(explanatory video) 
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9. Analogies - dynamic geometry versus physical entity Maybe imagination is unable to cope with the complex system of dynamically interlaced horn tori, but when we look at only one and later onto two, the mechanism turns into a simple principle with only few variables. The idea of revolution should be clear: the horn torus rolls with its longitudes along the symmetry axis, changing the latitudes. Different sizes of horn tori turn with different angular velocities when they roll with the same circumferential speed. Sure, comprehensible! - Changing size during rolling is possible without 'slippage' on the axis. Clear too? - Very big horn tori turn with very small angular velocity, extreme small ones with an enormous rate. And that's the first analogy to physical phenomenons: Extremely differing scales combined with their reciprocals appear in one single image. Now we look upon rotation: The horn torus spins with its longitudes around the symmetry axis. The angular velocity is totally random, but shall be constant first. Combined with revolution you always find a scale - a size of tori - where a certain ratio rotation to revolution occurs, e.g. 1:1, what would be called the standard dynamic horn torus (see old original, but better refer to newer animation which explains more). Whichever circumferential speed of revolution and angular velocity of rotation you choose, you always find yourself in exactly the same image. By resizing you always come upon the standard dynamic horn torus as unit for self-metrisation of the whole system. It's only a question of scale, and this is another analogy: self-similarity. Look at one horn torus, rolling from far away (big size) to a selected spatial spot (minimum size) or even to a point (size zero): the unrolling line, as described in the section 'Dynamic geometry' first curls around the surface many many times per one turn of revolution, then the number of windings decreases until you discover loops, blades, kinds of 'resonances', their number per one rotation increasing the smaller the torus becomes. This continuous unrolling line has plenty significant properties, nevertheless it is only one single object, based on one process. One can call this line an 'entity'. The process is the combination of revolution and rotation. Never view it as real existent! It is a mere visualisation of nothing else than numbers, analog to the well-known Riemann spheres: imaginary (revolution) and real (rotation) numbers! Combination of both kinds of numbers creates an incredible complex variety of properties, due to the dynamic, that is included in the horn torus image. The game is, to pick a property and look for an analogy with physical objects and their 'interactions'. Interaction is an important term in physics. All measurable phenomenons are based on interaction. In our model interaction only can take place in Point S, where all horn tori contact each other and touch the common axis t. Note that all lines on the surface of the horn torus pass Point S as parallels, regardless whether the line forms dense spirals or curls on the torus surface. The line may wind around the horns as often as you want, in Point S every line turns to a parallel, meets the tangent there, becomes part of the axis. In a rotating horn torus Point S is a singularity, and so our world of horn tori is a set of singularities. That doesn't make imagination and interpretation of interaction easier, I am afraid. But let ourselves be surprised. close |
