

1.  All simple and regular stereometric bodies (solids) which occur as holohedric forms of the inorganic Crystal Systems, for instance the holohedric form of the Isometric Crystal System (representing the most symmetric Class of that System), or the holohedric form of the Tetragonal Crystal System (representing the most symmetric Class of that System), can be found in certain organic forms.

2.  The Cube and the Octahedron, which are holohedric forms of the Isometric Crystal System, are realized in the organic hexahedral and octahedral forms of the Polyaxonia rhythmica.

3.  The Quadratic Octahedron, which is the holohedric form of the Tetragonal Crystal System (or Quadratic Crystal System), are realized in the organic forms of the Isostaura octopleura.

4.  The Rhombic Octahedron, the holohedric form of the Orthorhombic Crystal System, is realized in the organic forms of the Allostaura octopleura.

5.  The Hexagonal Dodecahedron, or, equivalently, the Hexagonal Bipyramid, the holohedric form of the Hexagonal Crystal System, is realized in the organic forms of the Isostaura dodecapleura (belonging to the Isostaura polypleura).

1.  The first-order form individuals, the Cells, can exhibit all possible basic forms. They, however, preferably show the lower basic forms, especially the Monaxonia (Haplopola and Diplopola).

2.  The second-order form individuals, the Organs, also can show every possible basic form, preferably, however, on the one hand, the lowest (Anaxonia, Homaxonia, Monaxonia), on the other hand the highest (Eudipleura and Dysdipleura).

3.  The third-order form individuals, the Antimers, exhibit exclusively the heteropolar forms (single pyramid), of them more seldom the homostaurotic forms (regular pyramid), while more often the heterostaurotic form (irregular pyramid). The latter most often is of the dysdipleural form, next of the eudipleural form.

3.  The fourth-order form individuals, the Metamers, most often possess, like the Antimers and the Persons, the heteropolar basic form (single pyramid). And from this category more often the zeugitic form (half amphitect pyramid), more seldom the homostaurotic form (regular pyramid). Among the Zeugites the most often is the Eudipleural form.

4.  The fifth-order form individuals, the Persons, also possess mainly the heteropolar basic form (single pyramid), namely that of the Homostaura (regular pyramid), or that of the Allopola (zeugitic form), while the latter either is the amphipleural, or the zygopleural form.

5.  The sixth-order form individuals, the Colonies, only seldom show higher basic forms, most often they have lower basic forms, either Anaxonia and Homaxonia, or Monaxonia (Haplopola, and mostly Diplopola). The diplopolar monaxon form seems to be the most common basic form of the Colonies (trees and the like).

Although we have succeeded to include all Crystals in our Promorphological System of Basic Forms -- whereby it was necessary to establish a number of new promorphological categories -- we will here (i.e. with respect to the ensuing tables) limit ourselves to ORGANIC basic forms.

1. Anaxonia acentra  (absolutely irregular)

1. Anaxonia centrostigma  (Forms with a center of symmetry, but lacking mirror planes).
2. Homaxonia   (Sphere).
3. Allopolygona   (Irregular Endospherical Polyhedron with unequal polygonal sides (faces)).
4. Isopolygona   (Irregular Endospheric Polyhedron with similar polygonal faces).
5. Icosaedra regularia   (Regular Icosahedron).
6. Dodecaedra regularia   (Regular Dodecahedron).
7. Octaedra regularia   (Regular Octahedron).
8. Hexaedra regularia   (Regular Hexahedron, Cube)
9. Tetraedra regularia   (Regular Tetrahedron).

1. Haplopola anepipeda   (Spheroid).
2. Haplopola amphepipeda   (Cylinder).
3. Diplopola anepipeda   (Egg).
4. Diplopola monepipeda   (Cone).
5. Diplopola amphepipeda   (Truncated Cone).
6. Isostaura polypleura   (Regular Bipyramid).
7. Isostaura octopleura   (Quadratic Octahedron).
8. Allostaura polypleura   (Amphitect Bipyramid).
9. Allostaura octopleura   (Rhombic Octahedron).
10. Homostaura   (Regular Pyramid).
11. Tetractinota   (Quadratic Pyramid).
12. Oxystaura   (Amphitect Pyramid).
13. Orthostaura   (Rhombic Pyramid).

1. Amphipleura   (Half Amphitect Pyramid).
2. Eutetrapleura radialia   (Bi-isosceles Pyramid).
3. Eutetrapleura interradialia   (Antiparallelogram Pyramid).
4. Dystetrapleura   (Irregular 4-fold pyramid).
5. Eudipleura   (Isosceles Pyramid).
6. Dysdipleura   (Irregular three-fold Pyramid).

1. Spiraxonia   (Spiral).

1. Anaxonia.
2. Homaxonia.
3. Allopolygona.
4. Isopolygona.
5. Icosaedra regularia.
6. Dodecaedra regularia.
7. Octaedra regularia.
8. Hexaedra regularia.
9. Tetraedra regularia.
10. Haplopola anepipeda.
11. Haplopola amphepipeda.
12. Diplopola anepipeda.
13. Diplopola monepipeda.
14. Diplopola amphepipeda.

1. Isostaura polypleura.
2. Isostaura octopleura.
3. Allostaura polypleura.
4. Allostaura octopleura.
5. Myriactinota.
6. Decactinota.
7. Octactinota.
8. Hexactinota.
9. Tetractinota.
10. Polyactinota.
11. Enneactinota.
12. Heptactinota.
13. Pentactinota.
14. Triactinota.
15. Octophragma.
16. Hexaphragma.
17. Tetraphragma.
18. Diphragma.
19. Heptamphipleura.
20. Hexamphipleura.
21. Pentamphipleura.
22. Triamphipleura.
23. Eutetrapleura radialia.
24. Eutetrapleura interradialia.
25. Dystetrapleura.
26. Eudipleura.
27. Dysdipleura.

AXONIA,   Forms possessing constant axes.
Homaxonia,   Sphere,   all axes equal.
Heteraxonia,   not all axes equal.

HETERAXONIA
Polyaxonia,   not one definite axis stands out as main axis.
Protaxonia,   one definite axis stands out as main axis.

POLYAXONIA
Arrhythma,   Irregular Polyhedra.
Rhythmica,   Regular Polyhedra.

ARRHYTHMA
Allopolygona,   irregular polyhedra with unequal polygonal faces.
Isopolygona,   irregular polyhedra with similar polygonal faces.

RHYTHMICA
Icosaedra,   Regular Icosahedron.
Dodecaedra,   Regular Dodecahedron.
Octaedra,   Regular Octahedron.
Hexaedra,   Regular Hexahedron, Cube.
Tetraedra,   Regular Tetrahedron.

MONAXONIA
Haplopola,   uniaxial forms with equal poles.
Diplopola,   uniaxial forms with unequal poles

HAPLOPOLA
Haplopola anepipeda,   both poles not representing tranverse planes. Spheroid.
Haplopola amphepipeda,   both poles representing transverse planes. Cylinder.

DIPLOPOLA
Diplopola anepipeda,   both poles not representing transverse planes. Egg.
Diplopola monepipeda,   one pole representing a transverse plane, the other not. Cone.
Diplopola amphepipeda,   both poles representing transverse (unequal) planes. Truncated Cone.

STAURAXONIA
Homopola,   Homopolar main axis. Bipyramids.
Heteropola,   Heteropolar main axis. Pyramids.

HOMOPOLA
Isostaura,   Regular Bipyramids.
Allostaura,   Amphitect Bipyramids.

ISOSTAURA
Isostaura polypleura,   Regular Bipyramids with 6, 10, 10+2n sides (faces).
Isostaura octopleura,   Quadratic Octahedron.

ALLOSTAURA
Allostaura polypleura,   Amphitect Bipyramids with 8+4n faces.
Allostaura octopleura,   Rhombic Octahedron.

HETEROPOLA, Stauraxonia with heteropolar main axis. Pyramids.
Homostaura,   Regular Pyramids.
Heterostaura,   Irregular Pyramids.

HOMOSTAURA
Isopola,   Regular Pyramids with 2n sides.
Anisopola,   Regular Pyramids with 2n-1 sides.

HETEROSTAURA
Autopola,   Amphitect Pyramids.
Allopola,   Half Amphitect Pyramids.

AUTOPOLA
Oxystaura,   Amphitect Pyramids with 4+2n sides.
Orthostaura,   Rhombic Pyramid.

ALLOPOLA
Amphipleura,   Half Amphitect Pyramids-with-4+2n-sides (n is not 2).
Zygopleura,   Half Rhombic pyramids.

This (whole) website focusses on the ONTOLOGY of natural material objects. These objects are supposed to be generated by certain dynamical systems (as is explained in the First Part of Website). This means that also all their properties are generated by those systems. The dynamical law of such a system is then considered to be the  E s s e n c e  of the generated object, and as such resides in the latter's ontological core. The properties of the object are the visible products of that dynamical system and thus are derived from the Essence, in the sense that they become manifest, and as such reside in the object's ontological periphery (but still being intrinsic properties) (In fact we should say that the Essence of the object becomes manifest by way of the object's intrinsic properties). One of these properties is STRUCTURE.  And two important aspects of structure are SYMMETRY and PROMORPH (= stereometric basic form). The latter accounts not only for symmetry, but also for the number and arrangement of antimers. The former is treated algebraically as  s y m m e t r y  g r o u p s  in the documents on Group Theory.
Now we could ask ourselves "what is the precise  o n t o l o g i c a l  s t a t u s  of  symmetry groups and of promorphs, i.e. do they exist as such independently of our thinking, and if so, in what way do they so exist?" And this, of course, leads us to the general question of the ontological status of whatever mathematical structure (geometric or algebraic).
Well, this question is anwered in a document within the Group Theory Series. In it we also elaborate more on what a promorph in fact is or should be, and in what it differs from the corresponding symmetry group. To see this document in a separate window (and then -- if you happen to land at the very beginning of that document -- scrolling down a bit till you read for the second time "Ontology of symmetry groups .  .  . "),  click HERE. After having consulted the document, close it, and you will be back where you were. If you want to see this document in its context, click on  SEQUEL  TO  GROUP  THEORY  in the left frame, and then go to  SUBPATTERNS  AND  SUBGROUPS  Part XIII, and scroll the document down a little, as indicated above.

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