Cross-roads :  What to do next?

Continue with Philosophy proper, or study Symmetry as one of the highly important features emerging from dynamical systems that generate intrinsic beings?


In all the foregoing documents of this First Part of Website we considered a possible metaphysics (ontology) of intrinsic beings, of which the most clear examples are crystals (as representatives of the inorganic world) and organisms. We found out that an intrinsic being is generated by a dynamical system, and that the dynamical law of the latter can be consisered to be the Essence of that being, i.e. its intrinsic whatness.  What is generated by the dynamical system is that intrinsic being, not its properties :  they are just co-generated, or better, implied.  This is so, because only genuine beings are generated, and properties are not genuine beings. Nevertheless we can loosely call the properties of a generated being (or intrinsic thing) the manifest products of the dynamical law (of the dynamical system that generated that being).


In the documents (essays) that follow -- the Special Series -- we investigate these properies, i.e. the properties as products (better :  implications) of the dynamical law, but, because of that : intrinsic properties. They form the being's ontological periphery, while its Essence is its ontological kernel. Of course there are many properties of intrinsic beings (crystals, organisms). From these we have selected for further ontological study the following :  Symmetry and Promorph .  These are very important aspects of the spatial structure of any given intrinsic being. So it is natural that the following documents extensively consider crystals .  Symmetry is well expressed in them. In the Special Series of the present website (i.e. First Part of Website) we then study the 32 Crystal Classes and geometrically investigate the symmetries involved. But as soon as we want to study the internal symmetries of crystals it becomes very complex and not easy to depict graphically on a two-dimensional drawing pad. So we will concentrate for the time being on the two-dimensional analogues of crystals (also done in First Part of Website), the study of which will already be very instructive to understand their three-dimensional counterparts. Then, in the Second Part of Website, after some general considerations concerning three-dimensional crystals, we will prepare for the study of their three-dimensional structure and symmetry, not by means of graphical representations, but by means of the mathematical theory that describes and elucidates their symmetry :  Group Theory .  In doing so we will concentrate on symmetry groups and give many examples. This culminates in an extensive group theoretical study of two-dimensional periodic patterns, which were studied earlier (First Part of Website) from a geometric-crystallographic point of view. After this we will turn to the internal symmetry of three-dimensional crystals, and describe it group theoretically, while making use of much that we have learned from their two-dimensional counterparts.
Then it will be time to turn to the symmetry of organisms .  If we look to organisms we will see that, unlike in crystals, most of them are -- strictly taken -- totally asymmetric. But on closer inspection most of them do display symmetry nevertheless, but in a more or less hidden fashion. We could say that in organisms, just like in crystals, intrinsic symmetry is generated during individual development, but is often more or less 'overruled' by other processes. So we must somehow extract the intrinsic symmetry that lies beneath the orgamism's asymmetric appearance. Such an extracted symmetry we will call the promorph of the given organismic individual. Most organisms possess such a promorph because they are regularly built up (but, also, again, often partly overruled) out of subordinated individuals. These subordinated individuals together constitute the whole organismic individual as it appears as such to us. Think, for example of a common starfish. It consists of a regular repetition of five similar parts or subordinated individuals. The study of these subordinated individuals, accounting for some general aspects of the organism's morphology, and with it for its intrinsic symmetry, is called Organic Tectology and is considered on the Second Part of Website.  It forms the prelude to the study of symmetry in organisms :  Promorphology ,  also on Second Part of Website.  In Promorphology we extract the intrinsic symmetry of given organisms (or of their parts, i.e. of their subordinated individuals) together with another aspect, closely related to this symmetry, namely the repetition not only of identical parts, but also of just similar parts, where these parts are ordered around an imaginary point, axis or plane within the organism. Such parts are called antimers .  Well, what we do in Promorphology is determining the promorph of a given organismic individual. And we do this in the following way :  The simplest geometric body geometrically expressing the ideal intrinsic symmetry of that given organismic individual, and, also geometrically, expressing the number and arrangements of that individual's antimers, is the promorph or ideal stereometric basic form of that given organismic individual. We will see that the Promorph generally digs a little deeper into the global structure of a given organismic individual then Symmetry alone does.
After having expounded the "Promorphological System of Basic Forms", i.e. systematically presented all the possible types of promorphs, we will apply this Promorphology to crystals as well. This demanded a lot of theorizing, because crystals do not as such possess antimers, which fact tends to forbid considering crystals in a promorphological way. It turned out, however, that crystals have something like a 'hidden' promorphology, and we succeeded to reveal it.

All this is quite a lot about symmetry :  Two-dimensional crystals, Group Theory (as prelude to the study of symmetry in three-dimensional crystals), Group theoretic consideration of two-dimensional periodic patterns (for its own sake, or also as prelude to 3-dimensional crystals, and still in progress), Organic Tectology (subordinated individuals), Organic Promorphology (Stereometric basic forms in organisms), and, finally, Inorganic Promorphology (Stereometric basic forms in crystals).
The reader can find all this, to begin with, in the last series (Special Series) of First Part of Website (present website), and the rest in Second Part of Website.  As such then, the reader studies one of the most important and general properties of intrinsic beings ,  beings, generated by certain dynamical systems, and as such directly implying these properties (which are aspects of the total spatial structure of such a being).

So what read next?

So this whole prospect as outlined above is very special (in contrast to general) indeed, because it elaborates on just a few aspects of spatial structure emerging from one or another dynamical system when this system generates an intrinsic being. It consists of a whole lot of crystallography, biology and mathematics, and the reader might become impatient as to when, if ever, the website continues the more general metaphysical approach of its very beginning (largely, First Part of Website, where philosophical issues like substance, form, matter, substrate, individuality, essence were pursued). So I can well imagine that the reader wishes to see a continuance of  p h i l o s o p h y,  i.e. a further elaboration of the Theory of Being, rather then the extensive treatment of symmetry involving Crystallography, Group Theory, Promorphology, etc.  ( The documents on these subjects, could, however, also be studied for their subject's own sake and intrinsic interest and importance, totally apart from philosophy).

Continuing Philosophy

Well, for such a continuance of philosophy (here almost exclusively natural philosophy) the website has much to offer :  The philosophically minded reader could skip the whole Second Part of Website, i.e. skip all the detailed treatments about symmetry and promorph, and directly proceed further with genuine philosophical issues, and choose between three possiblities :

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