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construction of a horn torus
(alternate method)
 
Take a sphere with longitudes and push in both poles until these two points meet themselves in the center and merge to only one residual point - our 'Point S'
 

If you consider the originating sphere as Riemannian, then the two points which represent zero and infinity are identical after transformation to the horn torus! That causes a strange, nearly contradictory topology: the horn torus appears as a simply connected clopen set* and as differentiable manifold with Riemannian metric, though the topological properties have to be reviewed (until now science ignored the horn torus completely),
 
as example we introduce a bijective conformal mapping / conventional method of constructing the horn torus
 
*) consider that a line surrounding the horn torus bulge only once (e.g. a longitude) is not a closed curve, it corresponds to a line with zero as boundary point, but when circling around twice or more, it is closed without border. This fact will become important later when identifying closed lines on the horn torus surface through the center ('resonances', Lissajous figures) as figurative analogues to elementary particles ...
 
clopen: closed and/or open, depending on interpretation of neighbourhoods of point S (disjoint or connected) ...
 
topology gets totally intricate when we incorporate inverse figures (as solids and as numbers) into our reflections (nice ambiguity!), e.g. by simply interchanging longitudes and latitudes. In the dynamic horn torus model we then change from an infinite number of particles within one universe to an infinite number of universes within one particle or 'entity', but that's another, exciting story ...
 
mathematicians, physicists, join in! the horn torus is worth and necessary to be treated scientifically: the horn torus is an excellent graphic representation of complex numbers, a compactification with considerable more properties than the Riemann sphere has, it connects zero and infinity in an amazing way, can be dynamised by two independent turns, rotation around the axis and revolution around the torus bulge, what creates an incredible complexity and forms a coherent comprehensive entity, also explaining the self-interacting capability and fractal order of complex numbers, and finally - most important - horn tori can be interlaced into one another and so symbolise interactions between entities, easily interpretable as physical objects: 'space', 'time', 'particles', 'forces' ...
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