That is what the title should actually be now, but I didn't realize that precessing--but otherwise "stable"--orbits would work in the even dimensions until after I had created the topic (see my 2

^{nd}reply for details).

/correction.

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I already talked about this idea in the post by Higher_Order, 4D Orbits, but, after looking around on here and elsewhere to see if anyone else had already thought about it, I found nothing, so I thought that this should be its own post. To be clear, I am not absolutely convinced that this is how things definitely work. I do not have a mathematical proof or anything like that, but things do appear to fit together extremely well.

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Here is the ELI5: the quality of the "oomph" of motion changes with the number of orthogonal rotations allowed by a space, but, when that space is curved toward a point within it, the quality of the resulting "oopmh" of apparent attraction changes with the number of dimensions; and these qualities match to allow stable orbits

Okay, what does that mean?

First, the quality of the "oomph" of apparent attraction created by curved space is referring to gravity and means that how its strength changes with distance depends on the number of dimensions. Specifically, gravity in 3D changes with the square of distance, but gravity in 23D changes with the 22

^{nd}power of distance. Generally, gravity in ND changes with the (N-1)

^{th}power of distance. This occurs because of spherical symmetry itself.

Next, the quality of the "oomph" of motion is referring to kinetic energy and means that how its value changes with velocity depends on the number of orthogonal rotations. Specifically, kinetic energy in 2D and 3D increases with the square of velocity, but kinetic energy in 22D and 23D increases with the 22

^{nd}power of velocity. Generally, kinetic energy in (2n)D and (2n+1)D increases with the (2n)

^{th}power of velocity (there are no odd powers of velocity). This occurs because of spherical symmetry, too, but in a much less obvious way.

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First, it is important to understand why exactly kinetic energy scales with the square of velocity in 3D, which is itself not obvious. I recommend reading this PhysicsForums comment on the thread, "The final explanation to why kinetic energy is proportional to velocity squared", where the OP (who also wrote that comment) asks for some sort of deeper, intuitive answer. Maths story short, the "oomph" that is carried away by a fragment after splitting an object into 2 pieces moving in opposite directions logically must work out to depend on the square of velocity of that fragment. In other words, the cost itself of creating motion increases with the motion (i.e. energy per velocity is proportional to velocity, which rearranges to energy being proportional to velocity

^{2}). I wrote some more about this in a comment in the post by Higher_Order, but that is the general idea.

^{2}/ time

^{2}, or more simply, mass * velocity

^{2}. For an orbitting planet, the reason that its orbital angular momentum remains constant (assuming no exchanges with its spin orbital momentum) whereas its orbital kinetic energy can vary according to its orbital velocity is that the angle covered by the planet varies in step with the orbital velocity. Another maths story short, this works out as a constant quanity with the same units, angular momentum.

Then, putting these points together in the context of general rotations leads to the kinetic energies of higher dimensions being qualitatively different from the kinetic energy of 3D. For example, in 4D, there are 2 planes of rotation. An object can spin around its center of mass in 2 orthogonal ways at 2 independent rates, which means that it will have 2 different spin angular momenta. Projecting the object onto either plane of rotation will show the points within it tracing circles; however, it is critical to understand that these points have only 1 actual path through 4D space which is confined to a 2-torus. Another key insight here is that even when the 4D object is "not rotating" in one of the planes of rotation, all of the points in that plane are still "moving" because they are rotating according to the other plane of rotation. This is just like how in 3D every point along the "axis of rotation" still rotates even though they are just points that do not translationally move anywhere.

Now, this can be translated from spins into orbits by looking at an orbital system itself as a object that is spinning. When a solid object rotates, it usually does not fly apart because of intermolecular forces. However, when an orbital system rotates, it does not fly apart because space itself is curved, meaning that the orbitting body is still following a "straight" path. This difference in underlying reason also means that the shape of the path does not have to remain circular. In fact, the eccentricity of an orbit can be thought of as a ratio of the components of the orbital velocity that are aligned with the semi-major and -minor axes of the ellipse. When the ratio is near 1, the orbit will be nearly circular; and when the ratio is farther from 1, the orbit is more elliptical. In both cases, the orbitting body is still following its own version of "straight" path. In other words, the orbital velocity skews the curvature, creating eccentricity.

Finally, putting these points together in the context of general rotations leads to the orbital systems of higher dimensions being qualitatively different from the simple orbits of 3D. Having only a single plane of rotation oversimplifies things and hides how to generalize angular momentum and how this would actually balance the qualitatively different gravities of higher dimensions when the dimension is odd. For example, assuming that angular momentum remains as it is in 3D for 4D leads to the conclusion that orbital paths still have to stay in a 2D plane where a steeper curvature is no longer compatible because orbital velocity cannot change quickly enough. However, the modified "double-angular" momentum of a doubly rotational orbit overcompensates for this steepness, and orbits are still unstable. But in 5D, curvature changes with the 4

^{th}power of distance, which is a steep enough curvature to perfectly match 2 orthogonal angular momenta guiding the orbital path. This results in a 3D region around the center of mass within which the orbital path can wind its way; it would have been a 2D region, but the 2-torus is defined by the elliptical orbits in either plane of rotation. In 6D, "triply-angular" momentum again overcompensates for the increased curvature steepness, but, in 7D, the curvature steepness catches up perfectly again. This results in a 4D region around the center of mass within which the orbital path can wind its windy way; it would have been a 3D region, but the 3-torus is defined by the elliptical orbits in each plane of rotation.

In fact, a fascinating way of thinking about this process is that the addition of planes of rotation modify the quality of motion, specifically, they add another time derivative of position into the mix, which can be visualized as vectors, originally pointing at the center of mass, being "peeled away" by the orthogonal motion of the new rotational plane, leaving a new time derivative of position to guide the orbital path; also, the number of decoupled vectors plus one determines the dimension of the region around the center of mass within which the orbital path remains (this becomes an increasingly "thinner" slice of the ambient space since it is scaling with the number of rotational planes). In 1D, there are no such things as orbits, and gravitational sources necessarily induce velocity vectors pointing at the center of mass. In 2D, the ability of a gravitational system to rotate means that it does not necessarily induce a velocity vector pointing at the center of mass, however, the acceleration vector is still stuck, and, in 3D, this becomes a balanced setup. In 4D, the ability to doubly rotate means that the change of velocity induced by one plane is itself changed by the other plane, pushing the acceleration vector off of the center of mass, but leaving the jerk vector, and, in 5D, this becomes a balanced setup. In 6D, the ability to triply rotate means that the change of acceleration induced by the second plane is itself changed by a third plane, pushing the jerk vector off of the center of mass, too, but leaving the snap vector, and, in 7D, this becomes a balanced setup. 8D pushes off the snap vector, leaving the crackle vector, 9D stabilizes it all, 10D pushes off the crackle vector, leaving the pop vector, 11D stabilizes it all, and so on (I think the next vector for 12D and 13D should be called zip).

Speaking of 13D, I want to live in a 13D space! I could orthogonally see the 27 exotic surfaces of the 8-ball, the 1 exotic surface of the 9-ball, the 7 exotic surfaces of the 10-ball, the 5 exotic surfaces of the 11-ball, and the 991 exotic surfaces of the 12-ball (there are no exotic surfaces for the 13-ball, but there are 16255 for the 16-ball!). I could also orthogonally see not only how to optimally pack n-balls up through 12-balls, but how the packing, which is perfect for 2-balls, grows looser and looser until 8-balls pack perfectly again, only to start loosening again (before returning to perfection for 24-balls, possibly for the last time). I could also orthogonally see the weird patterns of ways to "wrap" the surfaces of balls onto the surfaces of lower dimensional balls. And much more! 3D is so boring... Although, doubling my velocity would increase my kinetic energy by a factor of 4096, so... hmm. Maybe having 22 different directions to swerve to miss an oncoming car crash makes up for the sheer destructive "oomph" of not missing...

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Anyway! The reason I even thought about all of this is a comment by Quickfur in that 4D Orbits thread.

quickfur wrote:Since momentum varies with the square of the velocity, when gravity also obeys an inverse square law this allows certain subsets of trajectories to have momentum and force balance each other out in an equillibrium.

And then PatrickPowers made me think even more deeply.

PatrickPowers wrote:It's because the acceleration vector is always pointing at the center of mass, so the orbit is confined to a 2D plane defined by that acceleration vector and the velocity vector of the planet.

So... if this idea catches on and revolutionizes higher dimensional analysis, that is how it started. But, if it turns out to be nothing, it is all their fault.