(explanatory video) 
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7. Dynamic geometry - renunciation of dimensionality To replace the engrams in descriptions of nature, we introduce a purely abstract model, which shall represent fundamental physical objects. Trick is, to use the well-known three-dimensional space, but only as sort of crutch, not as space where objects and processes are embedded. Our model has no dimensions, but has, instead, a very active dynamic. As shown on front page of this website (more descriptive here), we put horn tori of many sizes into one another, nest or interlace them so that all have the same symmetry axis through their common center, every horn torus being inside the next bigger, all - more ore less - very close to another. As surface we imagine an infinitesimal thin 'membrane', actually nearly not existent. Same as we can (mathematically) imagine an infinite set of numbers on a limited line, we imagine a huge number of horn tori put together the described way. All touch one another in the same point, in the center, which we call 'Point S' - from symmetry or singularity. Now we pull the axis from outside in one of the two possible directions and see all horn tori rolling along this axis. Simultaneously they roll along each other, and all apply exactly the same circumferential speed, just the speed with which we pull the axis. Indeed this speed is the same for all horn tori, but their angular velocities differ. The smaller the torus is, the faster turns its bulge. The rolling along the axis by performing this torsion of the bulge we call 'revolution'. So all revolution velocities are different. Very big tori approach zero angular velocity, very small ones turn extremely fast. If one has difficulties to imagine that mechanism, take normal balls of very thin glass and with different sizes, every ball containing a smaller, laying on the 'bottom'. (You know the Russian nested Matryoshka dolls?) When you rotate the biggest around a horizontal axis, all enclosed rotate with the same circumferential speed, but with higher angular velocities the smaller the respective glass ball is, the innermost being the fastest. So far, so clear? (If not - take gear wheels :-) Now let the horn tori additionally rotate around the symmetry axis. Allow each to choose any of both possible directions for rotation. At the beginning we don't set a particular angular velocity, perhaps leave it constant first, and we will consider later, which different mechanisms could have an effect on rotation. The following is a really great challenge for imagination: move a very small - infinitesimal small - distance away from Point S. Hold there a pen, a marker, on the thin surfaces of all horn tori, let them run and then trace the lines, which are being drawn onto every horn torus. We get unrolling lines (cycloids, trajectories), the shape depending on the ratio of angular velocities, connected with torus size. On very small tori appear lines nearly not diverting from meridians (very fast revolution!), on 'medium' ones we see various loops with any number of 'blades' and 'coils'. At big tori rotation prevails and the cycloids converge as windings close to the torus latitudes. The array of curves and the particular (three-dimensional Lissajous) figures later will be objects of examination. We will find amazing things, that lead to obvious interpretations. (Yet again: more detailed explanations in the German version.) → example for marker trace (Lissajous figure, unrolling line, cycloid, trajectory) close |
