I am planning to build a special coil of my own design, one that I believe will have some remarkable properties.  I am calling on the community here to come along for the ride, and I would welcome your participation at any level.

This work is a spin-off of my theoretical work regarding the structure of the electron.
In my model, the electron is a tiny black hole, with a toroidal event horizon.  It has two distinct angular momenta, which combine to impart a helical frame-drag on the surrounding space.  For you relativity buffs out there who say that a toroidal event horizon is forbidden, I can assure you that this is a special case which elegantly evades this restriction.  The electromagnetic field is identified with the helical frame-drag.  Lightlike geodesics on the horizon twist once around, once through the torus, closing on themselves to form perfect circles.

It occurred to me that a conducting coil whose turns match the lightlike geodesics on the electron event horizon might have some interesting properties.  It turns out that there is a fascinating mathematical object which embodies this geometry:  The Hopf fibration.

I decided it might be helpful to model the magnetostatic field of my theoretical coil before beginning construction.  Using the software package Radia, which was used to design magnets at the European Synchrotron Radiation Facility, I have accomplished this and done some basic R&D regarding the coil parameters.  I found that when the ratio of (the distance from the center of the torus to the center of the circular loops) to (the radius of the loops) is exactly pi (!), the magnetic field inside the tube of the resulting toroid has nearly equal components in the axial and the toroidal directions.  The magnetic field inside the "donut hole" is purely axial at the midplane, and forms a potential well at the center.

Attached are some images.  The first is a basic visualization of the coil surface.  The second is a plot of surfaces of constant magnetic vector potential; the intersecting contours represent the toroidal and axial components of the field.  The third is a plot of the axial and toroidal field components taken along the x-axis,  notice the conjunction of the axial and toroidal components within the tube.

I would be interested in seeing graphic outputs of your simulation involving the golden mean, phi and Phi, in addition to PI.

The pi ratio was discovered by accident, while attempting to find some limiting behavior of the magnetic field for different tori.  I have done some simulations in which the golden mean enters as an underlying ratio;  I'll try to generate a few graphics from these for you.

As the coil makes one revolution around the circumference of the toroid, it could make one turn around the torus - or two - or three.

Different winding ratios are intriguing, but for the purposes of this project, I am sticking to a 1:1 winding ratio.  In the far limits, other ratios will degenerate into the common toroidal solenoid, or the common current loop.  I am splitting the difference, half-way in between these two extremes.  This ensures that the windings are perfect circles, and the current will emulate the spin structure of the electron, with each winding threading all the other windings.

There are, however, two ways to combine the rotations, producing a right-handed or left-handed twist.  Coils with opposite handedness could be wound on the same core, producing (or reacting to) added dipole fields and canceled toroidal fields, or vice-versa.  Also, multiple coils of varying toroidal parameter could be nested one inside the other, while preserving the winding ratio.  Differently scaled versions of the same coil could be brought into close proximity (e.g. stacked atop one another), to create a venturi-like effect.

Also, it occurs to me that there might be some benefit to situating an untwisted circular coil inside the tube.  So many possibilities!

...one of these windings could be excited with sine wave of frequency f1 and
the other winding could be excited with sine wave of frequency f2
whereby f1 and f2 could be exact multiples/sub-multiples of each other....

Yes, the coils will be pulsed.  When I get to the experimental phase of the project, I'll try out a whole range of pulsing methods.  I agree that pulses with short duration and even shorter rise-time are the most likely to produce interesting effects, especially with a multi-stranded coil.

Another question is where to locate the electronics?  It looks like for
the simulated case as shown, maybe the middle of the torus,
inside the winding?

Good question.  I imagine that a parallel-plate capacitor in the dipole field might display some interesting behavior.  I've got a long way to go before I can start to answer that question.

there is an artifact which I do not understand.

The B-field graph is a plot of the z-component (axial), and the y-component of the field, taken along the x-axis.  The y- component is positive on one side and negative on the other, because the B-field circulates inside the tube.  I'll post some vector field plots to illustrate this.

The second plot is in the x-z plane, and the first one is in a plane slightly above the equator, parallel to the the x-y plane.  We find that the field has a dipole characteristic outside the tube, and a mixed toroidal and dipole  field inside the tube.

ZT,

here is another image concerning an artifact.

Is the blue trace or the red trace an extraneous artifact,
which should not be there?

It disturbs me to not see perfect symmetry when everything else
is perfectly symmetrical.

Which should be removed, the red trace or the blue trace?

Earl

Earl,

Both traces should be there.  The field is sampled along the x-axis, which is the abscissa in this graph.

In your markup, you colored the red trace blue and the blue trace red, which is a bit confusing, but no matter, just follow along closely:  In my original graph, the red trace is the y-component of the magnetic field.  The blue trace is the z-component.  You can think about it like this;  on the right, the y-component of the field inside the tube is coming toward you, out of the screen, and on the left, it is going away, into the screen.  The z- and y- components have nearly the same magnitude inside the tube, so on the left in this graph their individual traces appear to merge.  Here's a close-up of the field on the left, showing the individual traces.

It is precisely this (near) convergence of the field intensity profile within the tube that led me to believe that this particular "pi-ratio" geometry is "special".  Other tori do not display this close agreement of the field intensities, instead either the axial or the toroidal field begins to dominate within the tube. The slight non-uniformity within the tube I think is due to the asymmetry of the fringe fields, which can exit the coil more easily on the outer perimeter of the torus, where the windings are farther apart.    When the ratio of the radius of the individual rings to the displacement of their individual centers from the origin is 3.141592653589793..., we get the closest agreement of the two traces.  This was discovered empirically, and it still baffles me as to "why pi?".

May I also enjoy the ride?

Hopf fibration is beyond my skills but maybe I can follow it. For now, I however fail to see a potential practical connection (ref 'interesting properties') between electron model and toroidal coil. Is it the potential well you envisage as giving hope or something else (i.e. singularities)? Can you detail on it in the available time? Also, it is not clear for me if you aim toward a superconducting coil or a regular one.

The topics look very promising.
Thanks for sharing it with us,
Tinu

May I also enjoy the ride?

Hopf fibration is beyond my skills but maybe I can follow it. For now, I however fail to see a potential practical connection (ref 'interesting properties') between electron model and toroidal coil. Is it the potential well you envisage as giving hope or something else (i.e. singularities)? Can you detail on it in the available time? Also, it is not clear for me if you aim toward a superconducting coil or a regular one.

The topics look very promising.
Thanks for sharing it with us,
Tinu

As to why I think that this coil geometry might produce something unusual, in the end, it's just a hunch.  But a good, educated hunch, I think.  I like the idea of taking a whole bunch of electrons and making them dance together as one giant electron. That's what happens inside a superconductor, as the wavefunctions of the electrons cohere and spread out across the entire material.  Magnetic flux is quantized inside a superconductor, and manifests itself as toroidal vortices of supercurrent within the material.  What would happen if we confined the flow of electrons to this primal, underlying geometry with a specially-designed coil?  I want to find out.

I would love to work with superconductors, but I think I'll try copper at room temperature first (unless some superconducting wire should happen to fall off a truck) ...

Can you please give a descriptive of the size, and configuration of the coil discussed ?
- Schpankme

For the prototype, I will wind copper on a torus with an outer diameter of about 250mm.
I will choose the toroid described above, whose windings thread the torus once for every revolution around.

I misrepresented the ratio of the individual rings' radii to the displacement of the rings' centers from the origin.  It is 1+Pi, not Pi as I stated earlier (I was using a differential ratio in my calculations).  The conjunction of the toroidal and poloidal B-field intensities within the tube motivates my choice of this particular torus for study.

64 rings, each composed of an as-yet undetermined number of copper windings, will be inclined at an angle of 13.972 degrees relative to the equatorial plane.  They will be spaced equally around the torus.

Appealing to the toroidal coordinate system,  I am choosing the torus of constant v = Log(1 + Pi + Sqrt(2 Pi + Pi ^2)), with scale factor a = 100mm.  The Hopf-rings on this torus have a radius of 103.049 mm.  Their centers are situated on a circle in the equatorial plane with radius 24.8815 mm.

I will use an air core.  The form for the windings will fashioned from of rings of non-magnetic material, probably wood, brass, or teflon.  They will have grooves to seat the windings.  There will be 64 grooves on each support, each accommodating a multi-turn ring of copper wire.

@zero,
if that is your'e question then steven mark has already answered it. it is eltecro magnetics of the earth.  what part of that do you not understand?

Of course!  It is all so clear to me now.  wha?  huh?

Looks good.
The pressure is on.
You got design, sims, 3dmodels, time to crank it out.

It can be a real meatfest if there is no physical production.
Any builds are good. Results vary as always. Any high speed current freqs produce.
Your model looks like a full winding of the Rodin coil too.

--giantkiller. Build it and they will come. I've always liked that saying.

So how is it going ZT?

A