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The setting is as follows: a 3-Dimensional-Space(3-DS) character/Spaceship has entered the 4-DS and has encountered a planet (from far away).

Now he wants to orbit around it, but the planet is a 4-DS planet, so either our character can't see 1 dimension or he can see it all well, but he can't make any sense of what he's seeing.

Imagine you are a 2-DS character and know how a circle looks like (from the side) but suddenly you enter 3-DS and you encounter things like a sphere, a cylinder, a cone or a donut/toroid and he can't make any sense of them. And that's only taking into account regular spheres. Imagine encountering a radiator or a pile of rubble.

For that he would need a flight computer to tell him an approximation of what he's seeing. The same for our 3-DS character.

Now for the problem at hand. Could the 3-DS pilot orbit and maneuver around a 4-DS planet with his spaceship?

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    $\begingroup$ There are no stable orbits in 4D space period. $\endgroup$
    – Daron
    May 20 at 17:10
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    $\begingroup$ @Daron What about for r^1 forces instead of r^-3 forces? $\endgroup$
    – wizzwizz4
    May 20 at 18:29
  • $\begingroup$ Technically we're in 4D space. Height, Width, Depth and Duration. $\endgroup$
    – Thorne
    May 21 at 5:31
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Your character won't be able to orbit around the planet, not because of their inaptitude, but because of physics

In 1920, Paul Ehrenfest showed that if there is only one time dimension and greater than three spatial dimensions, the orbit of a planet about its Sun cannot remain stable. Ehrenfest also showed that if there are an even number of spatial dimensions, then the different parts of a wave impulse will travel at different speeds.

enter image description here

There will be no planet to orbit around, no star for the planet to orbit around and so on.

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    $\begingroup$ That table is dazzling. How does an orbit, or movement of any kind, even work with zero time dimensions? $\endgroup$
    – KeizerHarm
    May 20 at 9:12
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    $\begingroup$ @4.12.22.4.18.0. You misunderstand the nature of the of the "unstable" here. It's not that a planet or whatever is unstable and would fall apart or something. It's that there are no stable orbits. The fact that there are stable orbits in 3d is directly tied to the fact that there are 3 dimensions and the gravity equation falls off like 1/R^2. In 4 dimensions, all paths through space with relation to another gravitational body either collide into that body or the fling off never to return. There are no stable circular orbits. $\endgroup$ May 20 at 19:27
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    $\begingroup$ Unstable doesn't mean you can't orbit though; it just means that you can't orbit perpetually. Even in 3+1 space, pretty much any system of more than 3 bodies is technically unstable. $\endgroup$
    – Gene
    May 20 at 20:20
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    $\begingroup$ A much bigger problem is what happens to the electron orbitals of the atoms in your body. Unlike planets, unstable (open) electron orbitals would continuously lose energy (as photons) and would quickly spiral into the nucleus turning all protons into neutrons, ending all chemistry and with it all life, $\endgroup$ May 20 at 20:30
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    $\begingroup$ @SurpriseDog "Ultrahyperbolic" and "elliptic" are two types of mathematical equations called PDEs that are used to model pretty much everything in physics. Essentially, if t is the number of spatial dimensions and d the number of time dimensions, the PDE is elliptic if t=0 or d=0 and ultrahyperbolic if t, d >=2. A lot of how we think about physics boils down to what's known as a Cauchy problem: I give you equations governing how physics works and initial conditions sampled from some initial subset of space and time, and you tell me how physical quantities will evolve. $\endgroup$ May 21 at 13:12
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Geometry

enter image description here

Yes, your pilot can navigate. Four dimensions of space can be projected onto a 2D surface, just as we see three spatial dimensions projected onto our 2-dimensional.

You can simulate this, even in current graphics engines by expanding quaternions into quintonians for all of your basic geometry. Quintonians are quaternions with one more rotational term added to them. Some folks are already looking at adding them to modern graphics engines.

If your curious, the above graph shows you the familiar coordinate axes, with a fourth spatial dimension added as "w", "x", "y", and "z". As you can see, it's typically impossible to see all four dimensions at the same time, because they just crowd one another out. As to smooth transitions to and from 3D to 4D space, I imagine compactified dimensions exist, and one of those compactified spatial dimensions "opens up" as you get closer to the body.

If your pilot knows that the 4D space is out there, then, it is possible for the pilot to use an appriately equipped simulator to train and build up experience flying in 4D. Another problem your pilot will have is that the controls for your spacecraft probably only provide control in the 3 familiar dimensions. It's possible to navigate safely in 4D by using a combination of 3D maneuvers to get to the same 4D attitude. That's the kind of thing that will require foresight, planning, and practice.

Physics

Let's reconsider this just for 4D :

The drop-off rate for forces is $F = {1 \over {r^{N-1}}}$, where r is distance and N is number of dimensions.

For a 4D universe then, the force of gravity is re-formulated as $F = {GMm \over r^3}$, where you can drop off the mass of the orbiting item m as trivially small and get the easier $F = {GM \over r^3}$

An orbit is where centripetal forces ${mv^2 \over r}$ match the force of gravity ${GM \over r^3}$, so that you neither fall or rise.

Dropping m again, $v^2 = {GM \over r^2} \rightarrow v = \sqrt{GM \over r^2}$

Escape velocity is computed differently, using potential and kinetic energy.

Orbital potential energy is the integral of force $W = {{GM} \over {3r^2}}$.

Kinetic energy ${1 \over 2}mv^2$ at escape is greater than or equal to potential energy ${{GM} \over {3r^2}}$.

  • ${1 \over 2} mv^2 = {{GM} \over {3r^2}}$
  • $v^2 = {{2GM} \over {3r^2}}$
  • $v = \sqrt{{2GM} \over {3r^2}}$

Tying the two together, escape velocity $v_e = \sqrt{{GM} \over {{2\over3}r^2}}$

Yet the orbital velocity is $\sqrt{GM \over r^2}$

Because of the $r^2$ term, the force of gravity is dissipating at a tremendous rate in the 4D world. Escape velocities are tiny. For an Earth-sized mass, escape velocity is ~3 meters per second. A Jupiter-mass planet in an Eart-sized frame would have an escape velocity of only ~80 meters per second.

These two numbers (stable orbit and escape velocity) are also very close to one another, and as "r" increases drive closer and closer to being the same number.

I think, then, that it doesn't really matter what the "orbit" is. You are probably following the trajectory of the 4D planet wherever it goes. It's gravitational effect on your ship is trivial.

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  • $\begingroup$ @4.12.22.4.18.0. Thank you for keeping the question open! I added a top section about geometry and getting around. $\endgroup$ Jun 2 at 13:11
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Strange, but visible

Your 2D example is actually perfect. Imagine this 2D character is seeing a pillar, a ball and a pencil. The moment the character starts moving through the 3D space, it'll still perceive 2D. That means some things will seem the same, or slightly different, while at other times they disappear or other things appear. The same is likely true for 3D to 4D space. There is a game in development that makes use of this concept, called Miegakure. It's quite interesting to see the landscape change based on this 4th dimension you move in.

The thing is, you can extrapolate this to a bigger scale. Imagine a 2D character in a spaceship that suddenly gets into 3D space. The spaceship will orbit if it'll have the right speed and direction. The human will probably not be able to understand, or understand enough and navigate this space. A computer however doesn't have these problem. It works with the parameters given. That is why computers can calculate with many more dimensions, as long as the parameters are given/calculated.

There are many assumptions here, like that it is actually possible to move in the new dimension. It might be that the ship can't navigate in the new dimension, being locked to the forces of nature. Like an airplane finding itself suddenly in space. However, it might be able to by chance or preparation, like a 2D thing suddenly in 3D space having surface area to move into wind and alter it's path.

The thing is, we don't know or understand how your 4D space works. That means no one can make a true prediction, except that in theory it isn't possible, as in L.Dutch his answer.

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    $\begingroup$ Oh, so when my character enters the 4-SD zone, if he's not a robot or an AI, he will die immediately because his fluids spill from his body in the 4th direction? $\endgroup$ May 20 at 13:00
  • $\begingroup$ Is Miegakure the game you are referring to? $\endgroup$
    – Brade
    May 21 at 2:31
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    $\begingroup$ @4.12.22.4.18.0. I think we can't really know. What woupd happen with a 2D character going into 3D space? Would the character be transformed, or just see the 3D results in 2D? How would such a character be transformed? Would would the transformation of 3D to 4D be much different? I think that part is Science Fiction and up to you. $\endgroup$
    – Trioxidane
    May 21 at 6:12
  • $\begingroup$ @Brade looks like it! Thanks! I'll add it to the answer. $\endgroup$
    – Trioxidane
    May 21 at 6:12

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