8. Spatial point - without dimensions - what in the world is that?|
For our common sense nothing is more self-evident than description of space as right and left, fore and aft, top and bottom, eventually combined with a point in time. So we need three resp. four coordinates: x, y, z and t. In our horn torus model it is infinitely more difficult to imagine spatial points and relation between different points. It needs effort and time - and much capability of abstraction - to learn about, exercise and handle horn tori. It won't work without. But it's really worth doing!
The most important message first: one single point, together with an infinitesimal neighborhood, contains the whole universe! Every particle of the world is represented by exactly one horn torus in the set, with one shared tangent in common Point S. The size of the horn torus represents something like 'distance' to the particle or to the 'location', where its size is zero or say minimal. To move that way, to that location, the respective horn torus has to diminish in size. Simultaneously (nearly) all other horn tori change their size, increase or decrease, according as they diverge or get closer. Only particles, that move on a 'parallel' with same 'velocity', keep their associated horn torus size constant.
The complete set of all spatial points in the universe is the set of combinations of all occurring horn torus sizes. - Wow! - Our imagined space has vanished. Only a infinite series of numbers is left. These series and permutations of them describe all the universe, but don't show any structure. Only a consciousness that picks out remarkable and significant patterns, will see laws and objects, properties and beauty, last mentioned being one of the most complex quality, that can be derived from mere numbers in these permutations. But that's another topic.
Next step of abstraction: We reduce every horn torus to the mentioned unrolling line (cycloid) on its surface 'membrane', described in the foregoing section 'Dynamic geometry'. That takes off the symmetry from the horn torus, because this line starts on one particular meridian. So the horn torus is determined by a combination of numbers: size, rotation velocity, including direction (+/-) and longitude of the meridian, the mentioned series of numbers in fact turn to matrices and the nested horn tori to a dense set of curling curves. Nevertheless, for visualisation we better remain in the picture of the complete dynamic horn tori including their virtual membrane. The term 'dynamic' comprises both kinds of turn and variation of size.
We keep this image, even when we undertake another rigorous reduction: we remove nearly all parts of unrolling lines with the exception of an infinitesimal neighborhood of Point S, because all relevant information about the further courses of all lines is included in the short snippets. We now have extended the spatial point from a set of coordinates (x,y,z,t) to a spot, that contains its relation to all other spots in universe. The spot is infinitesimal small, but much more than the accustomed point as usually defined in mathematical sense. Every spot represents nothing less than the whole universe itself! To move from one spot to an other only means to change the point of view on to the universe.
Too much for imagination? And worse things are yet to come ...