15. Metric - when dynamic processes induce discrete values|
Several times I used the term 'self-metrisation', but didn't explain it sufficiently. I will do it now: To undertake measurements within a space, be it spanned by coordinates or only a range of numbers, we need a unit to establish relations between occurring values. In a static continuous space formed by linearly independent coordinates no element can be distinguished as particularly eligible for that purpose. We have to choose units arbitrarily and artificially. Different with dynamic horn tori! They can relate revolutions to rotations and vice versa. Their cycloids produce patterns, self-contained curves, Lissajous figures on their surfaces, and they show significant dynamic properties: number of rotations per revolution and number of revolutions per rotation.
There is exactly one situation, where the turns are symmetric, namely when rotations per revolution is the same as revolutions per rotation. With this special case, i.e. revolution : rotation = 1:1, we have an unit, which all other situations and sizes can be related to. Coming from both sides of the numerically obvious symmetry we now have a smallest possible unit for revolution and for rotation: 1 (one) each.
After doing a couple of mental leaps and perhaps mathematical formulations nothing and nobody can inhibit myself to associate one full rotation with reduced Planck's constant ħ and one full revolution with Planck time or - because time is redundant - with the quotient Planck length / speed of light, LP/c. Speed of light c, as mentioned already, is represented by the circumferential speed of revolution. No matter what value one chooses for c, the geometric situation and all relations persist in a self-similar way, c always seems to stay constant, seen from inside the set of horn tori. And the well known relations between the mentioned physical objects and quantities achieve imaginable meaning.
Consequently fractions of one full rotation and fractions of one full revolution are not defined, don't occur in the dynamic processes. But that doesn't restrict the variability of the image. The opposite is true! And the analogies to other known and accepted models increase. Example: the electron resp. every horn torus with half-integral ratio of turns (fermion) needs two steps of rotation, two full turns, to match the original shape.
As already explained the complete unrolling line or cycloid forms one entity, starting with size zero (resp. minimum size), then along many 'blades', formed by very fast torus bulge revolutions, then passing striking resonances, especially as 3- and 2-blade horn tori, which we have identified as quarks and nucleons, continuing to the electron (two windings or rotations per bulge revolution) and farther to bigger horn tori with many windings, identified as photon. Every part of the entity, always emerging as horn torus, can be compared with the unit, the standard dynamical horn torus with ratio 1:1 for 'measurements': one rotation per one revolution and one revolution per one rotation. With constant circumferential speed c size of the tori is small for ratio revolution : rotation > 1:1 and big for < 1:1. All existing entities, perhaps an infinite number, have this ('conical') shape, all exactly the same, plus the respective mirror images. Every horn torus of an entity with a specific size has its defined place within the entity. And in every spatial point all entities are represented by one horntorus each, most of them with different sizes.