⇐  gifs     ⇑   text contained in ‘The Big Hiss’ video:
 
 
... imagine:
Hypothetic Big Bang didn't take place at one single point as a zero-dimensional origin of further developments, but it happened all over a pre-existing one-dimensional infinity simultaneously. It was not a singularity, not an ‘explosion’ with subsequently occurring ‘inflation’ of space, it was an abrupt transformation of a linear, infinite quantity into an infinite pattern. The appearance of space and matter wasn't a creation but ‘only’ a transition from one state of infinity into another, from smooth, steady, linear and continuous conditions into distinguishable discrete parts with the capability of mutual interactions between all the newly emerged components. The decay into discrete constituents, sudden loss of a pre-existing continuum and drastic depletion of spatial points simulates an inflationary scenario in the new space. That way we avoid the epistemological problem ‘creatio ex nihilo’ - and the pre-existing quantity might be called ‘time’.
 
The simultaneously occurring annihilation of points all over the pre-existing infinity during the transition from continuum to a discrete structure isn't a big bang - it is: The Big Hiss
 
... imagine: an infinite emptiness without any structure or obvious property - except one: a one-dimensional order relation which distinguishes between ‘nothing’ and ‘infinity’ – mathematically already a vast foundation for generation of further relations, laws and properties!
 
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the red line symbolises the one-dimensional infinity, the black background stands for all the rest, be it ‘really nothing’ or all infinitely many properties, which the red line doesn't have ...
 
the order relation within the one-dimensional infinity allows to select one singular ‘point’ or ‘location’ on the red line, which differs from all other points or locations clearly, and every point is unique
 
the order relation also allows to determine a second point on the line and to identify and allocate a quantity ‘distance’ or ‘length’, which can be assigned to the set of these two selected points
 
to draw a line as image of the one-dimensional infinity isn't the only way for visualisation, and we choose a different graphical method by bending all ‘distances’ PS to circles, with S as boundary point: ...
 
note: the ‘length’ maintains constant during bending and the procedure – a mathematical mapping – is ‘bijective’, what means that it is a one-to-one transformation, the line section and circle being absolutely equivalent!
 
now let's continue with a series of more points ...
same procedure for all points on the red line!
this procedure (a bijective mapping) generates a uniform ‘area’, but with an intrinsic structure (the circles, nested into one another), the mathematical ‘cardinality’ of our line and this area is the same
 
... next step:
back to the red line, the one-dimensional infinity ...
we give a second ‘dimension’ to the line, but not in usual manner by adding another line to construct a new coordinate system, but in such a way that it remains an intrinsic property of the line and the points on it, e.g. as ‘turns’ of line and points around themselves
turns of points and lines can not be recognized, but with our trick, the bijective mapping of points on a line to circles, which all touch the line at one point, we make such turns visible, knowing full well, that all this is allegoric, not happening in three-dimensional space!
 
now let the circle turn around the line ...
the turn around the axis (the red line) is called ‘toroidal rotation’ or ‘rotation’ in short, but that's not the only turn, which a horn torus can perform: the ‘poloidal revolution’ or ‘revolution’ in short is a torsion of the torus bulge around itself, quasi a rolling along the axis, and additionally the torus can change its size during this rolling in such a way, that the unrolled distance equals the circumference of the bulge perimeter (longitudes), just as we have learned in the first step as a mapping of the distance between points to a circle
 
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note: the unrolling speed – identic to the circumferential speed – is constant, but the angular velocity decreases with increasing size of the horn torus, they are inversely proportional
similar situation with decreasing size of horn tori: constant speed, bulge perimeter (= horn torus longitude) equals unrolled distance, and angular velocity of revolution is inversely proportional to size
 
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warning: all the images and animations are allegoric only - they symbolise 2-dimensional entities, e.g. complex numbers, and you must not interpret them as figures within our normal 3-dimensional space!!
nevertheless, we use the geometric figure horn torus as analogue visualisation, since it illustrates in an abstract, but easily intelligible pictorial way many big mysteries in our comprehension of reality ...
 
... and now we let the horn torus turn: it shall perform both toroidal rotation around the main symmetry axis and poloidal revolution of the torus bulge around itself. In the following example we choose a rational ratio of the angular velocities, revolution : rotation = 1 : 2, watch how the combination of both turns looks like in motion and then we hold a marker pen on the surface of the turning horn torus. The line which the pen draws is called ‘trajectory’ and for rational ratios we get ‘Lissajous figures’ on the horn torus surface
we turn the axis vertical, and then: ... let's roll!
 
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now the matter is about to get exciting !!
we already have learned that the horn tori, while rolling along the axis with constant unrolling speed, change their size and angular velocity according to position and unrolled distance – size and velocity being inversely proportional – and we have noticed, that small horn tori turn extremely fast (poloidal) and big ones very slow. At constant rotational speed (toroidal) the ratio revolution to rotation is high for small and low for big horn tori. Small tori show trajectories with many ‘blades’ (turns around the bulge), on big ones these lines are ‘spirals’, winding around the axis very often.
 
For rational ratios v = revolution : rotation the trajectories are closed lines (for v ≥ 1 after one rotation, for v ≤ 1 after one revolution), they become so called ‘Lissajous figures’. In the following animation we trace a developing trajectory from beginning on a very small horn torus up to big size and recognize these Lissajous figures clearly as ‘resonances’, first very distinct, then, with increasing size, as short flashes only, lessening on bigger horn tori and disappearing finally:
 
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this depiction of our trajectory is a rather simplified illustration of the supremely complex continuous uncoiling process (presumably our known mathematics isn't suitable to describe it properly), but nevertheless it is, as dynamical coordinate, base for a dynamical geometry, providing plenty of analogies to physical phenomena
 
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first obvious physical interpretation: sharp resonances represent fermions, the sections between them bosons and the lines without any resonances on bigger horn tori are photons (ratio << 1). In this way all kinds of elementary particles are beaded on one thread, within one single coordinate, forming a unique fundamental entity
 
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thoughts – playfully developed further as intellectual game:
after first ‘turn’ of our one-dimensional infinity (red line, occurring earlier in this video), all congruent fundamental entities combined to one each, left wide gaps in the preceding continuum and transformed the smooth, continuous, linear infinity into a complex, discrete, chaotic, infinite pattern – abruptly and ‘simultaneously’ all over the pre-existing infinity ...
 
as described on the related website https://www.horntorus.com/ in every spatial point all fundamental entities are represented by one particular horn torus, belonging to one entity each, they are nested into one another, just as symbolised in the following image ...
 
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as example of dynamically nested horn tori we take four of their 2:1-Lissajous trajectories and two with ratio 1:2, what very roughly symbolises four nucleons and two electrons – a He-4 atom, but neither in scale nor regarding speed or direction of their rotations, only to demonstrate the principle of dynamically nested horn tori. Don't interpret it as figure in the three-dimensional space – all that, the horn torus model of space, time and particles, only serves as a figurative allegoric visualisation of the mathematical situation!
 
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