Twisters

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

Twisters

Postby anderscolingustafson » Wed Mar 05, 2014 2:29 am

How would a Hurricane or Tornado work in 2d and how would a Hurricane or Tornado work in 4d?
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Re: Twisters

Postby Keiji » Wed Mar 05, 2014 7:00 am

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.
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Re: Twisters

Postby wendy » Wed Mar 05, 2014 7:22 am

We don't know enough about how the curl function works in 4d, to see how this works.
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Re: Twisters

Postby ICN5D » Wed Mar 05, 2014 7:43 am

What does curl function mean? I've seen it represented in magnetic fields, and now weather. I'm interested now. Seems like something to do with circular motions, or bisecting rotations.
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Re: Twisters

Postby wendy » Wed Mar 05, 2014 8:01 am

Curl is the cross or vector product between the del pseudo-vector and a different vector. It is a description of eddy currents, including magnetic currents and winds. However, it's particular to three dimensions, unlike the scalar product, which exists in every dimension.
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Re: Twisters

Postby quickfur » Wed Mar 05, 2014 3:28 pm

My intuition is that in 4D, any twisting motion will disperse quickly due to the extra dimension -- it will not be self-reinforcing as in 3D tornadoes. The thing about 3D is that when you have vertical motion of air, the remaining 2 dimensions constrain the motion of air surrounding the column into circles, which gives rise to a kind of "constructive interference" that causes a persistent rotation. In 4D, however, given vertical motion of air, there are 3 remaining dimensions around it, so any rotational motion of the surrounding air would interfere with each other because they are not constrained to be parallel to each other. So it seems likely that they will destructively interfere, and no persistent rotation will arise (although turbulence will certainly result).
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Re: Twisters

Postby ICN5D » Wed Mar 05, 2014 6:26 pm

Hmm, that's interesting. Trying to visualize 4D atmosphere dynamics now...

Well, when I look at the 3D mag field lines around a bar magnet, it sort of resembles the grain of a 4D rotation. The way the lines converge along a narrow path down the middle of the magnet, and the way it curls back on itself, like a sock rolling back, around the outside of the magnet. The flux lines are tracing out the path of a 4D rotation, and showing how a cube rotates into a cubinder, or cylinder rotates into a duocylinder. Quickfur, you mentioned before some relationship with rotating fields. That's what it looks like, a continuously flowing field like that of a smoke ring vortex. By rotating a 4D shape around in 4D, its cells are performing the same motion along the grain of the flux lines. It's the inside out rolling of a rotation.

Image

Image

So, perhaps that's what is happening with flux lines, the pattern simplifies when thinking in 4D. Maybe the field is in the shape of an actual duocylinder, in 4D.
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Re: Twisters

Postby quickfur » Wed Mar 05, 2014 6:58 pm

That's an interesting concept... so according to this idea, there will be no electric/magnetic "poles", per se, corresponding with a single vector, but magnetic/electric "planes", corresponding with bivectors. That sorta seems to make sense in its own way, and fits rather well into 4D, because every plane naturally has an orthogonal plane (its orthogonal space), so this induces a kind of "splitting" of 4D into 2D+2D with a particular orientation.

One peculiarity is that you can't have a planar flow, since particles can only move in a single direction simultaneously, so perhaps magnetic/electric effects don't arise unless there's rotation in one, which induces rotation in the other?

Not sure where this leads, but it sure seems interesting! In any case, it sure looks nothing like 3D electromagnetism. :P
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Re: Twisters

Postby ICN5D » Wed Mar 05, 2014 7:44 pm

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.

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.

Remember this list? This is what's happening, I believe:

In 3D Magnetics, + and - , 2 unique magnetic poles on 3D planet, used with 2D vector

+ : positive charge
- : negative charge

+- : neutral charge


In 4D Magnetics, + and - and ^ , 3 unique magnetic poles used with bivector

+- : plus-minus charge
+^ : plus-carat charge
-^ : minus-carat charge

+-^ : neutral


In 5D Magnetics, + and - and ^ and *, 4 unique magnetic poles used with trivector

+-^ : plus-minus-carat charge
+-* : plus-minus-asterisk charge
+^* : plus carat-asterisk charge
-^* : minus-carat-asterisk charge

+-^* : neutral
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Re: Twisters

Postby anderscolingustafson » Wed Mar 05, 2014 8:47 pm

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


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?
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Re: Twisters

Postby Keiji » Wed Mar 05, 2014 11:15 pm

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...
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Re: Twisters

Postby quickfur » Thu Mar 06, 2014 12:05 am

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.

Assuming it's possible to generalize electromagnetism to 4D at all, yes.

I think we're going a little too far with generalizing the extrinsic phenomena caused by the electromagnetic force, like the number of charges, rather than studying the intrinsic nature of the force itself that might actually lead to a plausible 4D counterpart. This is exactly the kind of dimensional analogy that Coxeter warned against, like predicting the formula for the volume of the 3-sphere by randomly incrementing various elements of the 2D and 3D formulae A=pi*r^2 and V=4/3*pi*r^3, which does not lead to the right answer. :P

(The correct volume for the 3-sphere is actually 1/2*pi^2*r^4, and can only be arrived at by analysing the underlying n-volume integrals, not by wild guessing based on the 2D and 3D formulae. If you don't believe me, try guessing the volume of the 4-sphere... the correct answer is 8/15*pi^2*r^5, which is basically impossible to deduce from looking at the formulae of earlier dimensions alone. You have to understand the integrals that produce it. :nod: )

But woah, the idea that the electromagnetic field could be in the form of a duocylinder...

It's certainly an attractive idea, and one might even go all the way and say that the electromagnetic lines of force follow the Hopf fibration of the 3-sphere... :P 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? IMO that's a more profitable direction of research than randomly incrementing things in various electromagnetic phenomena, fun as it might be (hey, I like the idea of duocylindrical 4D electromagnetism too, but I'm not confident it will actually lead to anything consistent!).
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Re: Twisters

Postby ICN5D » Thu Mar 06, 2014 12:14 am

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.


Then, granpa says:

granpa wrote:In 4d the magnetic field is no longer a vector field but rather becomes a bivector field


Which reinforces my feeling about three poles. Of course, given that this would be in 4D space, it would flow like a 5D open toratope. I see some connection in this logic with a cylspherinder. A neutral charge has the symmetry of an n-sphere. Removing a component makes it unstable, and it seeks out to neutralize itself. Removing one component in +-^ makes a +- or +^ or -^ charge, with two polar components. This reflects the bivector, as having two parts as well. The cylspherinder has a torisphere ortho bound to a spheritorus. The torisphere-cell has spherical symmetry, like the two components. All three together as neutral is like a pentasphere. By removing a component, we get a symmetry break, and turns into cylspherindric symmetry. The spheritorus-cell reflects the flow direction I guess. Just pure speculation applied to the duocylinder idea.


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.
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Re: Twisters

Postby ICN5D » Thu Mar 06, 2014 12:20 am

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?


We're trying to figure out the principles first. Hang on, bud :) It may not even work that way...
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Re: Twisters

Postby quickfur » Thu Mar 06, 2014 12:43 am

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.

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). This means much of the interplay of charge balance in 3D electromagnetism -- y'know, what makes atoms possible and what drives chemistry -- will not be replicated, and thus the resulting "electromagnetism" (if we can call it that!) will look nothing even remotely resembling electromagnetism!

[...]
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.

Actually, I suspect they will look more like the Hopf fibration of the 3-sphere, because the orthogonal rotation only applies to points that are perfectly aligned with the two orthogonal planes of rotation. Any oblique points will follow a spiralling path that essentially traces out the fibers of the Hopf fibration. The duocylinder is but the simplest subsymmetry of the Hopf fibration; other examples include BXD, which sports 8 interlocking rings of 6 cells each, and Jonathan Bowers' spidrox (spiral-diminished rectified 600-cell), which sports 12 interlocking rings of alternating prisms and antiprisms (and also 20 interlocking rings of twisting square pyramids -- see the Boerdijk–Coxeter_helix). There's also Bowers' regular polytwisters, which have curved surfaces that follow these same spiralling paths. These are all discrete subsymmetries of the Hopf fibration, of course. The full fibration is a continuous map that consist of a continuum of these interlocking circular fibers.

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. :o It would be a screwy kind of gravity (literally! :lol: ), but it would be globally homogenous and affine, yet locally directed, so perfect circular orbits would be a natural consequence of its intrinsic structure. :o_o: Hmm... this leads to very interesting ideas!!
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Re: Twisters

Postby ICN5D » Thu Mar 06, 2014 1:04 am

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. :o It would be a screwy kind of gravity (literally! :lol: ), but it would be globally homogenous and affine, yet locally directed, so perfect circular orbits would be a natural consequence of its intrinsic structure. :o_o: 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 :)
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Re: Twisters

Postby quickfur » Thu Mar 06, 2014 2:46 am

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. :o It would be a screwy kind of gravity (literally! :lol: ), but it would be globally homogenous and affine, yet locally directed, so perfect circular orbits would be a natural consequence of its intrinsic structure. :o_o: 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 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). It has many interesting characteristics, both mathematically and geometrically.

One is that these circles are all mutually disjoint, so none of them intersect each other; in other words, you can cut up the 3-sphere into these circles, so to speak, or conversely, the union of these circles equals the entire surface of the 3-sphere. This in itself isn't that remarkable, but what makes it remarkable is the way these circles are laid out on the 3-sphere. Unlike the 3D case, where you can cut the 2-sphere into, say, latitude circles, all the circles in the Hopf fibration are great circles, or geodesics (their radius is equal to the radius of the entire 3-sphere). Furthermore, they are laid out in a spiralling, chiral fashion. Suppose we identify one of them as our starting point, say it's a circle in the XY plane. Around it we find circles that swirl around this XY circle, interlocking with our designated circle, and around them are more swirling circles, at an incrementally flatter angles to our XY circle, wrapping themselves around it in an increasingly thick bundle. As we approach 90°, these circles start to swirl around the ZW plane, until they converge upon a single circle in the ZW plane, which is perfectly orthogonal to the original XY circle.

Now, it may not be immediately clear from this description, but all of these circles are actually transitive: pick any one of them, and you find an identical structure of swirling circles around it, just like around every other circle. So it forms a continuous symmetry group, in which all the circles are equivalent via a rotation. What kind of rotation? Precisely a Clifford double rotation -- that is, a simultaneous rotation in two orthogonal planes with the same rotation rate.

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. So you may think of it as a kind of "unzipping" of the 2-sphere, where you stretch it open at the south pole point so that it becomes an equatorial circle, and then spinning the rest of the sphere around in 4D to cover the surface of the 3-sphere exactly once (under said Clifford double rotation). This in itself may not adequately convey the rich structure that's produced: Jonathan Bowers had the idea to examine what happens when our starting 2-sphere has a tiling imposed upon it -- i.e., what happens if we "apply the `Hopf function'" to a 3D polyhedron -- that is, the spherical tiling corresponding with that polyhedron. When we do that to, say, a dodecahedron, we find that the top pentagonal face (say) maps to a pentagonal torus that wraps around a section of the 3-sphere, and the adjacent 5 pentagonal faces map to 5 pentagonal tori swirling around this first torus, and then the adjacent 5 other pentagons map to another 5 pentagonal tori swirling around the previous 5 tori, and the bottom pentagon maps to the last torus that wraps around the plane orthogonal to the first torus. This is what Jonathan Bowers calls a "dodecahedral regular polytwister": these 12 tori are all equivalent to each other under a certain symmetry, which is variously known as swirlprism symmetry. Other polyhedra undergo similar mappings to 4D, each Platonic solid producing a regular polytwister. All of their symmetries are subsymmetries of the Hopf fibration; just as the polyhedra themselves belong to a subsymmetry of the 2-sphere.

When we apply this mapping to the dihedral tiling of the 2-sphere (i.e., paint the top hemisphere red and the bottom hemisphere blue, for example), then what we get is none other than the duocylinder itself. So you see, duocylindrical symmetry is but a subsymmetry of the Hopf fibration -- the simplest non-trivial one!

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.

In any case, all of these rings have that characteristic interlocking structure, of which the Hopf fibration is the continuous version. The duocylinder is just the simplest discrete subsymmetry of it; and even within the duocylinder itself you can already spot some of the larger symmetry group: if you pick a point on the duocylinder's ridge (where the two tori touch each other), then it's possible to pick out a great circle (i.e., a geodesic path on the surface of the 3-sphere) that lies entirely on this ridge: this circle would form a swirling pattern around the ridge, and it corresponds with one of the Hopf fibers that isn't exactly on one of the two planes of the duocylinder's symmetry. Put another way, you obtain duocylindrical symmetry if you color one of the circular Hopf fibers red, and the orthogonal fiber blue, and then color the other fibers according to whether they are closer to the red or blue fiber. So, duocylindrical symmetry is just where you specially mark out a pair of mutually-orthogonal fibers in the Hopf fibration of the 3-sphere. If you were to mark out other sets of fibers, say 4 fibers in tetrahedral symmetry, then this produces the tetrahedral subsymmetry of the Hopf fibration. Or mark out 20 of them in icosahedral symmetry to get an icosahedral subsymmetry.

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! :lol: :XD:

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.

Yes. :)

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 :)

Which would jive with how quantum chromodynamics works with a 3-charge system. :)
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Re: Twisters

Postby Polyhedron Dude » Thu Mar 06, 2014 2:59 am

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.


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.
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Re: Twisters

Postby ICN5D » Thu Mar 06, 2014 3:41 am

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.



Yup, I can agree with that. That's what I was thinking, too. Or, maybe a spherical one, that floats around and crashes into the ground.




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?
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Re: Twisters

Postby anderscolingustafson » Thu Mar 06, 2014 6:02 am

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.


So basically the magnetic north pole could be orientated around one of the circles of rotation and the magnetic south pole could be orientated around the circle of rotation perpendicular to it. I was thinking that the magnetic field on a 4d planet could be orientated around two of the circles of rotation as well.

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! :lol: :XD:


So one way to get stable orbits in 4d would be for Gravity and Electro Magnetism to be combined into one force? If there was a continuum between the two forces would that cause any problems for atoms at the subatomic level considering that in our universe atoms are made from 3 particles and two particles that have a charge?

I was also thinking about how to get stable orbits in 4d and was wondering if it would also be possible to get stable orbits by adjusting the equation for centripetal force so that the velocity would be cubed instead of squared but you never commented on my topic on changing the equation for centripetal force viewtopic.php?f=27&t=1875.
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Re: Twisters

Postby quickfur » Thu Mar 06, 2014 6:13 am

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.

When I first started learning about 4D, the farthest I got was the 5-cell and the tesseract. It took me a good long while to be able to visualize the 16-cell -- back then the only thing I knew about it was that it had 4 tetrahedra folded around an edge: which was true, no doubt, but completely unhelpful in visualizing the shape as a whole. It wasn't until I discovered its equivalence to the octahedron bipyramid that it clicked for me. And then the next milestone was the 24-cell: I had read all about its unusual properties by then, but I had no picture in my mind to assign it to. It took me another good while before I finally worked out the exact decomposition of its rhombic dodecahedron projection into 3D into 6 octahedral cells (with another 12 cells projected to the rhombus faces of the rhombic dodecahedron) -- then finally I understood what it was. Shortly after, I (re)discovered the structure of its cuboctahedral projection. So finally I could visualize the 24-cell.

But the 120-cell and 600-cell: I didn't even know where to begin. The sheer count of cells was already enough to convince me that they're impossible to visualize. But one day, I stumbled across a site that explained the structure of the 120-cell. It didn't quite cinch it for me at the time, but it was a start. I began to have some vague idea about what the 120-cell looked like. Then began the long quest of writing my polytope viewer program that could finally render projections of the 120-cell that I could understand. Well, that took years, not just because the program took a while to write, but also because I didn't know where to even begin looking for data files that correctly represented the 120-cell's structure -- it is way too big to code by hand, as I'm sure you know (I mean, it has 600 vertices!!). In the interim, I sorta gave it up as being too complicated, but one day, I chanced upon, of all things, a screensaver that showed an animated 120-cell. It wasn't rendered in the most understandable of ways, but by looking at it, I suddenly realized that the 120-cell was actually not that complicated after all: it consisted of only about 2 to 3 layers of cells surrounding a central cell before you get to the limb, and the far side is essentially a repetition of the same structure on the near side. The large cell count was really only because the 3D surface of the polytope had so much more room to be filled with cells, but globally speaking, it was just approximately 3 layers of cells on one side and 3 layers of cells on the other side. It was a good while before I could realize this vision of mine about the 120-cell, though. I had to perfect my polytope viewer program, and then add a convex hull algorithm (since inputting the entire structure by hand was completely out of the question), and then find reliable coordinates for the 120-cell.

Finally, after much effort, I managed to produce an image of the 120-cell that represented, for the first time, my vision of what it would look like to a hypothetical 4D being. The first images weren't that great, but they captured the essence of it that later on, based on accumulated experience in rendering these things, I remade in this favorite image of mine, that I'm quite proud of to this day:

Image

The 600-cell, however, remained beyond my grasp. Eventually I did do a piecemeal rendering of it, which eventually culminated in this image of it:

Image

I have to admit, though, that I still have trouble fully grasping its structure in my head; at least, not to the level I can "see" the 120-cell in my mind. I kinda lose track of things shortly after the first layer of 20 tetrahedra surrounding a vertex in icosahedral formation, 'cos the number of cells just grows out of control quickly. :P The images I do have, though, at least do help me realize that the vertices of the 600-cell form great circles around it, as you may notice somewhat in the above image. After much study, I've come to see the 600-cell as a highly-augmented version of a series of monostratic layers, consisting of an icosahedral pyramid stacked on top of an icosahedron||dodecahedron segmentochoron stacked on top of a dodecahedron||icosidodecahedron segmentochoron, and then the same layers in reverse. These layers by themselves are not the 600-cell, and in combination they are non-convex, but by adding suitable additional tetrahedra over the gaps between the layers, the thing closes up into a shape in which every tetrahedron is equivalent. So the 600-cell is a kind of glorified 4D equivalent of an augmented icosidodecahedron, if you will. (The usual analogy is that it's a 4D icosahedron, but IMO that doesn't quite capture its global structure.)

And so I came to some understanding of the structure of these things, upon which I embarked on the project to render projections of every 4D convex uniform polychoron. The process has been very enlightening, especially among the 120-cell/600-cell family members, which shed light on many details of how the 120-cell and 600-cell are put together. And, I'm proud to say, I finally completed this project this month with the biggest of them all: the omnitruncated 120-cell. :) Of course, in the meantime I've also started exploring the CRF polychora, and that has also been very enlightening into the nature of 4D geometry. :) It's certainly very mind-expanding, as you say.


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?

I'm not 100% sure about this one. I think the 600-cell can also be decomposed into rings of 12 tetrahedra each, alternately connected by vertices and faces, and there should be 50 such rings. But I'm not 100% sure if the rings are fully disjoint, since otherwise it would not represent a subsymmetry of the Hopf fibration. In any case, the 600-cell and the 120-cell are dual to each other, so the two share the same set of Hopf fibration subsymmetries, if you invert the corresponding elements. Similarly, the tesseract and the 16-cell are dual to each other, and so share their mutual set of subsymmetries. The 24-cell and the 5-cell are both self-dual, though the subsymmetry in the case of the 5-cell isn't particularly enlightening as it corresponds with the Hopf fibration of the entire 2-sphere, so it doesn't quite show the spiralling structure of the Hopf fibration, even though it does show the 5 tetrahedra in a single Boerdijk-Coxeter helix that covers the entire surface of the polytope. The 24-cell is a pretty unique thing in being self-dual yet not a simplex. But I'm not sure if there's any unique interplay with the Hopf fibration here.

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.

Perhaps. The thing about the Hopf fibration, though, is that the circular fibers are not parallel, unlike the atmospheric bands on planets like Jupiter. In fact, none of them are parallel; they are all "skewed" relative to another in a spiralling fashion.

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?

Not sure... the Hopf fibration applies to the 3-sphere, and thus to all convex polychora (since they essentially serve as tilings of the 3-sphere), but I'm not sure how (or whether) a connection can be drawn to non-convex things like toratopes.
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Re: Twisters

Postby quickfur » Thu Mar 06, 2014 6:20 am

One thing that suddenly occurred to me about the Hopf fibration: it is chiral, as I pointed out, which means that there are two distinct mirror images of it! So if somehow derive 4D gravity from a Hopf fibration structure, it would induce a universal chirality to the 4D universe (only one of the mirror-images would occur), which would be rather strange. Almost like how in biology only one of two possible stereoisomers in certain molecules occur naturally.

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!
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Re: Twisters

Postby ICN5D » Thu Mar 06, 2014 7:17 am

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.
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Re: Twisters

Postby quickfur » Thu Mar 06, 2014 4:05 pm

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.

But that depends on what you mean by "charge"... :) For all you know, 3D electromagnetic charge may as well be caused by opposite chirality in some deep underlying symmetry. :P Since we can't look into electric charge with a microscope, we can only build models based on external observations, so we don't really know for sure.


Have you ever thought about doing any toratope renders? Like the amazing complex tigroids?

The main problem is that currently my polytope viewer cannot handle non-convex objects. :( Well, it can, if you make the polytope definition files describe a non-convex object, but the renders will come out all wrong because it assumes convexity when doing visibility clipping.

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.

I haven't been able to keep up with the toratopes thread, 'cos you guys have been going way too fast for me. :P (Though I suspect you'd say the same about the CRF thread. :lol: )
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Re: Twisters

Postby ICN5D » Thu Mar 06, 2014 7:26 pm

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. :P (Though I suspect you'd say the same about the CRF thread. :lol: )


Yep, you're right about that. When you say something like " bi-icositetra-diminished " and related, there's nothing but a fog that obscures the gem. I'm sure that at first glance of those terms, you get an instant visual. But, in contrast, something like the 220210-(tiger,torus) tiger is a fog to you! Those CRFs are amazing in their own respect, no doubt. The twisting helical property is strange, not what I would have expected could occur. The fact that all of them have unit edges is wild. I haven't done too much research into pentagons, icosahedra, etc. Only the rotatopes, tapertopes, toratopes, and duoprisms have been my study so far. Actually, I find the high-D toratopes a bit easier, in that they have no sharp edges. So, learning the cuts allows you to mentally trace out their structure in its entirety. Still grasping it, of course! But, within a month or so, I've come really far in my understanding. As you can tell, I love playing with computing algorithms. It's what has revealed all of high-D to me, is the math system I created. Even though I don't know calc or parametric equations, I still understand the logic of high-D shapes. I know what I'm seeing in my head is right. So, I created a language and computation that described what I saw happening with linear operations. It's very powerful now, with the new adaptation. It's doing real math amazingly quick and accurately. In still have yet to dissect the new process for everyone, to see how it works. After that, I will be compiling the full list of 6D shapes that can be made with my notation. There's over 350 of them, not including toratopes! Then, compute their n-cells.
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Re: Twisters

Postby Keiji » Thu Mar 06, 2014 8:56 pm

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. :D 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!
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Re: Twisters

Postby quickfur » Thu Mar 06, 2014 9:11 pm

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. :D 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!

Improved the wording some more. ;)
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Re: Twisters

Postby ICN5D » Fri Mar 07, 2014 4:42 am

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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Re: Twisters

Postby quickfur » Fri Mar 07, 2014 6:28 am

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

Hmm. This is interesting, but not quite what I was looking for. I was looking more for an interpretative treatment of the curl operator and what intrinsic underlying symmetry causes it to arise (I vaguely recall reading something like that before, but I don't remember what it was). The problem is that by the time you get to equations dealing with the curl operator, you're already squarely stuck in 3D with no obvious way to get beyond without causing ripples of unexpected (and probably unwanted) side-effects and unforeseen consequences to propagate through the whole system. To truly get something that "fits" well in 4D, we need to find out what causes electromagnetism to manifest itself the way it does in 3D, and then figure out what would happen if it were to manifest itself in 4D.
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Wendy on Hopf Fibulation, etc

Postby wendy » Fri Mar 07, 2014 8:09 am

Hopf fibulation, or clifford parallels, form an interesting feature of four dimensions, that is not replicated in three dimensions.

Great Arrows

A great arrow is a great circle, with a direction. A great circle, then is counted twice. 'Phase space' is a physicist's term, where every phase is represented by a point.

The phase-space of great arrows in three dimensions, is a sphere. You place your right hand on the surface of the sphere, with the fingers pointing in the direction of movement, the thumb would point to the north pole. The north pole then, is a the centre of rotation. Every point of the sphere is the centre (as in N pole) of a great arrow. Note the south pole represents the great arrow reversed - ie if N is +X, then S is -X.

Since in three dimensions, one only has great-arrow rotation, the rotation-direction is represented by a point on the surface of the sphere, and the intensity is represented radially: The polar axis of the sphere passes through the surface at +1 at the N pole, and -1 at the south pole, if one supposes that rotation is to the east.

In four dimensions, you still have great arrows, but the phase-space is now a six-dimensional thing. It's actually a "bi-glomohedrix prism", or the limit of the toratope ((iii)(iii)), as the outer brackets go to zero. The full phase-space of lettable rotations is a six-dimensional sphere, ie (iiiiii).

The "bi-glomohedrix prism" is a prism-product of two sphere-surfaces. Every possible great arrow corresponds to a pair of points, one on each sphere. Let's call these spheres, for no obvious reason, L and R. If we fix a point on L, and take all the points on R, each circle in this set corresponds to a series of great arrows forming a set 'clifford parallel', which we can tell is a right hand set, because the great arrows near a given set swirl as the fingers of the right hand point, when the thumb points in the direction of the arrow. So a variation of R for a fixed L is a right-hand clifford parallel. Likewise, for a fixed point R, every point on L corresponds to a set of clifford left-parallels.

A point X on L, and Y on R, gives a particular great arrow. We write it as +X, +Y. The same great arrow -X, -Y gives the same great circle, reversed. The point +X, -Y and -X, +Y, are the two rotations in the completely orthogonal plane.

If you think in terms of the duocylinder, it works like this. The circle in the wx hedrix we shall suppose is a great arrow, with a particular direction (w to x to -w to -x). The circle in the yz hedrix is completely orthogonal, and can go y to z to -y, or z to y to -z. These circles run along the middle of the faces, and it is the margin between the faces that interest us here. This shape is a 'cifford torus', or ((ii)(ii)) as the outer brackets go to zero. The clifford torus is 'equidistant' from the great arrows. When you unwrap the clifford torus, you get a rectangle. One side is marked in eg w,x,-w,-x. The other corresponds to y,z,-y,-z. As you go at even speed along one axis, you go at the same angular speed on the other one too. So even though you're travelling on a straight circle, you are (according to a fixed coordinate system), going around the clifford arrow.

The clifford torus then corresponds to a grating of lines running diagonally, for the L parallels, and a similar grating running parallel to the right diagonal, for the R parallel. A L progression that carries w to x, would carry y to z, while a R progression would carry w to x as z goes to y.

In terms of our six dimensional system, the common great arrow is a point on L,R. The left-parallels are then all the points l,R (for all l), and the right-parallels, are L,r for all r. That is, we get just two spheres crossing at a point, much as the x=a, y=b lines are two lines crossing at a point (a,b). The point opposite L,R is +L,-R or -L,+R, represents the orthogonal great circle, rotating in different directions. No other orthogonal points exist.

Symmetries: (f)

Let's plot some symmetries on this. It's quite interesting. (all numbers are decimal :) ). Quickfur is indeed correct when he supposes that various symmetries form hopf fibulations, but it is far more interesting.

The {3,3,5} has 1444 great arrows among its symmetries. 900 are of order 2, 400 of order 3, and 144 of order 5. All of these numbers are square, but we're not really interested in 38² = 1444, but the other three: 30, 20 and 12. These great arrows map onto the space as o3x5o2o3o5o, o3o5x2o3o5x, and x3o5o2x3o5o. That's ID × ID, D × D, and I × I. But these are applied to the same symmetry group.

Against the {3,3,5}, the digonal symmetry runs from a vertex, along the height of a triangle, then from edge to opposite edge of a tetrahedron, then along the opposite triangle's height, back into a vertex. This is repeated four times. The {3,3,5} has 600 faces, each with three edge-pairs, 1200 triangles with 3 heights, and 120 vertices, each with 15 opposite edges. All together, we get 1800 tetrahedra crossings, 3600 triangle crossings, and 1800 vertex crossings. A great circle crosses 4 tetrahedra, 4 vertices, and 8 triangles, represents two great arrows, so one sees there are 900 such great circles.

The triangle group runs from a vertex, down the vertical height of a tetrahedron, across the centre of the triangle, and up another tetrahedron, repeated six times. We have then n/2 great arrows have 600*4 tetrahedron crosses, 120*20 vertex crossings, and 1200*1 triangle crossings, This gives in each case 400 great circles.

The pentagon group runs along the edges and vertices: each great arrow has ten edges and ten vertices.

For example, this means, that the rings of ten dodecahedra fall into twelve groups, which exactly match the twelve vertices of the icosahedra. Likewise, you can group the 120 faces of the 5,3,3 into sets of six (the faces are connected by an edge pointing out of a vertex of the dodecahedron), and the whole set form the vertices of a dodecahedron. And if you follow from a dodecahedron, along the height of a pentagon, you will come to another height of a pentagon, which makes then a ring of 4 dodecahedra, and this can be repeated 30 times to give the icosadodecahedron.

The groups h, hr

The symmetries that correspond to the triplets x3o3o, o3x3o, and o3o3x, and x3o4o, o3x4o, and o3o4x are a little more involved.

The "simplest" group corresponds to the 32 great circles, formed by a bi-tetrahedron prism, and the resulting central inversion of this. In six dimensions, this corresponds to the vertices of the half-cube. It is represented the set of octahedra, six at a time, on the 24-chora, and a different set is represented by the 24 vertices of the 24-choron.

If you merge two of these together, you get the 64 vertices of the bi-cube prism. This found in the eight sets of six faces of the octagonny, where the arrows pass through the triangle.

There is a group represented as a digonal group (o3x3o), or a tetragonal group (x3o4o), with six members. These are the six sets of four cubes around the tesseract, or the six sets of four vertices of the 16-cell, or the six sets of four vertices and faces along a set of diagonals of the 24choron.

The set o3x4o, representing a set of 12 digonal rotations, is found in the octagonny. These run across the octagons, and across the diagonal symmetries of the truncated cubes that form the octagonny's face.

3,3,3

This does not have a hopf symmetry.

Subsymmetries

When you set the icosahedral symmetry over the octahedral symmetry, then these symmetries match: o3a5e and a3o4e. That is, the ID becomes an octahedron, and the D becomes a tetrahedron or cube. The linkage is actually more facinating than it looks.
The dream you dream alone is only a dream
the dream we dream together is reality.

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