## 4D Vorticies

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.

### 4D Vorticies

In 4D one may have spherical vorticies. In water and air they would be less dense than their surroundings and hence tend to rise like bubbles.

A 4D superfluid could have spherical rotational vorticies. It appears that there is no tight packing of spheres in 4D, so unlike 3D the vortices could more easily move around relative to one another.

How about the magnetic vorticies that appear in Type 1.5 and 2 superconductors. In a gravitational field the 4D vorticies would rise. When they get to the surface would they pop like champagne bubbles?

On the other hand I would think that the vortex that forms when you drain the bathtub would be cylindrical like ours. It's purpose is to move water. A spherical vortex doesn't do that. A cylindrical vortex has a 1D core, a spherical one has a 0D core. Tornadoes, same. Storms can't be 4D vorticies because the atmosphere is too thin.

How about the magnetic vorticies such as the "flux tubes" that the Sun is full of. I would guess there would be both. The 4D vorticies wouldn't matter much because they aren't connected to anything. They would rise and dissipate.
PatrickPowers
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### Re: 4D Vorticies

Why would 4D vortices be spherical? If there is rotation involved, then it cannot be spherical, because of the so-called hairy ball theorem.
quickfur
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### Re: 4D Vorticies

quickfur wrote:Why would 4D vortices be spherical? If there is rotation involved, then it cannot be spherical, because of the so-called hairy ball theorem.

It only applies in odd dimensional spaces like ours.
PatrickPowers
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### Re: 4D Vorticies

You missed my point.

According to Wikipedia, a vortex is a "region in which the flow revolves around an axis line". Now obviously, in 4D there is no axis line but there is a stationary plane, so you have fluid revolving around a plane. This produces a cubindrical vortex. You may also have a second revolution in the stationary plane, orthogonal to the first revolution, so you could have a double-vortex of sorts. In that case, you'd have a duocylindrical vortex.

There is no case in which you might obtain a spherical vortex.
quickfur
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### Re: 4D Vorticies

quickfur wrote:You missed my point.

According to Wikipedia, a vortex is a "region in which the flow revolves around an axis line". Now obviously, in 4D there is no axis line but there is a stationary plane, so you have fluid revolving around a plane. This produces a cubindrical vortex. You may also have a second revolution in the stationary plane, orthogonal to the first revolution, so you could have a double-vortex of sorts. In that case, you'd have a duocylindrical vortex.

There is no case in which you might obtain a spherical vortex.

In math I just accept whatever definition the author is using. I say that in an even dimensional universe there can be a vortex that revolves around a point. Scientists who work with 2D dimensional surfaces use this definition.
PatrickPowers
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### Re: 4D Vorticies

PatrickPowers wrote:
quickfur wrote:You missed my point.

According to Wikipedia, a vortex is a "region in which the flow revolves around an axis line". Now obviously, in 4D there is no axis line but there is a stationary plane, so you have fluid revolving around a plane. This produces a cubindrical vortex. You may also have a second revolution in the stationary plane, orthogonal to the first revolution, so you could have a double-vortex of sorts. In that case, you'd have a duocylindrical vortex.

There is no case in which you might obtain a spherical vortex.

In math I just accept whatever definition the author is using. I say that in an even dimensional universe there can be a vortex that revolves around a point. Scientists who work with 2D dimensional surfaces use this definition.

Well, yes. But that's not spherical. In 4D I can indeed conceive of a vortex undergoing isoclinic rotation. But that occupies the volume of a glome; you can't have a spherical vortex. (Unless by "sphere" you mean a general n-dimensional sphere, not just a 3D sphere.)

In a 4D bathtub, if we assume a spherical drain hole, you'd run into the hairy ball theorem: there is no uniformly non-diminishing vector field over the spherical cross-section of the descending water, so any vortex that might form cannot be spherically-symmetrical. I'm not sure exactly what will happen (3D fluid dynamics is complex enough, I can't imagine solving fluid equations in 4D ), but you won't get a spherical vortex out of it. Perhaps the excess angular momentum will create a cubindrical vortex, but the spherical confinement of the drain hole would force it to have two poles where angular momentum vanishes; I'd imagine the water flow at these poles would be unconstrained by the rotation of the rest of the vortex, so it'd drain out faster. Not 100% sure how this would affect the vortex and its stability. I wonder if this means a draining 4D bathtub would always have two antipodal points where water flows out the fastest? It's pretty mind-bending to try to imagine what might happen.
quickfur
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### Re: 4D Vorticies

The field of water momentum vectors is not constrained to the 2-sphere that is the opening of the drain. They might all be perpendicular to that surface, which is a solution with no poles.

As the vectors get close enough to being tangential to the surface then air sucks down and penetrates the surface. Then as far as water momentum vectors are concerned that surface has a hole in it so it is no longer a 2-sphere.

I once did an experiment with a drain vortex. The water quite forcefully insisted on a certain direction of rotation. I suppose the configuration of drain pipes favored that direction.

I don't know how a bathtub would drain in 4D. My guess is that it is pretty much the same as 3D but the vortex is unstable and snakes around. There might numerous 2-planes in which the angular velocity is maximal. The vortex sucks out the angular momentum in that 2-plane, but then that plane no longer has maximal angular velocity so the vortex will tend to move to some other plane that does. But its an elastic rotation instead of a rigid one. That complicates things too much for me.
PatrickPowers
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### Re: 4D Vorticies

PatrickPowers wrote:The field of water momentum vectors is not constrained to the 2-sphere that is the opening of the drain. They might all be perpendicular to that surface, which is a solution with no poles.

Hmm. I never thought of that! That's a neat way of looking at it. If there's any rotation, though, assuming the water is actually draining and not staying in place, the vector field must intersect the 2-ball in some way. What would that look like? No idea, but it's something interesting to explore.

[...]
I don't know how a bathtub would drain in 4D. My guess is that it is pretty much the same as 3D but the vortex is unstable and snakes around. There might numerous 2-planes in which the angular velocity is maximal. The vortex sucks out the angular momentum in that 2-plane, but then that plane no longer has maximal angular velocity so the vortex will tend to move to some other plane that does. But its an elastic rotation instead of a rigid one. That complicates things too much for me.

I wonder if there's a way to solve 4D fluid dynamics for this specific case. Might come up with some very interesting flow patterns!
quickfur
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### Re: 4D Vorticies

I'm thinking of vortices and other non-zero curl fluid flows as having an imaginary "axis." nD fluid will flow about a n-2D axis. In two dimensions the axis is either a point or a point pair. The point pair is like a 2D smoke ring. In three dimensions the axis is either a line or something homeomorphic to a circle (knot axes are unstable and quickly collapse into circular axes). In four dimensions the axis could be a plane or a tube, but I think it could also be like a sphere, which I think is what Patrick meant. I find it easier to only imagine the realm that the axis resides in and then to color the realm on either side of the axis to visualize the 4D fluid flow. For example with vortex rings I would imagine a plane with a circular axis within. I'll color the plane blue inside the circle for going "down" into the screen, and red outside the circle for going "up." Hence in four dimensions it's easy to imagine a planar axis splitting a realm into red and blue sides. I imagine a sphere in the realm, with the inside blue and the outside red. I suppose that the axis could also be torus-shaped, but I don't know how stable this vortex would be, I assume that it would be more and more unstable with increasing genus. If a torus axis vortex is stable, a Klein bottle axis vortex might also be stable, as might a projective plane axis vortex. I don't know if 2-knots are possible as vortex axes.
개구리
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### Re: 4D Vorticies

This response is for a general audience, so kindly do not be offended if I'm saying things you already know.

Here in plain old 3D toroidal vorticies are not unusual. There is the smoke ring. Scientists have made smoke ring cannon. Dolphins make toroidal vorticies of water with a core of air and use them for toys. I have also seen a video of such a toroidal vortex in the shape of a trefoil knot. It wasn't very stable but it did exist for a second or two.

There are also electromagnetic toroidal vorticies. Such are predicted to be rife in the superconductive cores of neutron stars. These vorticies have a twist which is topologically preserved so they should be fairly stable, though no one knows how much.

In some Type 1.5 superconductors it has been calculated that polygonal braided quantum toroidal vorticies will exist. That is, a "square" vortex has four subvorticies each with energy which is a fraction of a quantum. The four subvorticies are wrapped in a larger vortex which also has energy which is a fraction of a quantum. The sum of the energies of the five vorticies is an integer. It is then difficult for a subvortex to break down and release its energy, as the system cannot release a fraction of a quantum of energy. Such vorticies should be stable.

I learned all this from the work of Egor Babaev, who has had a great many articles published in Nature and Physical Review.
https://en.wikipedia.org/wiki/Egor_Babaev Among other things, he led experimentalists to prove that the Type 1.5 superconductor existed here on Earth.

Toroidal vorticies of light are theoretically possible. It has been claimed that they have been produced.

In 4D I would expect that toroidal vorticies would be more stable than in 3D. Going back to those 3D vorticies made of water and air, there is compression and expansion going on during the rotation. Indeed I find it surprising that such vorticies can persist for many seconds, but they do. In 4D there is no such compression and expansion, so it is a more favorable environment for such things.

In 4D it is normal to have two planes of rotation whose intersection is a point. The rotation in each plane is independent of the other. It seems that in this case one would need to think of two planar axies. In 4D a 2-plane does not partition the space, so you can't say whether a point outside the 2-plane is on one side or the other.
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### Re: 4D Vorticies

Patrick,

By "toroidal axis" I am referring to the axis of the fluid flow, not the general shape of the fastest flowing fluid itself. In 3D, assuming an infinite fluid, the axis of a regular vortex is a line. A smoke ring's axis is a circle. Vortices are cylindrical because for each point along the linear axis, there is a circle of flowing fluid. For every point on the circular smoke ring's axis, there is a circle of flowing fluid, resulting in a toroidal shape. For simplicity I omit the fluid flow entirely and only think of the vortex axis. When the axis itself is toroidal (only possible in 4D) then I believe the vortex would be ditorus shaped.

PatrickPowers wrote:There are also electromagnetic toroidal vorticies. Such are predicted to be rife in the superconductive cores of neutron stars. These vorticies have a twist which is topologically preserved so they should be fairly stable, though no one knows how much.

...Toroidal vorticies of light are theoretically possible. It has been claimed that they have been produced.

What are electromagnetic vortices? I've heard of knotted up magnetic field lines but that just sounds like a sustained trefoil smoke ring to me.
What is a light vortex? Is that related to circular polarization?

PatrickPowers wrote:In 4D a 2-plane does not partition the space, so you can't say whether a point outside the 2-plane is on one side or the other.

An infinite or closed 2-surface doesn't bisect a flune but it does bisect a realm. The realm is the middle of the vortex. When you look at a smoke ring face-on, you see its circular axis in a plane, with fluid flowing along the direction of your vision into (or out of) the inside of the circle and out of (or into) the exterior of the circular axis. A tetronian could look at a vortex face-on to see some 2-surface axis bisecting a realm, with the fluid flowing in on one side and out on the other, parallel to her line of vision. The 2-surface exists as an axis in a flune the same way a 1D line or ring exists as an axis in a realm. This is what I mean by "inside" or "outside" an axis, a dimension is subtracted. A planar axis in a flune has fluid flowing around each of its points similar to how a linear axis in a realm has fluid flowing around its length (producing a smoke column).

I don't think that 4D bathtubs would drain double-rotation fashion along a pair of planar axes, because one of the planes would have to be perpendicular to gravity's direction, meaning the flow around that plane would require going up and down repeatedly in the direction of gravity. I'll need to think about it more, but perhaps they'd rather tend to drain around tubular axes?
개구리
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### Re: 4D Vorticies

개구리 wrote:
What are electromagnetic vortices? I've heard of knotted up magnetic field lines but that just sounds like a sustained trefoil smoke ring to me.
What is a light vortex? Is that related to circular polarization?

Charged particles moving in a magnetic field produce a magnetic field. This feeds back on itself and creates a vortex. Minuscule electromagnetic vorticies occur in Type 1.5 and II superconductors, big ones known as "flux tubes" in plasmas like the Sun. Sunspots are where a gargantuan flux tube passes through the surface of the Sun.

There are solutions to Maxwell's equations that are vorticies. That's all I know about vorticies of light. It's just something I read about in scientific journals and find amazing. There is no apparent use so little effort goes toward study. I believe is different than circular polarization in that such vorticies must be toroidal, but I don't have much confidence holding forth on this subject.
개구리 wrote:
I don't think that 4D bathtubs would drain double-rotation fashion along a pair of planar axes, because one of the planes would have to be perpendicular to gravity's direction, meaning the flow around that plane would require going up and down repeatedly in the direction of gravity. I'll need to think about it more, but perhaps they'd rather tend to drain around tubular axes?

Right, it would have to be a cylindrical vortex similar to what we see. The vertical dimension is for the draining of the water, so there are only three left. Same with tornadoes and typhoons. To have a double rotation vortex that translates matter (non-zero net momentum) you need five spatial dimensions.

It seems to me that that in 4D turbulence and eddies in general would tend to be ellipsoidal instead of cylindrical. The rule of thumb I'm using is "if it can move in some way then it will."
PatrickPowers
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### Re: 4D Vorticies

I thought about 4D bathtub drains a bit more. So far, we've been assuming that the drain hole is roughly in the shape of a sphere (i.e., the tube underneath is an extended spherinder). If the tube has a different shape, e.g., a cubinder, the drain hole would correspondingly be cylindrical, and any vortex that forms, I'd assume, would be symmetrical along the axis of the cylinder. The resulting vortex would be roughly in the shape traced out by tracing a cylinder along the 4th direction while gradually shrinking it. Basically, it'd be a prism of a 3D vortex. Such a vortex would be stable, and would have a fixed plane of rotation.

Other shapes are possible. If the drain hole is in the shape of an ellipsoid (the tube being an ellipsoinder -- the extrusion of an ellipsoid) with two equal axis (the third being unequal), I'd assume the resulting vortex would prefer an orientation along the unequal axis, as it would minimize friction if the fluid rotates in the perpendicular plane. I'm not sure what exactly would be the effect of the ends of the ellipsoid gradually converging to a point (as opposed to terminating in circular lids as in the cubindrical case), but I'd expect the vortex would be more-or-less just like a 3D vortex, extruded and shrunk accordingly according to the curvature of the ellipsoid. It may wiggle a little, but I'd expect it would more or less prefer a fixed orientation along the preferred plane of rotation, thus minimizing friction.

Now imagine if we shrink/stretch the unequal axis of the ellipsoidal drain hole so that it becomes spherical. I'd imagine the behaviour wouldn't be much different, except now there's spherical symmetry, so there's no longer a preferred plane of rotation. So the vortex would likely switch its orientation frequently, as small deviations in fluid motion perturb the orientation of the plane of least friction.

Furthermore, upon further consideration I think what I said about the poles of the sphere being points of fastest flow was wrong: on the contrary, assuming that the angular momentum of descending fluid is more-or-less evenly distributed, this would mean that closer to the poles of the sphere the fluid would rotate faster; so a most of the velocity of the fluid would be in the plane of rotation, i.e., the poles would be where downward flow is slowest. Meaning, the primary region of fluid descent would be around the equatorial region where rotation is slowest. So the rate of draining would be highest around the equatorial region, and slowest around the poles. I haven't decided yet what would happen exactly at the poles. Perhaps the fluid would be practically stationary? Or perhaps it would induce a jet of non-rotating fluid shooting into the drain? Not sure at the moment.
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