14. Patterns and strings - winding cycloids and tiny snippets of them|
Looking at one single horn torus, we see various patterns, formed by the cycloid during unrolling along an axis: coils, spirals, loops, following strictly deterministic rules. Two interacting tori combine their patterns to a different kind of dynamically changing patterns when mutually unrolling their cycloids, still in deterministic manner. The more horn tori we add to the set, the more complex become the patterns, turning into a fractal quality, dynamically swirling around, and still deterministic. With increasing number of horn tori the complexity grows as well, and it becomes difficult or impossible to detect any determinism. But the patterns persist respectively turn into a higher level of fractal complexity.
The patterns not only occur in the order of elementary particles, where the ratio of angular velocities (revolution : rotation) is bigger than ½, they continue into the macroscopic world, where rotation prevails, range over all scales and leave visible marks everywhere, as molecular and crystal structures, in organic matter and organisms, in snow flakes and great many more synergetic effects. Even - only as an example - R. Sheldrake's term morphogenetic or morphic field could be brought to life again, now seen from a completely different angle of view and released from the stigma of inexplicability. The same with so called 'instincts' of lower animals. Maybe they have unknown special senses to detect such patterns? And considerations concerning the multiple dynamic feedbacks from the 'neighborhood' - indefinite many from the whole universe - to every single spatial point take the weight from the complexity argument in so called 'proofs of god' and strengthens position of evolutionists in a further way.
We already have touched upon the fact that the whole cycloid of an unrolling horn torus can be reduced without loss of information to the part within a small neighborhood of Point S. Even an infinitesimal small piece of the line is sufficient to determine the whole curve. The patterns then equally are being reduced to an infinitesimal small bundle of snippets, characterising one particular spatial point (better: spatial 'spot'). Change of the patterns means movement to a different spot.
These snippets are not the well-known strings - whereas parallels in behavior exist! - Strings are embedded in a multidimensional space and show complex apriori properties, so that they, in my opinion, don't seem to be candidates for fundamental entities, created at the beginning of the universe. They describe much, perhaps all, but they are - likewise as horn tori - a mere model of nature, as every physical theory is. I admittedly admire their mighty potency, but for outside observers the theory sometimes impresses as play castle of brave formula heroes and climbing wall of extravagant number crunchers. (Sorry, I wrote this 1996, in German. Today I perhaps better should attend more and pay my respects to the meanwhile widely accepted theories, especially the newer and auspicious loop quantum gravity, but here and now I just like to promote the horny tori - they really have deserved this attention!)
Here is the opportunity to emphasize again, that the horn tori don't exist as such, they only are a tool for imagination, to illustrate properties of numbers. They indeed only - as (dynamic!) Riemann surface resp. manifold - represent numbers, complex numbers, rotation standing for the real, revolution for the imaginary part. The values are equivalent to the longitude (real, number of rotations/spin and the particular meridian) resp. size (imaginary, effect of unrolling/revolution). These two properties already are implicated in the short snippet around Point S. The full horn torus is redundant, but very helpful for visual associations without added information ballast. The whole model only shall highlight the capability of complex numbers, to count themselves mutually, creating manifold patterns hereby.
Last sentence should be worth to think and reflect about thoroughly! - - - (fractal iteration patterns are well known examples)
My personal conclusion (despite being perpetually and marveling in close contact with beauties of nature and able to revel in surprises of life): our world is self-creating, basically consists of complex numbers, is arithmetic, is mathematics ... ( hope not to be stoned for that - blasphemous - reductionism ;-)