Are three dimensions trivial?

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.

Are three dimensions trivial?

Postby insignificant » Sun Jul 23, 2017 11:53 pm

There are somethings you can always do when you move up a dimension, like gaining a new direction to move in or being able to rotate in one more way at once (if you move up two dimensions). But some things are only gained by moving between certain dimensions. Going from 2 to 3 dimensions allows overtaking people, going from 3 to 4 dimensions means you can go round roads instead of crossing them.

Is there anything gained only from going from 4 to 5 dimensions or higher? Are there any 5d shapes with no 4d analogue?
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Re: Are three dimensions trivial?

Postby Mercurial, the Spectre » Mon Jul 24, 2017 9:33 am

In the sense of geometry, for every object, there is an analogue in a higher dimension.
You have more degrees of freedom if you go up one dimension, and you can use that to pass through obstacles (walking through walls in 3D space is trivial in 4D)
The sense is, it depends on what "analogy" you are using.

For example, the self-dual 24-cell exists in 4D. It can be constructed as a rectified 16-cell. Its analogue would either be the cuboctahedron (rectified octahedron) or the rhombic dodecahedron (dual cuboctahedron). Thus, they are analogues but not directly to each other.

Some, such as the family of simplices, hypercubes, and cross-polytopes, have analogues in all dimensions.

But some, such as the physics of double rotation, require 4 dimensions or higher. Same goes for triple rotations, which exist only in 6 dimensions or higher.
SárHwās Hnáras táygʲasi Hr̥táy kʲa waĉatás samHatás kʲa ĵn̥Hyántay. Táy kʲitˢtáH mánasaH kʲa dádHn̥tay Hántaray práti bʰraHtrásya pn̥tHáH kriHánt.

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Re: Are three dimensions trivial?

Postby gonegahgah » Mon Jul 24, 2017 12:23 pm

insignificant wrote:... some things are only gained by moving between certain dimensions. Going from 2 to 3 dimensions allows overtaking people, going from 3 to 4 dimensions means you can go round roads instead of crossing them.
Is there anything gained only from going from 4 to 5 dimensions or higher? Are there any 5d shapes with no 4d analogue?

Good question. Off hand I can only really offer that things like chains work better in odd dimensional spaces.
So 4Der's can walk around a road without the need to cross it but their chains get a bit awkward to construct.

In 2D they can only use S or C-hooks.
In 3D we have the chains we know with opposing O rings.
In 4D you probably need a combination hook and ring; a combination of the 2D & 3D approach.
In 5D I think opposing sphere rings will do the trick. Someone will correct me if I am wrong...

If that is the case then we really need one forward direction and opposite opposable directions to allow the chain sections to have all perpendicular directions covered.
If true I guess 5D has that advantage over the 4D guys of having legitimate chains... Is it incorrect our more knowledgeable hosts here?
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Re: Are three dimensions trivial?

Postby quickfur » Mon Oct 30, 2017 6:15 pm

Yeah, tying knots in 4D is tricky business. Any linear knots, i.e., made with a linear rope, that is, a flexible object that's very long in one dimension but small in the other dimensions, will easily fall apart when you tug the ends of it, because the extra dimension allows the rope to basically un-knot itself just by sliding over itself / any obstacles in the way. In order to tie a knot in 4D, you need "ropes" that are long in two dimensions, i.e., more like sheets than ropes. (Although in 4D, 2D sheets behave more like ropes than actual sheets, which would have to be long in 3 dimension!) But this is a lot more complicated than just tying a linear (1D) rope because you have to bend a 2D surface into a right shape to make a knot, whereas in 3D, it's just a matter of threading a 1D rope through the right loops.
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Re: Are three dimensions trivial?

Postby quickfur » Mon Oct 30, 2017 7:19 pm

But to answer the OP: this page lists a bunch of features special to each dimension, including what becomes possible/impossible as you go higher.
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Re: Are three dimensions trivial?

Postby Teragon » Tue Dec 12, 2017 10:49 pm

There are many cool things that are possible in 4D that are not possible in 3D. You can play the drums like a guitar. Rivers are less likely to join in 4D. You don't have to cross rivers and streets or descent into valleys. There are also some things that are possible in 3D, but not in 4D, for example tornados or stable gyroscopes. Going higher just allows for more and more crazy things to happen. In 5D, gyroscopes are stable again when they do a double rotation, you could use one and the same cog wheel as part of two different motors that function independently. In 6D you could build 2D screws that you can tighten along both dimensions. As the number of dimensions increases, the number of possible geometric objects increases without bounds.

Besides those things that just become possible at a definite number of dimensions there is a range of things that change gradually. Hypervolumes get more and more difficult to estimate. In 70 dimensions, a jar that is twice as high as another one, but only 1% shorter in the other directions, would actually have the larger volume. The amount of space around you increases, cities grow closer and closer together. In 2500 dimensions, all the people of my village could be direct neighbours. Still I wouldn't be able to hear them across a few meters, because sound waves dillute so fast.
Last edited by Teragon on Wed Dec 13, 2017 8:49 am, edited 1 time in total.
What is deep in our world is superficial in higher dimensions.
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Re: Are three dimensions trivial?

Postby quickfur » Wed Dec 13, 2017 12:54 am

Haha, in 2500D, would you be able to hear anything at all?? You'd have to put your ear next to your neighbour's mouth in order to pick up the sound before it completely dissipates! :lol:

Because of the same phenomenon, though, the higher up the dimensions you go, the more likely your world will be a very dark one. I think I did a short derivation of this before, somewhere on this forum, that making the (very big!) assumption that you live on an n-spherical planet in n dimensions, for large n the vast majority of the surface of the planet would be perpendicular to any single light source, like a nearby star. So only a tiny fraction of the planet's surface would be illuminated, (and conversely only a tiny fraction would be completely dark, i.e., opposite the single light source), whereas most of the planet would be in perpetual dusk. Seems to be a pretty miserable way to live.

Of course, all of this is probably moot because AFAIK, stable orbits only exist in 3D and below, so if life exists in a higher dimensional space at all, it must be live in a completely alien environment than our poor 3D brains can imagine. And it must also take on completely foreign forms too, because in 4D (and I assume higher dimensions too, though I don't have proof of that) the Schroedinger equation for the hydrogen atom has no local minima besides r=0, meaning electron orbitals cannot exist and atoms as we know them would instantly collapse and would not form the basis of matter and chemistry as we know them. So if matter exists in 4D and above at all, it must be of a form that's so foreign that it would resemble nothing like how we perceive our world.
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Re: Are three dimensions trivial?

Postby Teragon » Tue Dec 19, 2017 9:07 am

quickfur wrote:Yeah, tying knots in 4D is tricky business. Any linear knots, i.e., made with a linear rope, that is, a flexible object that's very long in one dimension but small in the other dimensions, will easily fall apart when you tug the ends of it, because the extra dimension allows the rope to basically un-knot itself just by sliding over itself / any obstacles in the way. In order to tie a knot in 4D, you need "ropes" that are long in two dimensions, i.e., more like sheets than ropes.

It doesn't have to be that complicated. It's enough to connect the ends of the sheet in one dimension to form a tube, which can be quasi one dimensional. A knot can then be tied by threading one part of the tube through another.
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Re: Are three dimensions trivial?

Postby quickfur » Tue Dec 19, 2017 7:10 pm

Are you sure that's sufficient?

Let's say you have a 2D sheet rolled up into a tube (i.e., a long cylinder shape). Since the resulting space can be embedded in a 3-manifold, we can make the simplifying assumption that we can embed it in a 3D subspace. Now consider a second rolled up sheet that you wish to tie with this one. Since that's also a 3-manifold, it can also be embedded in a 3D subspace. If it lies in the same 3D subspace as the first tube, then sure, we can easily knot it. But since we have a 4th direction of displacement, it's trivial to translate one of the tubes out of the 3D subspace it's embedded in to a different 3D subspace which does not intersect the original one. This then separates any knots that may have involved them. Therefore, they have been unknotted. QED.
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Re: Are three dimensions trivial?

Postby Klitzing » Tue Dec 19, 2017 7:45 pm

By the same argument even 2 cords within our 3D world ought fall apart, as those locally can be described everywhere by 2 non-coincident parallel planes.
But we know that in 3D knots are possible. So it might be your simplification of embeding globally into a hyperspace, which goes wrong here.
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Re: Are three dimensions trivial?

Postby quickfur » Tue Dec 19, 2017 9:00 pm

Haha, OK. So I've oversimplified. :oops:

But can two rolled up tubes knot in 4D? I still don't quite see it.
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Re: Are three dimensions trivial?

Postby student91 » Tue Dec 19, 2017 9:14 pm

Can you knot two 3-dimensional manifolds in 4d?
I guess you can embed a 3d-torus , i.e. R^3/Z^3 (so take a 3-dimensional real space R^3, then say two points are "the same" when their difference is in Z^3, i.e. all three components are integers. You obtain a torus, it can be embedded in C^3 by f(x,y,z)=(e^2ipix,e^2ipiy,e^2ipiz), guess you can do it in R^4 as well??) in R^4. Then I guess you can knot these tori in 4D, and then I guess you can put a 2-manifold in these cubes such that te result is still knotted. Take for example this surface, and wrap it up as a torus as I just described (did I, is it clear?). Then knot these tori, and you have a knot. (I don't know this for sure, but at least your logic breaks in this example, so you might get further here)
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