 • I think, from your answer, that you calculate cross product of X3 and X4 to lie in the X1, not the X2 direction.

I did never say anything about X1,2,3,4 and how we should draw it.

Going to > dimensions 4 needs more abstraction than you are obviously used too. The drawing we discuss here has only one purpose. To show that the mutual perturbation of field vectors leads to a synchronous rotation.

Physically, as said there are no ring currents. We also can not properly draw planes in 2D from at least 5D space what is needed for 4 rotations. Further charge is topological.

So all boils down that either you can grasp the difference between 3D and > 3D or not. The real questions are much tougher than the ones you ask.

• The real questions are much tougher than the ones you ask.

I absolutely agree. I had thought that trying to understand Figure 1 from your ResearchGate paper would be a way to understand your SO(4) theory at an entry level. I was surprised that this turned out not to be the case and that there are even aspects of it that seem to be in error.

One such aspect is your seeming contention that a cross product (such as is required by a Boit-Savart type operator) even exists in 4 dimensions. I say no and have outlined how your proposed 4D cross-product procedure that requires the assignment of an arbitrary order to spatial dimensions leads to a physical contradiction. Do you now agree with me on this?

• I say no and have outlined how your proposed 4D cross-product procedure that requires the assignment of an arbitrary order to spatial dimensions leads to a physical contradiction.

Why should it?

Did you ever look at SO(4) or octonions? As said at the end there are no more fields just the operator survives at resonant radius.

What you know from classic physics is only the time space of the basic forces. But the origin of forces is the phase/frequency space.

I also first did believe that we can just use this basic drawing and to compute the final state. But the fifth rotation does kill such a simple attempt as we not solely have the strong force equation we also have the charge coupling equation with these orbits in the fifth rotating dimension. Just to name the simple problem.

What I so far did not fully model is the external potential coupling as it can be added to the 4 rotation mass. See the overall precision that gets reduced to 7+ digits by this.

But as long as people do not understand the gravitation equation I see no value in investing any more time for small improvements. Modelling isotopes is much more fun and helps LENR.

• I ... have outlined how your proposed 4D cross-product procedure that requires the assignment of an arbitrary order to spatial dimensions leads to a physical contradiction.

Why should it?

Why should your proposed cyclic procedure for calculating cross products in 4D lead to a physical contradiction? For exactly the reason I carefully outlined in post #742 (here). If there is something wrong with my reasoning then please let me know. I think that something more specific than "Why should it?" is required.

• Mods. Is there a better thread for the SO(4) stuff? I have no objection to all the recent SO(4) material on this thread being moved elsewhere.

• Official Post

Here it is...just like the 4th dimension, you are in it and you didn't know..

• If a charge running on an X3,X4 circle produces a field that points to X1 then only the vector X2 of the charge running in X1,X2 gets affected. But the true neighbour of X3,X4 is X4,X1.

That's why the picture shows arbitarry coordiantes just to illustrate the effect. In "reality" I should draw 3 planes that show the circularity. But this is not discussed in this drawing!

The other thing we can't draw is the fact that the "field" aligns (is contained) with the flux tube. Thus even if two vectors point in the same direction these do not add/subtract if they lie outside the tube or have no intersection.

Thus again do not deduce anything other from this drawing, than such a configuration induces symmetric rotation.

Did you remember that fields degenerate to "approx. infinite" thin surface flux?

• If a charge running on an X3,X4 circle produces a field that points to X1 ...

Why would a charge running on a circle in X3,X4 produce a magnetic field in the direction of X1? What law are you using here? Mathematically, there is no such thing as a cross product in 4D, so where do you derive this field from?

... a field that points to X1 then only the vector X2 of the charge running in X1,X2 gets affected.

I am absolutely fine with this as long as we are in 3 dimensions. Its the Lorenz force. But in 4 dimensions it again makes no sense because -- again -- no cross product in 4D

But the true neighbour of X3,X4 is X4,X1.

No it isn't. X4,X2 is just as much a neighbour of X3,X4 as X4,X1 is. Are you using "true" or "neighbour" in some new way that I am not understanding?

• Why would a charge running on a circle in X3,X4 produce a magnetic field in the direction of X1? What law are you using here? Mathematically, there is no such thing as a cross product in 4D, so where do you derive this field from?

You talk nonsense!

• Mathematically, there is no such thing as a cross product in 4D,

Baffled baloney..

"The vector cross product function in 4D involves 3 vectors to produce a resultant vector that is orthogonal to all three. partial cross-product, and then multiplying the third initial vector to this matrix to complete the cross-product function. ... result that is orthogonal to that 3D subspace."

• Yes. What you are referring to is a sort of triple product in 4 dimensions. If anything deserves to be called a cross product in 4D then this does. But it is not what Wyttenbach is using. To find a magnetic field corresponding to moving charge, Wyttenbach claims to be using a Biot-Savart like function that maps pairs of 4D vectors (i.e., velocity and displacement) to a single vector (magnetic field). The triple product you have found doesn't do that. It maps 3-tuples of vectors to vectors. Different thing. If Wyttenbach were to show that he is using some sort of triple product then I would have much less problem with what he is doing. But the type of 2 --> 1 cross product he claims to be using simply doesn't exist in 4 dimensions.

• You talk nonsense!

You will have to be more specific. Once again, for a current loop in the X3, X4 plane, why do you think that the resulting magnetic field is in the X1 direction? Why shouldn't it be completely orthogonal to that and lie in the X2 direction?

For the benefit of Alan Smith I can cast what I just said as much as possible in the usual engineering language. Let's call the 4 spatial directions X, Y, Z, and U. Wyttenbach is asserting that if there was a current loop somewhere above Alan's head lying in the Z, U plane, it would generate a magnetic field in the X direction but not the Y direction. Alan may now try and imagine this and point in the X direction. But why isn't the resultant magnetic field in the Y direction? Y is just as much orthogonal to the Z, U plane as is X. Why isn't Alan pointing in the Y direction? Wyttenbach offers no answer other than that he calls the X direction "X1" and the Y direction "X2" and 1 comes before 2. I don't get that. The laws of nature should not depend on Wyttenbach's choice of which axis to label 1 and which to label 2.

Alan should be wondering what's going on here. In fact, Alan, I would like you to point in whatever direction you think the magnetic field from a Z,U current loop should lie. I can't do it but you appear to accept Wyttenbach's methods so maybe you can do it. Let me know what happens.

• The vector cross product function in 4D involves 3 vectors to produce a resultant vector that is orthogonal to all three.

It's definitely something for a computer...You also see this in calculations for GR...

The main problem in physics is that all basic actions only can be 2:1 or 1:1. This is covered by pretty old math. But we have two more complete division algebras the quaternions and octonions, that are the product of 2 or 3 circles. So we basically can have a 4:1 action as usually due to symmetry this reduces to 22:1.

Despite this classic physics invented (QM)-QED & GR that both work with fringe 3:1 action symmetry only to notice that all tractable solutions finally are 2:1 symmetric systems...( As they use a point field they can go with the 3:1 wedge product for many simple solutions) On Quantum Mechanics basic missconceptions 9805079.pdf

Most people fail to understand that a field that spans 3D, like the magnetic field, is not a 3D symmetric field nor has it 3D freedom. always two gradients are the same (magnitude that defines the action/ Energy) due to the rotation along a circle. So the field of a ring current is given by the torus surface topology and a radial function the action is 2D, where as for a point field (charge/mass) the induced action is 1D.

But if you follow researchgate you will find that the majority of people does not know the very basics.

• The laws of nature should not depend on Wyttenbach's choice of which axis to label 1 and which to label 2.

Never heard of symmetry? Nature is way more complex than you think. Any 4D space has 4 3D subspaces that each can share one action plane. This is exactly what we see in the final particle structure, where 4 protons that act (rotate) independently combine their action(s) over a common manifold and form the alpha particle.

There is one really big difference with a 3D subspace and a GR/QED 1:3 metric. The 3D subspace is allowed to freely rotate along 3 axes inside 4D where this in a 3:1 metric leads to fake solutions because mass then rotates in time...

• Never heard of symmetry? Nature is way more complex than you think. Any 4D space has 4 3D subspaces that each can share one action plane. This is exactly what we see in the final particle structure, where 4 protons that act (rotate) independently combine their action(s) over a common manifold and form the alpha particle.

There is one really big difference with a 3D subspace and a GR/QED 1:3 metric. The 3D subspace is allowed to freely rotate along 3 axes inside 4D where this in a 3:1 metric leads to fake solutions because mass then rotates in time...

For a current loop in X3, X4. Why is the resultant magnetic field in the X1 direction and not the X2 direction?

• SO(4) symmetry group is just a simplification of reality. You may think about vacuum like about kind of elastic foam: if we squeeze it somewhere, it will expand in another PERPENDICULAR direction - something like this:  This explains, why we always have two mutual components of vacuum stress field, i.e. electrical and magnetic vector components, which are phase shifted by 90°. The same concept can be applied recursively, which would bring us the Abellian SO(4) symmetry group. This concept doesn't follow just from elasticity of vacuum foam, but also from its inertia, when the foam also starts to behave like turbulent fluid. When we jump on foam, then it deforms itself in circular motion like vortex ring. The resistance against such a motion is given by its inertia momentum, i.e. when we would jump fast enough, then the foam would resist to deform in original direction - instead of it it will start to deform in perpendicular directions. We can observe pretty much the same effect even in fluids, which are forced to deform fast: the system of parasitic vortices oriented in perpendicular direction emerges.   So that when vacuum foam deforms, it always does so in system of nested directions, i.e. dimensions, which brings the symmetries of positive and negative charge, between others. The nested dimensions are denoted by different color:  The inertia of vacuum is expressed by weak structure constant which says, how much energy of vacuum with electromagnetic field will get translated into another dimension. Unfortunately the SO(4) model is not omnipowerful, because vacuum can undulate in another, non-abellian ways, for example in scalar way like this, which this symmetry group cannot describe:  Another problem follows from fact, that at high energy density the vacuum foam becomes nonuniform, because it tends to thicken under shaking in similar way, like soap foam and its fine structure constant converges to unity. Which also means that mutual undulations of electric and magnetic field aren't perpendicular each other anymore and their symmetry becomes more complex: the nested vortices must also follow dodecahedral geometry of hypersphere packing. It manifests itself for proton and neutron composed of quarks, which don't have half-integer spin anymore, but merely 1/3 or 2/3 charge, because vacuum density at their centers gets higher, than at their perimeter.

• Unfortunately the SO(4) model is not omnipowerful, because vacuum can undulate in another, non-abellian ways, for example in scalar way like this, which this symmetry group cannot describe:

The limit of SO(4) is the proton De Broglie radius. It describes what happens inside the limit we have in 3D,t physics world where the De Broglie radius is equivalent to the Heisenberg uncertainty (E/B) of measurement. So outside we need other models of interaction. The effective end of SO(4), where no more strong magnetic bonds are active is the quantum orbital state n=3. The state n=2 is somewhat special as a single wave bond is more or less the same as a potential.

• I agree with you that Cavendish influenced Maxwell ... But now it doesn't matter ... And what is important - the following - the modern formula of the "Coulomb's law" does not belong to Charles Coulomb! The modern formula of the so-called "Coulomb's law" was invented by the mathematician Maxwell, who in his work made unacceptable "tolerances", made unacceptable mathematical transformations that gave rise to the modern "electric charge". With his 1873 treatise, Maxwell actually perverted the teachings of Charles Coulomb, Poisson, Weber, Ampere and Thomson ... Let me remind you that Thomson gave the following definition of "charge" - On page 387, Thomson gave a definition of charges -

"Charges. - En vertu des theorems fondamentaux, on devra distribuer sur la surface de la sphere A une masse qaa = ∑An d’electricite, et sur B une masse de signe contraire qab = ∑Bn pour produire les memes potentiels sur ces spheres. "

“Charges. "According to fundamental theorems, we will have to distribute the mass qaa = ∑An of electricity on the surface of the sphere A, and the mass of electricity of the opposite sign qab = ∑Bn on B, in order to create the same potentials on these spheres." Thomson's symbol "q" denotes the mass of electricity ... In the same place he writes - "we can also notice that the distribution on A is the sum of distributions equivalent to each of the masses of An."

Source - SURFACE OF TWO ELECTRIC CONDUCTIVE SPHERES - https://drive.google.com/file/…La-Nz2Gj/view?usp=sharing

SURFACE OF TWO ELECTRIC CONDUCTING SPHERES - https://cloud.mail.ru/public/36jS/5fWpDQ6Ta

A year before Maxwell's publication, Thomson made his publication in 1872 -

Electrostatics and Magnetism, W. Thomson, 1872 - https://cloud.mail.ru/public/djfA/BGMcxAtQY

Electrostatics and Magnetism, W. Thomson, 1872 - https://drive.google.com/file/…2t51CoMv/view?usp=sharing

It is not difficult to understand the discrepancy in the approaches of Thomson and Maxwell ...

In 1872, Thomson followed Coulomb's theory and Coulomb's definitions - and this is clearly confirmed by the phrase - "... having designated Q, Q 'the amount of electricity that makes up charges ...", i.e. the term “charge” was understood as the mass of electricity (and mass is the amount of matter - the amount of electricity) ... On page 25, Thomson points out this directly - “... can be seen from the following examples of his results:

a) When the distance between the bodies is large in relation to their linear dimensions, the repulsion is inversely proportional to the square of the distance and directly proportional to the product of the masses. "...