Stability of 4D atoms

[quickfur]

I know this has been discussed before, that 4D atoms are unstable because there are no stable orbits in 4D. At least, no orbits in a sense analogous to 3D orbits. But it just occurred to me that even in our 3D world, electrons don't really "orbit" around the nucleus. That's sometimes a useful mental image of an atom, but in reality, or at least reality as described by quantum mechanics, electrons are standing waves centered on the nucleus in constructive interference with themselves. Or nuclei, in the case of molecules.

Now standing waves are stable in any number of dimensions because they are by definition self-reinforcing. A vibrating guitar string is essentially a 1D standing wave; you can create standing water waves (2D) and electrons are 3D standing waves. So what's to stop a 4D world from having 4D electrons that are 4D standing waves around the nucleus? Now that I think of it, this wouldn't be unstable at all. Am I missing something??

Of course, such 4D atoms would be very, very different from what we're used to, because there is no equivalent of the vector cross product (at least not involving only two vectors) in 4D, so Maxwell's equations and most of physics is out the window. I don't even know where to begin to imagine what 4D physics might be like. Nevertheless, I'm starting to think that stable 4D atoms are indeed possible!

However, I don't know if we can even begin to calculate what the shapes of 4D orbitals might be like... the Schroedinger equation is difficult enough to solve for 3D already (it can't be solved analytically except only in the simplest case); 4D would be a serious pain.

What do you people think?

[pat]

I agree... there may be stable orbitals even though there aren't stable orbits.

[quickfur]

I agree... there may be stable orbitals even though there aren't stable orbits.

Any idea how one might go about calculating them or finding out their shapes, though?

[pat]

Well, I should have also said that I agree the Schrodinger equation is hard enough in two-dimensions with one particle. For even the simplest atom in 4-D, you'd need two particles and four dimensions.

I'll try playing with it sometime. But, I probably won't get to it for a few weeks.

[quickfur]

Well, I should have also said that I agree the Schrodinger equation is hard enough in two-dimensions with one particle. For even the simplest atom in 4-D, you'd need two particles and four dimensions.

Yeah, not at all trivial. Even more non-trivial is the fact that we need a working definition of angular momentum in 4D. I'm still struggling to understand how to derive 4D equivalents of rotational motion from Newton's 3 laws (which appear to be easy to generalize to 4D). You can't use the 3D cross product, so how do you define angular momentum? Plus, in 4D you can have two orthogonal rotational planes simultaneously. Can a unique angular momentum even be defined? This is not even anywhere near considering how one might quantize 4D angular momentum yet.

Perhaps a better approach is to start by considering 4D wave equations rather than to try to quantize a particle-based model. Just as 1D standing waves have 0D (point) nodes, 2D standing waves have 1D (linear or curvilinear) nodes, 3D standing waves (electrons) have 2D nodes (e.g. the plane dividing the antinodes in a p-orbital), a 4D electron wave would have 3D nodes. These 3D nodes are not confined to 3-hyperplanes; they may curve in 4D. Hmm, I'll have to think about this more.

[wendy]

Hi

I think this has been already attempted.

The s orbital is the same as our s orbital, viz 2 electrons.

The p orbital would take 8, not 6 atoms. So the noble gases would start off as 2, 12, 22 The carbon would be element 7, not element 6. This is 2+vertices of a simplex.

I am not sure what the d orbitals amount to, but most of the atoms that fall in nature are in the first three rows.

W

[quickfur]

Hi

I think this has been already attempted.

Really? That's interesting. Is there any info on this online?

The s orbital is the same as our s orbital, viz 2 electrons.

So I assume it would be a tetraspherical distribution centered on the nucleus?

The p orbital would take 8, not 6 atoms. So the noble gases would start off as 2, 12, 22 The carbon would be element 7, not element 6. This is 2+vertices of a simplex.

Interesting. What would be the shape of the p orbitals? I suppose two approximately hyperspherical lobes?

I am not sure what the d orbitals amount to, but most of the atoms that fall in nature are in the first three rows.

This depends on the relative energies of the d orbitals vs. the s orbitals in the next period. I'm not sure if the energy distribution would be exactly analogous with 3D. I suspect not, because of the inverse cube law in 4D. But I'm not sure whether the energy of the d orbital would be lower or higher relatively. If it's lower, we might see a different kind of arrangement in the periodic table, in that the d orbitals will fill up in the "normal" order just like s and p, but the f orbitals will fill up between the s and p orbitals of the next period, giving rise to an f-transition series. Or maybe the transition is between p and d. Anyhow, this would give us much larger periods, and a lot more interesting elements. :-)

Alternatively, the energy of the d orbital may be higher than it is in 3D, and may give rise to yet another different arrangement of the periodic table: the d orbital transition may take place after the p orbitals of the next period. This would have profound implications on which elements are the noble gas analogs.

Unfortunately, I suspect that the calculations required to derive all this will be formidably impenetrable... IIRC even for the 3D case, the fact that the s orbital of the next period fills up before the d orbitals is attributed to shielding from lower-level electrons, and this cannot be calculated exactly, as the equations become unmanageable unless we make a simplifying assumption that the effect of shielding is to reduce the nuclear charge felt by outer electrons. I don't remember if the d-s split can be derived this way, or more exact calculations (which means much harder analogs in 4D) are needed. I don't remember if relativistic considerations are needed in this particular derivation. (I know the behaviour of gold and mercury are related to relativistic effects of the particular electron configurations of each atom.)

Anyway, on another note, I read somewhere yesterday that somebody claimed that the 4D atom was unstable because of the Schroedinger equation. But I fail to see why... is it because of the breakdown of angular quantities in 4D, since there is no unique angular momentum in 4D? (At least, none exactly analogous to 3D angular momentum.) As far as the orbitals themselves are concerned, I think the standing wave idea is definitely valid. What is it about the Schroedinger equation that might make it not work for 4D atoms?

[jinydu]

Who claimed that 4D atoms are unstable? Do you have a link? Just curious...

Anyway, I should be in a much better position to comment on this in about a year and a half, after I've taken courses in multivariable calculus and differential equations...

[quickfur]

Who claimed that 4D atoms are unstable? Do you have a link? Just curious...

It's actually something alkaline himself read somewhere, although he doesn't mention where. Look in this page: http://www.physicsforums.com/archive/t-8643_4d_sponges.html and search for "schroedinger".

Anyway, I should be in a much better position to comment on this in about a year and a half, after I've taken courses in multivariable calculus and differential equations...

I don't know... I've taken multivariable calculus and differential equation courses, but I have to say that Schroedinger's equation to me is still a totally different kettle of fish from the sort of equations we were dealing with in the course. Now I admit that the differential equation course I took was only introductory, and we focused mainly on first-order equations only. But nevertheless, Schroedinger's equation is not something people would deal with or be able to deal with unless it were a physics-oriented course.

[jinydu]

Currently, I know enough to understand what the Schrodinger equation says in 1D, and I can solve it in cases where the potential is constant.

Some time ago, there was a thread here where a user called pat claimed to prove that planetary orbits are unstable in 4D. He made a PDF document about it, but I didn't know enough to understand it...

[quickfur]

Currently, I know enough to understand what the Schrodinger equation says in 1D, and I can solve it in cases where the potential is constant.

Cool. So maybe you're more capable of dealing with physics equations than I am. :-)

Some time ago, there was a thread here where a user called pat claimed to prove that planetary orbits are unstable in 4D. He made a PDF document about it, but I didn't know enough to understand it...

I've read about that, that there are no stable planetary orbits in 4D because there is no solution to the 2-body problem. I.e., in 4D, if you have two objects that mutually attract each other, they either fling apart after a while, or smash into each other, but they'll never stay in an equillibrium (stable orbit) like in 3D.

However, in real-life, planets aren't in "perfect" orbits, because they feel the little gravitional perturbations from nearby objects, esp. passing planets nearby. So although 3D planets have the advantage that theory is on their side, the reality is that they aren't stable either, as the perturbations over time will cause the system to lose energy and the planets will eventually spiral into the star. It might be possible to have short-lived systems in 4D where planets orbit a central star, but there the math is not on their side, so they'll likely be much more susceptible to be thrown off their pseudo-orbit by nearby perturbations, and will lose their pseudo-orbital path much faster.

OTOH, maybe in 4D orbits aren't preferred, because of their relative instability, but more complex systems of planetary motion are possible when you have more than 2 significant objects in mutual interaction. I'm not good enough at the math behind all this to derive some systems myself, though, so I don't have any idea what these systems might look like.

But regardless of all this, electrons in real atoms don't orbit the nucleus in the same way as planets orbit the sun, but instead exist as standing waves around the nucleus, so AFAICT 4D atoms should be just as stable in this respect. I don't know if this is still true wrt the Schroedinger equation in 4D, though.

[jinydu]

However, in real-life, planets aren't in "perfect" orbits, because they feel the little gravitional perturbations from nearby objects, esp. passing planets nearby. So although 3D planets have the advantage that theory is on their side, the reality is that they aren't stable either, as the perturbations over time will cause the system to lose energy and the planets will eventually spiral into the star. It might be possible to have short-lived systems in 4D where planets orbit a central star, but there the math is not on their side, so they'll likely be much more susceptible to be thrown off their pseudo-orbit by nearby perturbations, and will lose their pseudo-orbital path much faster.

Actually, if I remember correctly, there was one 4D orbit that would be stable for all eternity: a circular orbit. However, the orbit would have to be exactly circular, one nanometer of deviation, and the orbit is unstable.

However, there was one idea that maybe there could be orbits that, although unstable as t --> infinity, would still be stable for billions of years, long enough for life to develop. But apparently, this idea was not studied in detail mathematically.

Another idea was that maybe general relativity effects would be much more prominent in 4D, and general relativity may allow more stable orbits. However, it seemed that nobody knew enough general relativity to try and tackle this problem.

Of course, if all else fails, we could try tampering with Newton's Second Law...

[skydive]

hi guyz

[quickfur]

Actually, if I remember correctly, there was one 4D orbit that would be stable for all eternity: a circular orbit. However, the orbit would have to be exactly circular, one nanometer of deviation, and the orbit is unstable.

Ah, so it's not that there are no unstable orbits, just that they must be perfectly circular? So planets in 4D aren't impossible, they'd just be very rare.

However, there was one idea that maybe there could be orbits that, although unstable as t --> infinity, would still be stable for billions of years, long enough for life to develop. But apparently, this idea was not studied in detail mathematically.

Certainly, if you were "close enough" to a perfectly circular orbit, it would take a while before the orbit either collapses or diverges. Still, even in our world where elliptical orbits are possible, orbits also degenerate after a while because of perturbations from the stable state. So in 4D this would happen on a much shorter time scale.

I'd expect things like comets or meteors to be virtually non-existent... and I wonder how a multi-planet system would work, since the planets could easily knock each other off their orbital paths.

Another idea was that maybe general relativity effects would be much more prominent in 4D, and general relativity may allow more stable orbits. However, it seemed that nobody knew enough general relativity to try and tackle this problem.

Of course, if all else fails, we could try tampering with Newton's Second Law...

Relativity is a whole 'nother can of worms... before I get there, I have to try to make some sense of rotational dynamics in 4D first. :-)

[pat]

I think there's some confusion about what it means for an orbit to be stable.

For an orbit to be stable, small perturbations from the orbit cannot progress into big deviations from the orbit. In 4-D, a circular orbit doesn't degenerate (unless it is perturbed out of circular). But, it's not stable.

You could balance a golf-ball on the head of pin. It would, in theory, balance. But, it would not be stable.

[quickfur]

I think there's some confusion about what it means for an orbit to be stable.

For an orbit to be stable, small perturbations from the orbit cannot progress into big deviations from the orbit. In 4-D, a circular orbit doesn't degenerate (unless it is perturbed out of circular). But, it's not stable.
[...]

You mean in 3D, small perturbations merely results in a slightly different elliptical orbit? Whereas in 4D as soon as there is a small perturbation it would cease to be circular and so degenerate quickly because there are no elliptical orbits in 4D?

How quickly would it degenerate w.r.t. the amount of perturbation? I'm wondering if it's very sensitive so that even a very small perturbation would quickly cause a large deviation. If small perturbations take a long time before it accumulates into a large deviation, then it should still be relatively stable, IMHO. (But of course, nowhere near as stable as 3D orbits.) It'd kinda suck if somebody jumping up and down on his planet would eventually cause it to crash into the star. :-)

[jinydu]

It'd kinda suck if somebody jumping up and down on his planet would eventually cause it to crash into the star. :-)

I don't think that could happen, since if we consider the person and the planet together as a system, the force of the person pushing "down" on the planet would be an internal force; hence there could be no acceleration.

By the way, I found the orbits thread!

http://tetraspace.alkaline.org/forum/vi ... sc&start=0

[wendy]

The terminology of the PG is

PATCH noun: hedron, plural hedra, adjective hedral
CLOTH noun: hedrix, plural hedrices, adjective: hedric
MEASURE: hedrage

One cuts hedra (patches) from hedrix (cloth) and sews them together to make a polyhedron. (many patches sewen together to give closure).

Likewise one has 2D hedr 3D chor 4D ter 5D pet 6D ect 7D zett 8D yott
and downwards 1D latr and 0D teel.

glomo/hedr/ix 2d cloth in the shape of a sphere
horo/chor/ix 3d cloth in a horizon-centred space (ie E3)
bollo/ter/ix 4d cloth as pseudosphere = H4.
poly/chor/on many 3d patches sewen to closure
sur/ter/on surface + 4d + patch
hedr/age 2d measure = "area" [always L^2]
chor/age 3d measure = "volume" [always L^3]
tegma/chor/age 3d measure measured in tegum-units.

The orbitals in 4d are modeled on the 3d versions.

The s orbital is in the shape of a sphere.

The p orbital is in the shape of a dumbell, at right angles to each other. The three angles are right to each other, so in 4d, one might have 4 right to each other.

The electrons are taken as spin-up and spin-down. In four dimensions, the space of rotations is much more complex, i have enumerated it as a seven-dimensional polytope, called a "bi-glomohedric pyramid". The relative part is the five-dimensional surface formed by the pair pyramid product of the surfaces of 2 3d spheres.

When there is a balance of energy, the mode of rotation is represented by either of the extreme glomohedrices: this makes all points rotate the same way.

The whole point about 4D physics is that the model that we're using for it gives an unstable result and is probably incomplete.

In 3d, one does get stable elliptical orbits, which allows one to shift the orbit of planets without crashing into the sun. But the unassisted radiant law (ie in 4d the inverse-cube law) does not make for this, probably because the ellipse sweeps a hedrage (ie a 2-content), where the required element is a chorage (3-content).

So what we're essentially doing now with 4d is what we were doing with 3d in the renansance, except we have no means to do experiments.


Wendy

[quickfur]

It'd kinda suck if somebody jumping up and down on his planet would eventually cause it to crash into the star. :-)

I don't think that could happen, since if we consider the person and the planet together as a system, the force of the person pushing "down" on the planet would be an internal force; hence there could be no acceleration.

That's true. But that would mean anything falling onto the planet, like meteors and stuff, will drastically affect its orbit, as will doing things like sending rockets out of orbit. In 3D, this just means the earth shifts (ever so slightly) to a slightly different elliptical orbit; but in 4D, based on pat's derivation, this introduces a constant displacement towards/away from the central star, which accumulates every year. At orbital rates comparable with Earth's, this means that in the 4 billion years of Earth's history, a tiny perturbation of 1 meter on the equivalent 4D planet's orbital radius would accumulate to difference of 1 million km. This would imply significant changes in climate, average temperature, etc..

By the way, I found the orbits thread!

http://tetraspace.alkaline.org/forum/vi ... sc&start=0

Cool. I glanced over pat's second PDF which describes the 4 types of "orbits". Now I'm starting to understand why orbits in 4D are unstable... the extra power in the inverse cube law introduces a constant factor into the differential equation, which translates to a constant displacement towards/away from the central star every revolution. It makes sense, mathematically.

Has anybody attempted the 3-body problem yet? :-) By the looks of the PDF, the mathematics itself isn't that hard since we're mainly dealing with vector equations rather than with individual coordinates. The only difference is the inverse cube law, really.

[quickfur]

[...]The orbitals in 4d are modeled on the 3d versions.

The s orbital is in the shape of a sphere.

The p orbital is in the shape of a dumbell, at right angles to each other. The three angles are right to each other, so in 4d, one might have 4 right to each other.

Is this just a direct generalization from 3D atomic orbitals, or do 4D wave interference patterns show these shapes?

I guess we can't say for sure unless we can solve the Schroedinger equation in 4D.

The electrons are taken as spin-up and spin-down. In four dimensions, the space of rotations is much more complex, i have enumerated it as a seven-dimensional polytope, called a "bi-glomohedric pyramid". The relative part is the five-dimensional surface formed by the pair pyramid product of the surfaces of 2 3d spheres.

When there is a balance of energy, the mode of rotation is represented by either of the extreme glomohedrices: this makes all points rotate the same way.

Strictly speaking, electron spin is a misnomer: the electron isn't spinning about a spatial axis (in the 3D case). I.e., it's not a rotation. Physicists interpret electron spin as the spinning of the phase of the electron wave, rather than a spatial rotation. It cannot be a rotation because that would require the electron to be under constant acceleration, and Maxwell's equations would predict that it is continually radiating magnetic energy, and so cannot exist in stable atomic orbitals.

The whole point about 4D physics is that the model that we're using for it gives an unstable result and is probably incomplete.

Well, I agree that most of physics would have to be suitably adapted for 4D; blindly generalizing theorems to 4D will give us contradictions. For one thing, we have to re-derive how angular momentum works, because there is no cross product in 4D involving only 2 vectors.

In 3d, one does get stable elliptical orbits, which allows one to shift the orbit of planets without crashing into the sun. But the unassisted radiant law (ie in 4d the inverse-cube law) does not make for this, probably because the ellipse sweeps a hedrage (ie a 2-content), where the required element is a chorage (3-content).

That's a nice way of looking at it. :-)

So what we're essentially doing now with 4d is what we were doing with 3d in the renansance, except we have no means to do experiments.

Yeah, we can only rely on the math here. :-)

[jinydu]

In 3d, one does get stable elliptical orbits, which allows one to shift the orbit of planets without crashing into the sun. But the unassisted radiant law (ie in 4d the inverse-cube law) does not make for this, probably because the ellipse sweeps a hedrage (ie a 2-content), where the required element is a chorage (3-content).

Unfortunately, I don't really understand the mathematics of your posts (hopefully, I will after a few more courses at university). But are you saying here that some kind of stable orbit is possible?

[quickfur]

In 3d, one does get stable elliptical orbits, which allows one to shift the orbit of planets without crashing into the sun. But the unassisted radiant law (ie in 4d the inverse-cube law) does not make for this, probably because the ellipse sweeps a hedrage (ie a 2-content), where the required element is a chorage (3-content).

Unfortunately, I don't really understand the mathematics of your posts (hopefully, I will after a few more courses at university). But are you saying here that some kind of stable orbit is possible?

She is stating a possible interpretation of what pat derived, that in 3D, an orbit can be shifted slightly and still remain elliptical, i.e., not on a collision path with the star or flying off out of orbit, but in 4D this is no longer true, probably because in 3D, sweeping out a 2D area ("hedrage") is sufficient to maintain orbital stability, but in 4D, you need to sweep out a 3D area ("chorage") to attain to stability, which you can't since an orbit is only in a plane. This is probably related to the fact that in 3D, the strength of gravity is an inverse square law whereas in 4D, it is an inverse cube law.

[jinydu]

Would it be possible for a 4D orbit to not stay within a plane, perhaps by modifying some other physical law(s)?

[quickfur]

Would it be possible for a 4D orbit to not stay within a plane, perhaps by modifying some other physical law(s)?

Depends on how far you're willing to go in modifying physical laws. Well, it's not like you're really modifying them per se, since we only know laws of physics in 3D, so it's more a matter of which laws we decide to generalize to 4D and how we go about doing it. It's easy to set down a law that says a smaller body will always orbit a larger body in a stable orbit, but that's cheating and doesn't really give us any interesting insights into 4D. We're more interested in something that resembles 3D but is suitably modified to give rise to stable systems in 4D.

I don't know if an orbit that doesn't stay in a plane would help very much... the path of a small body around a larger one is necessarily 1-dimensional, and when there are only 2 bodies in the system, there really isn't any non-arbitrary way of making them move outside a planar configuration. I mean, you can say that while in orbit the smallar body also bobs up and down in a perpendicular direction like a sine wave, but you'd have a hard time explaining how it does that without making arbitrary rules about how things move in that state, and whether that even helps make the orbit stable (it probably doesn't).

Now, one way to stabilize an orbit might be to have a 3-body system where the 2 smaller bodies are in harmonic orbital motion (sorta like some of the shepherd satellites of Jupiter and Saturn). I don't know if there is any 3-body configuration in 4D which may actually turn out to be stable; pat or somebody with more grasp of orbital mechanics than I do should step forward and do the 3-body calculation. This is probably quite hard, since even in 3D, the 3-body problem is non-trivial. In 4D, you have many more possible relative motions of the 2 smaller bodies, so it may be even more difficult to work out. Perhaps.

I'm still trying to work out a usable 4D rotational dynamics model.

Another idea I have is to calculate whether ring systems are stable in 4D. This is a longshot, because in 3D, rings aren't stable (Saturn's rings are ephemeral, they are much shorter-lived than Saturn itself). But who knows, maybe with the extra r in the equation they might turn out to be more stable in 4D than in 3D. (Any takers? :-P) But then again, perhaps circular orbits are OK after all... Saturn's rings are ephemeral but in our perspective they are still pretty much "permanent" features. Perhaps 4D planets are unstable but last long enough you could live on them for more than several lifetimes.

[pat]

Would it be possible for a 4D orbit to not stay within a plane, perhaps by modifying some other physical law(s)?

I don't think so. You've got your planet and your sun. You've got one vector for how the planet is moving and one for the line of force from the planet to the sun. Without some other force to lift the planet out of the plane of these two vectors, there's no way to get out of that plane.... unless you want to modify Newton's First Law. But, if you do that, you're on very, very thin ice.

Of course, that said... consider a moon orbiting a planet orbiting the sun. The force of the sun acting on the moon would serve pull the moon around a bit. In 3-D, this ends up stabilizing so that the moon's orbit of the planet is roughly in the same plane as the planet's orbit of the sun. In 4-D, it wouldn't really stabilize, but it would probably still end up in the same plane.[/i]

[jinydu]

This reminds me of something else:

On a physics homework assignment, I had to look at a three body system where all three masses were equal, they all moved in circles, and at any instant, the distance between any two masses is the same as the distance between any two other masses. It didn't take too long for me that at any instant, the masses are on the vertices of an equilateral triangle, which rotates about its center, the center of mass of the system, at constant speed. There are essentially three variables: the mass of each body, the distance between the bodies and the speeds of the bodies. Given any two quantities, I know how to solve for the third.

However, to me, this seemed like a question that was begging a generalization. How could I get four mutually equidistant equal masses moving in circles around their center of mass? I think that the masses would have to lie at the vertices of a regular tetrahedron, at any one instant. Of course, the masses are moving in circles, so I would expect the tetrahedron to be rotating, but I don't know what the rotation would be. In the (3 masses, 2 dimensions) case, it seems like there would only be two possibilities, counterclockwise and clockwise. Also, I haven't been able to work everything out completely:
http://www.sosmath.com/CBB/viewtopic.php?t=14451

If I could get the tetrahedron to work, the next logical step would be to examine what happens with 5 mutually equidistant equal masses. I don't think its possible to have 5 such points in 3D space; but it should be possible in 4D (although I haven't been able to prove either 3D impossibility or 4D possibility). Presumably then, the masses would be on the vertices of some regular 4D polytope. But once we've thrown in the 4th dimension, its a good bet that we'll have to start using an inverse-cube law for gravity. What happens then?

[quickfur]

[...]
However, to me, this seemed like a question that was begging a generalization. How could I get four mutually equidistant equal masses moving in circles around their center of mass? I think that the masses would have to lie at the vertices of a regular tetrahedron, at any one instant. Of course, the masses are moving in circles, so I would expect the tetrahedron to be rotating, but I don't know what the rotation would be.

Hmm. Are you sure such a system is actually possible, and stable? I can't picture any possible rotation of the tetrahedron that would work, at least not in 3D. No matter what plane it is rotating in, at least one of the masses will have an unbalanced force inwards, which will lead to the collapse of the system.

In the (3 masses, 2 dimensions) case, it seems like there would only be two possibilities, counterclockwise and clockwise. Also, I haven't been able to work everything out completely:
http://www.sosmath.com/CBB/viewtopic.php?t=14451
[...]

It seems that you are merely describing the movement of the vertices of a rotating tetrahedron. How do you set it up so that there is an outward force on the mass(es) outside the plane of rotation so that it doesn't fall inwards?

[PWrong]

Another idea I have is to calculate whether ring systems are stable in 4D. This is a longshot, because in 3D, rings aren't stable (Saturn's rings are ephemeral, they are much shorter-lived than Saturn itself). But who knows, maybe with the extra r in the equation they might turn out to be more stable in 4D than in 3D. (Any takers? ) But then again, perhaps circular orbits are OK after all... Saturn's rings are ephemeral but in our perspective they are still pretty much "permanent" features. Perhaps 4D planets are unstable but last long enough you could live on them for more than several lifetimes.

A ring might be interesting. Has anyone read Ringworld, by Larry Niven?
You have a huge ring around the sun, with the same radius as the distance from the earth to the sun. You can calculate the orbit by dividing it into blocks of width dx.

If a roughly circular ringworld occured in 4D, the problems with gravity would be overcome by tension in the ring.

For instance, if each individual part tries to fall towards the sun, the ring gets compressed against itself. Otherwise, the ring pulls away from the sun, and gets tense.

If it turns out to be unstable anyway, you adjust the laws of tension in 4D.
I'll try to work something out tomorrow.

[quickfur]

[...]
If a roughly circular ringworld occured in 4D, the problems with gravity would be overcome by tension in the ring.

For instance, if each individual part tries to fall towards the sun, the ring gets compressed against itself. Otherwise, the ring pulls away from the sun, and gets tense.
[...]

That suggests an even more interesting idea to me: what if you have a spherical shell instead of a ring? (As in, 3D spherical, not glomar.) So the "ring" ("sphring"? :-)) would actually occupy a 3D volume around the central star, not just 2D area. Seems like that might be what we need to balance out the extra r term caused by the inverse cube law.

[jinydu]

How do you set it up so that there is an outward force on the mass(es) outside the plane of rotation so that it doesn't fall inwards?

An outward force on the masses would not be necessary, just as no outward force is necessary to prevent the Earth from falling into the Sun.

However, in practice, I know that such systems would be extremely improbable. All the masses would have to have exactly the right mass, and be in exactly the right position at the right time, with the right velocity. It is almost certain that no such system exists in the Universe.