transformation of a sphere to a horn torus and vice versa,
 maintaining length of circumferences (cross section)
    ↓ context

There is an important difference between the horn torus model used by V. Puha and S. Saitoh on one side and my (W. Daeumler's) model on the other. The first is a static one, a mapping* from the complex plane via Riemann sphere to the horn torus, using stereographic projections within the three-dimensional space, as pictorial representation of the set (body) of complex numbers, whereas 'my' horn tori are dynamic: they execute rotations around the main symmetry axis and revolutions of the torus bulge around itself. They change their size and they are many, as many as 'particles' in our universe, all nested into one another, all intertwining at one ('spatial') point. They are not embedded in any three- or more-dimensional space, but every horn torus 'entity' embodies one coordinate of an infinite-dimensional space. The static horn torus is a rather simple figure with little properties. Its incredible complexity and creative capability only arises when the dynamic is added, as described and illustrated on these webpages. ( visualisations e.g. explanation , trajectories , resonances )
Mathematically associable English texts e.g. can be found on pages starting here, but to treat the matter really mathematically, one first of all has to conceive the principle, what in my view only can be transported as colloquial speech. I know that is not the mathematician's way to cognition, but I treat it as a more philosophical and fundamental physical topic, though: 'my' horn tori represent complex numbers too - only differently. Their dynamic shows the capability of complex numbers 'quasi to calculate with themselves and with each other' - as creators of our world.
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*) Proposal regarding conformal mapping from Riemann sphere to horn torus (as approximation only - and without relevance for the dynamic model!):
Bend a half longitude continuously to a circle, preserving its lenght, as shown in the figure below for two opposite ones (treat all half longitudes in this manner simultaneously and recognize that they pass through a series of different spindle tori). The finally obtained cross section of the horn torus bulge has doubled the original angles (from π to 2π), so conclude (fig. 2): to project a point P' from the Riemann sphere onto the horn torus first double the angle α between closer pole (here N) and the point's radius, then draw the perpendicular line from the horn torus bulge (!) center M to this leg of the doubled angle and intersect with horn torus (choose the intersection P" on the same 'hemisphere' on which the original point P' is located on the Riemann sphere). It is the same point as the intersection with the original radius! That was submitted by Vyacheslav Puha already. Now consider and compare the lengths of corresponding latitudes on Riemann sphere and on horn torus: on sphere (radius = 1 instead of 1/2) we have  2π * sin(α), on horn torus  2π * 1/2 * (1 - cos(2α)) = 2π * sin2(α), the ratio of both (horn torus : sphere) thus is sin(α). But now we have two projected points (P", P"'). As first approximation we shift both to one point P between, search the factor x in  arc SP = arc SP" * x = arc SP"' / x  and find  x = √sin(α)  (nota bene: this is not yet an exact solution! - neither a stereographic projection as compass and straight-edge construction is in sight - but the story goes on ..... )
  click animation to stop



... we have:
arc SP"' = arc SP" * sin(α)
SP" = SP' * sin(α) = sin(α)
SQ  = SN  * sin(α) = sin(α)
QP" = NP' * sin(α)
arc QP" = arc NP' * sin(α)
arc NP'  = arc SP"
arc QP" = arc SP" * sin(α)
arc SP"' = arc QP"
surprisingly simple relations
(the Old Greek already knew)
so it shouldn't be too difficult ...

proposal for mapping α → φ :
( φ / 2α = arc SP / arc SP" )
  φ = 2α * √sin(α)
(approximation only! - search
 for a solution still in progress.
 note: WD's horn torus model
 is inevitably non-conformal !)