11. Examples - a sneak peek as teaser|
Until now the length of the cycloid ('unrolling line') didn't appear, whereas it is the 'source of all the magic'. Because it gives us the opportunity to undertake mathematical computations and to establish a metric. Furthermore the fact, that the line is longer than a meridian of the horn torus, causes the existence of speeds that differ from the speed of light. When you look closely at Point S and a small neighborhood, you will see that a rotating horn torus does not unroll its meridians but - to avoid slippage - just this cycloid instead, that winds around the horns. So the effective speed of rolling along the axis is smaller. Fast rotating horn tori move slower - the faster the rotation the slower the locomotion along the axis.
But now - for physicists - two haphazard examples of computation: we calculate the length of the cycloid between significant resonances, noting the fact that the dynamic horn torus diminishes in size, when its angular velocity of revolution increases. Related to the standard dynamic horn torus (ratio of velocities 1:1 and circumference 1) we start with the loop of the electron (ratio 1:2, size 2) and end with ratio 4:1(!). We get, with help of computer, as length of the cycloid the value 2.07944, corresponding a rotation angle of 748.6°, effective (-2x360°) 28°36', not far from Weinberg angle. Is there any connection to the four(!) bosons of electroweak interaction?
Similar computation, equally random: when describing the doublets of weak isospin - for quarks (u,d), (c,s), ... - in some 5% of cases s replaces d and vice versa, and so d and s are superpositions of both, d and s. Why that? When calculating the respective length of the cycloid (direct corresponding to rotation angle!) we find: the three 'blades' of quarks u and d are twisted 44.7°, 164.7° and 284.7°, in relation to the 'zero-meridian' of the electron, the four of c and s are twisted 28.6°, 118.6°, 208.6° and 298.6°. In (298.6-284.7)/360 of cases the next figure is the four-bladed, coming from the three-bladed, and in the other direction, coming from the four-bladed, in (44.7-28.6)/360 of cases the three-bladed is next. That are 3.9 resp. 4.5 %. Maybe that leads to an attempt to answer the open question. - And maybe the cycloid will disclose some other secrets. Values of physical constants perhaps? Fine-structure constant e.g., as length of a section? - But, please, never forget: we are playing a game - just for fun!