ICN5D wrote:Yep, nice connection with the splitting in to 2D+2D. If one 2D plane is rotating, then the other 2D plane rotates with it. Which also means the two simultaneous rotating planes is mirroring the rolling ability of a duocylinder, being a product of two solid orthogonal disks. The two magnetic poles seem to correlate with the stationary plane of rotation into 4D. The flux lines outside the bar lie on the bisecting plane of this rotation. The direction of the flux current follows along the path of the moving axis, as the non-bisecting rotation here makes a torus.<br abp="676"><br abp="677">So, this also helps explain the whole 4D magnetic field as having three poles, with three charges sharing two potentialized components. All three charges combined become neutral. Removing one creates the potential, and it goes out seeking to neutralize itself. The three poles comes from the three stationary axes during a 5D rotation of a 4D shape. One is left over as the moving axis, the non-bisecting rotation. This is the flux current in toroidal form, as it flows through the system of three poles. If the 3D mag field is really in the shape of a duocylinder, then a 4D mag field may be a cylspherinder or related.<br abp="678"><br abp="679">Remember this list? This is what's happening, I believe:<br abp="680"><br abp="681">In 3D Magnetics, + and - , 2 unique magnetic poles on 3D planet, used with 2D vector<br abp="682"><br abp="683">+ : positive charge<br abp="684">- : negative charge<br abp="685"><br abp="686">+- : neutral charge<br abp="687"><br abp="688"><br abp="689">In 4D Magnetics, + and - and ^ , 3 unique magnetic poles used with bivector<br abp="690"><br abp="691">+- : plus-minus charge<br abp="692">+^ : plus-carat charge<br abp="693">-^ : minus-carat charge<br abp="694"><br abp="695">+-^ : neutral<br abp="696"><br abp="697"><br abp="698">In 5D Magnetics, + and - and ^ and *, 4 unique magnetic poles used with trivector<br abp="699"><br abp="700">+-^ : plus-minus-carat charge<br abp="701">+-* : plus-minus-asterisk charge<br abp="702">+^* : plus carat-asterisk charge<br abp="703">-^* : minus-carat-asterisk charge<br abp="704"><br abp="705">+-^* : neutral
Keiji wrote:Ok, I still don't think the whole "three magnetic charges" thing makes any sense, and I think there would still be only + and - no matter the dimension.
But woah, the idea that the electromagnetic field could be in the form of a duocylinder...
Keiji wrote:Ok, I still don't think the whole "three magnetic charges" thing makes any sense, and I think there would still be only + and - no matter the dimension.
quickfur wrote:That depends on how electromagnetism even works above 3D. I don't think we figured that out yet. For one thing, magnetic fields in 3D involve a cross product between two 3D vectors derived from the circulation of the electric field, but this is very specific to 3D. Only in 3D do you have a binary cross product. In 2D, the cross product has only a single argument -- and it's not clear how electromagnetism would even work like that (which of the two vectors should be chosen for the cross product?). In 4D, the cross product requires 3 arguments, but there are only 2 vectors available. What then? Where does the 3rd vector come from? And the further up the dimensions you go, the more complicated this problem becomes. In 5D, for example, the cross product requires 4 arguments but only two vectors are available! So where would you get the other 2 vectors from?
granpa wrote:In 4d the magnetic field is no longer a vector field but rather becomes a bivector field
But woah, the idea that the electromagnetic field could be in the form of a duocylinder...
anderscolingustafson wrote:If in 4d there were three electromagnetic charges in 4d then how would atoms work considering that 3 electromagnetic charges would mean that there would be 3 elementary particles that have a charge?
ICN5D wrote:Keiji wrote:Ok, I still don't think the whole "three magnetic charges" thing makes any sense, and I think there would still be only + and - no matter the dimension.
I borrow my instinct from quickfur's statement:quickfur wrote:That depends on how electromagnetism even works above 3D. I don't think we figured that out yet. For one thing, magnetic fields in 3D involve a cross product between two 3D vectors derived from the circulation of the electric field, but this is very specific to 3D. Only in 3D do you have a binary cross product. In 2D, the cross product has only a single argument -- and it's not clear how electromagnetism would even work like that (which of the two vectors should be chosen for the cross product?). In 4D, the cross product requires 3 arguments, but there are only 2 vectors available. What then? Where does the 3rd vector come from? And the further up the dimensions you go, the more complicated this problem becomes. In 5D, for example, the cross product requires 4 arguments but only two vectors are available! So where would you get the other 2 vectors from?
Which causes me to make the connection with multiple poles and multi-component charge potentials, like +^ charge.
[...]But woah, the idea that the electromagnetic field could be in the form of a duocylinder...
Yep, cool huh? The flux lines flow like a rolling duocylinder, in my opinion.
quickfur wrote:The neat thing about this is that all of these fibers are transitive, so the resulting symmetry is continuous: you could in theory rotate one of the rings to its orthogonal ring just by a simple 4D rotation. Which implies that if 4D electromagnetism were to exhibit this kind of structure, there would not be 2 or 3 charges but one single continuous charge!
Thinking about this now, perhaps this is just the thing we need for 4D planets to be workable... by postulating that this single-charge system isn't 4D electromagnetism, but 4D gravity. It would be a screwy kind of gravity (literally! ), but it would be globally homogenous and affine, yet locally directed, so perfect circular orbits would be a natural consequence of its intrinsic structure. Hmm... this leads to very interesting ideas!!
The problem with >2 charges is that the resulting force potentials will behave in a radically different way from 3D electromagnetism
the force will be strongly biased toward attraction (because for a 3-charge system, say, every charge feels the repulsion on one like charge and the attraction of two unlike charges, so assuming a macroscopic balance of all 3 charges, the net behaviour is an excess of attraction).
ICN5D wrote:quickfur wrote:The neat thing about this is that all of these fibers are transitive, so the resulting symmetry is continuous: you could in theory rotate one of the rings to its orthogonal ring just by a simple 4D rotation. Which implies that if 4D electromagnetism were to exhibit this kind of structure, there would not be 2 or 3 charges but one single continuous charge!
Thinking about this now, perhaps this is just the thing we need for 4D planets to be workable... by postulating that this single-charge system isn't 4D electromagnetism, but 4D gravity. It would be a screwy kind of gravity (literally! ), but it would be globally homogenous and affine, yet locally directed, so perfect circular orbits would be a natural consequence of its intrinsic structure. Hmm... this leads to very interesting ideas!!
Hmm, indeed! Gravity? One single continuous charge? Very strange and interesting connection. Gravity is, after all, a single continuous attraction. Have you read up on the donut world http://io9.com/what-would-the-earth-be- ... 1515700296 , that Marek found? If the flow of 4D gravity is like this Hopf fibration, then it could form something similar, naturally. I'm not sure what a hopf fibration is. I could look it up, but you would be better at translation.
The problem with >2 charges is that the resulting force potentials will behave in a radically different way from 3D electromagnetism
Well, it does kind of look like quark interaction. Some have a 2/3 charge, and it takes three to satisfy the bound up system, making protons and neutrons. That 2/3 charge could be the bivector between three arguments.
the force will be strongly biased toward attraction (because for a 3-charge system, say, every charge feels the repulsion on one like charge and the attraction of two unlike charges, so assuming a macroscopic balance of all 3 charges, the net behaviour is an excess of attraction).
Perhaps an excess of attraction makes stronger bonds, as in the strong nuclear bond
Keiji wrote:Surely they couldn't work at all in 2D, since you would need a third dimension for the helical (cylindrical formed) wind path to even exist in.
In 4D, I'm not really sure but I guess the helix would have to take the form of a cubinder, rather than a spherinder, as you can't have a helix around the extruded surface of a 3D sphere. That would leave a free dimension for it to move on. Perhaps it would align itself with a more macroscopic wind path, so instead of having a hurricane that focuses on a particular "point" of the planet's surface (albeit a point that moves over time), it would focus on a great circle of the planet's surface (that moves over time), or a portion of this great circle.
PolyhedronDude wrote:I imagine a 4-D tornado having a horizontal cross section that looks like a smoke ring that could be miles across, it would twist and undulate and sometimes twist into a figure eight pattern and forming two rings and split into two smaller tornadoes. Hurricanes would be larger versions that could circle the globe.
quickfur wrote:You can spot some of these subsymmetries in various 4D regular polytopes: the tesseract, for example, can be decomposed into 2 orthogonal rings of 4 cubes each, corresponding with duocylindrical symmetry, which is the same as the Hopf fibration of the dihedral tiling of the 2-sphere. The 24-cell can be decomposed into 4 rings of 6 octahedra joined at opposite faces, corresponding with the Hopf fibration of the tetrahedral tiling of the 2-sphere. The 24-cell can also be decomposed into 6 rings of 4 octahedra joined at opposite vertices, which corresponds with the Hopf fibration of the cubical tiling of the 2-sphere. The 120-cell can be decomposed into 12 rings of 10 dodecahedra joined at opposite faces, corresponding with the dodecahedral tiling of the 2-sphere. The 600-cell can be decomposed into 20 rings of 30 tetrahedra each, in an interesting formation that exhibits a local 3-fold twisting (known as the Boerdijk-Coxeter helix), corresponding with the icosahedral tiling of the 2-sphere. Interestingly enough, none of the regular polychora correspond with the octahedral tiling of the 2-sphere, but there is a CRF polytope that does: the BXD, or bi-icositetra-diminished 600-cell, a curious non-uniform yet cell-transitive and vertex-transitive polychoron consisting of 48 tridiminished icosahedra that form 8 rings of 6 cells each.
What's even more remarkable about this structure, is the Hopf fibration itself -- that is, the 1-to-1 mapping from the 2-sphere. It so happens, that if you mark out a point on the 2-sphere, say at the north pole, then that north pole point maps to some particular circle on the 3-sphere -- let's say the XY circle that we marked out above (it doesn't really matter which as long as we consistently choose the circles, since the structure is symmetric). The south pole point, then, maps exactly to the orthogonal ZW circle.
Coming back to the topic of electromagnetism/gravity, if we look at the "field lines" induced by duocylindrical symmetry (or, for that matter, any of the even subsymmetries, say the cubical subsymmetry or the dodecahedral subsymmetry), we find that actually, the specially-marked out directions (the pair of orthogonal circles) aren't special at all, since the other, oblique field lines are themselves circles which are symmetrically equivalent to the marked out pair! So there's really nothing special about those two particular chosen circles; they are equivalent to any of the other "field lines". Thus, instead of two distinct, opposite charges, what we get is a continuum of one "charge" to the opposite "charge". But since we can't tell which one is which, due to the symmetry, and in fact, we can't even tell if any given "field line" is the specially marked ones or not, since they are all equivalent under the Hopf fibration's symmetry group, we might as well just call everything one and the same single charge!
And so we arrive at this radically different version of 4D gravity, where the force does not act radially, but circumferentially, and perfect circles (since all of the Hopf fibers are geodesics) are a natural result of this kind of structure. It looks nothing like 3D gravity, to be sure, but it does promise stable planetary orbits!
ICN5D wrote:[...]quickfur wrote:You can spot some of these subsymmetries in various 4D regular polytopes: the tesseract, for example, can be decomposed into 2 orthogonal rings of 4 cubes each, corresponding with duocylindrical symmetry, which is the same as the Hopf fibration of the dihedral tiling of the 2-sphere. The 24-cell can be decomposed into 4 rings of 6 octahedra joined at opposite faces, corresponding with the Hopf fibration of the tetrahedral tiling of the 2-sphere. The 24-cell can also be decomposed into 6 rings of 4 octahedra joined at opposite vertices, which corresponds with the Hopf fibration of the cubical tiling of the 2-sphere. The 120-cell can be decomposed into 12 rings of 10 dodecahedra joined at opposite faces, corresponding with the dodecahedral tiling of the 2-sphere. The 600-cell can be decomposed into 20 rings of 30 tetrahedra each, in an interesting formation that exhibits a local 3-fold twisting (known as the Boerdijk-Coxeter helix), corresponding with the icosahedral tiling of the 2-sphere. Interestingly enough, none of the regular polychora correspond with the octahedral tiling of the 2-sphere, but there is a CRF polytope that does: the BXD, or bi-icositetra-diminished 600-cell, a curious non-uniform yet cell-transitive and vertex-transitive polychoron consisting of 48 tridiminished icosahedra that form 8 rings of 6 cells each.
That right there just cleared up an incredible amount of things on this website. I remember an earlier post of yours, that showed the two ring structures of cartesian products. And, how they form a duocylinder when the two reach infinite edges. Comparing that concept to all of those crazy shapes I see, like the 120-cell, 600-cell, 24-cell was crucial for me. Do you have any idea what those things sound like to an outsider? They sound like horrendous, over-complicated shapes that I could never know.
But the analogy is amazing, and mind expanding. The strange dual property of the 24-cell with the 4x6 and 6x4 rings, is that one of the special abilities that no other has?
When you described the nature of the multiple torii on swirlprisms, it reminds me of the toroidal convection belts in the atmosphere as a whole. Jupiter is much larger, and thus has many more bands. Just speculating, but maybe the jetstream is some sort of atmospheric relic, some mathematical emergence out of this hopf fibration. And the 4D hurricane as a colossal vortex ring follows the same principle.
Other than that, has there been any application with tiling a 3 or 4-sphere? Is there any kind of pattern seen in other toratopes, particularly those with holes?
quickfur wrote:However, since there are two possible Hopf fibrations which are mirror-images of each other, it would seem to suggest that we're back to a two-charge system, where one charge is the mirror-image of the other! So perhaps we can have a kind of 2-charge pseudo-"electromagnetism" in 4D after all... I've no idea how such a thing would even work, though, since it would be unlike anything that even remotely resembles 3D electromagnetism, in spite of having two charges!
ICN5D wrote:quickfur wrote:However, since there are two possible Hopf fibrations which are mirror-images of each other, it would seem to suggest that we're back to a two-charge system, where one charge is the mirror-image of the other! So perhaps we can have a kind of 2-charge pseudo-"electromagnetism" in 4D after all... I've no idea how such a thing would even work, though, since it would be unlike anything that even remotely resembles 3D electromagnetism, in spite of having two charges!
Well, how about that? It seems like a charge-anticharge entity, self neutralizing, but two components. Or, they are all universally attracted to each other in some way. But, I suspect they may be all neutral.
Have you ever thought about doing any toratope renders? Like the amazing complex tigroids?
I've learned a crap load from Marek about the cut algorithm, and how to build cut arrays. With your rendering skill and practice, there are some cool things that can be made. Those shapes, in addition to all of the rest, have been mentally conceived by my third eye, because I don't have any cool rendering programs. I think that's why I've gotten really good with understanding the toratopes. They're complex in the opposite way to polychora, in that they have only one surface. But, it is this surface that is super complex, most notably in the tigroids. I've been slicing up some 7D ones lately, but you already saw that.
quickfur wrote:I haven't been able to keep up with the toratopes thread, 'cos you guys have been going way too fast for me. (Though I suspect you'd say the same about the CRF thread. )
quickfur wrote:The Hopf fibration is a very interesting 1-to-1 mapping between points on the 2-sphere (i.e., 3D sphere) and circles on the 3-sphere (4D sphere)...
Keiji wrote:quickfur wrote:The Hopf fibration is a very interesting 1-to-1 mapping between points on the 2-sphere (i.e., 3D sphere) and circles on the 3-sphere (4D sphere)...
That was an extremely enlightening post. I've created a wiki page about the Hopf fibration, essentially a copypaste of your post with some minor rewording. Hope you don't mind!
quickfur wrote:But I think we should rather be looking into the intrinsic nature of the electromagnetic force, to understand why it works the way it does, before jumping to conclusions. Does anyone know of a description of the electromagnetic force in general terms, that doesn't refer to vectors of specific dimension?
Barring that, if we have to deal with the curl operators in Maxwell's equations, does anybody understand what it is about the curl operator that imparts the right properties for the equations to "make sense" together?
ICN5D wrote:quickfur wrote:But I think we should rather be looking into the intrinsic nature of the electromagnetic force, to understand why it works the way it does, before jumping to conclusions. Does anyone know of a description of the electromagnetic force in general terms, that doesn't refer to vectors of specific dimension?
Barring that, if we have to deal with the curl operators in Maxwell's equations, does anybody understand what it is about the curl operator that imparts the right properties for the equations to "make sense" together?
Check this one out, the part about continuous charge:
http://en.wikipedia.org/wiki/Lorentz_fo ... stribution
And this:
http://en.wikipedia.org/wiki/Gravitomagnetism#Equations
And especially this:
http://en.wikipedia.org/wiki/Gravitomag ... er_effects
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