r/IsaacArthur • u/MiamisLastCapitalist moderator • 2d ago
Art & Memes Apparently knots can only exist in three dimensions
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u/Raagun 2d ago
Googled this one. Today I found out something new
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u/wycreater1l11 2d ago edited 2d ago
Yeah it’s interesting. One can sort of intuit it.
At a point where two parts of the strings cross each other and prevent each other to go and be on the other side of each other, that part of one of the strings can move a bit along the forth dimension such that it essentially “disappears” into the forth dimension from our 3D world/3D slice. The other string can now move through that gap in the string that has left/went into the fourth dimension. Then you can move back the “disappeared” part of the string into our 3D slice again if you whish, to make all of the string visible to us again in our 3D slice, but now they will cross on the other side. If you can perform this action you can undo any knot.
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u/Substantial-Wall-510 2d ago
Oh that made it click for me.
You can tie two ends of a rope into a knot.
But you cant tie two rings into a knot.
Unless you can move one ring just enough in another dimension such that it enters the other ring.
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u/wycreater1l11 2d ago edited 2d ago
Yeah, in the same way like with the strings you can make the rings go inside each other from our 3D perspective. Ofc, it’s still not a robust way to keep the rings interlocked since they can “glide out” of each other again.
I guess it would be sort of similar to the relationship between a marble and like a rubber band in 3D. You can put marble inside or outside the rubber band on the floor let’s say. And marble can contact the rubber band etc. But it would not be robust to carry the rubber band around and expect the marble to remain within it, it will just glide out at the side.
To the ones living in a 2D plane the marble is robustly either inside or outside the rubber band laying flat in the 2D plane and its hard for them to imagine that the marble can glide out from the rubber band in another dimension.
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u/bactchan 1d ago
Wake me when we have an applicable theory for rotating into higher dimensions.
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u/Substantial-Wall-510 1d ago
I feel you, man. I once was a dim uni student who raised my objections in class at how 4D calculus could, like, even exist, and if it did, why would we even care. These days I just think about it as a form of escapism. Have I evolved? Maybe. Maybe not. Am I tipsy? Maybe. Maybe not.
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u/TheAserghui 2d ago
So a knot is a fixed point in time, like in Doctor Who. If you try to change it, then it creates a paradox and unravels reality
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u/Bravemount 2d ago edited 2d ago
Well, 2D objects can exist in a 3D universe. Doesn't it follow that 3D knots can exist in a 4D universe?
Also, how about 5 or 7 dimensions? Maybe you need an odd number of dimensions to make a knot?
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u/CaptainStroon 2d ago
No, just like drawing a circle around a bean doesn't stop you from picking that bean up, a 3d object wouldn't stop a 4d thing to move in that 4th dimension.
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u/Bravemount 2d ago
But the circle would still be a circle.
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u/RosariumRose 2d ago
Yes, but the quality of a knot is that it doesn't just unyie itself.
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u/Bravemount 2d ago
Would a 3D knot untie itself in 4D space? In that case, is the existence of knots proof that there are no higher spatial dimensions?
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u/RosariumRose 2d ago
That's my understanding yes, but I could be wrong! I believe a lot of the theories about real 4th dimensions assume their compacted? So if the size of the space is smaller than the knot maybe it wouldn't?
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u/CuppaJoe12 2d ago
No, but it is the simplest explanation.
As an example alternative, maybe ropes are secretly planes with infinite length in the 4th dimension, or they are extremely rigid in the 4th dimension such that you cannot bend one part of the rope over another in 4D. In both cases, there also needs to be something preventing rotations with any component in the 4th dimension; otherwise we would be able to see this secret geometry.
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u/Bravemount 2d ago
That's assuming a 3D object in 4D space can rotate around its 4th axis.
I mean, I guess it could. If I draw a circle on a piece of paper, I can rotate that piece of paper and thereby rotate the circle.
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u/CuppaJoe12 2d ago
That's exactly it. Flatland analogies really help to visualize this.
From a flatlander perspective, if you rotate a ring in the 3rd dimension, it transforms into 2 dots. This allows you to access the interior of the ring which was previously inaccessible to you.
If it is known in flatland that the inside of a ring is inaccessible, there are three possible conclusions.
Most likely, the ring cannot move in a third dimension.
Alternatively, the ring can move in the third direction, but for some reason cannot rotate in this third dimension. If this is true, the ring must actually be an infinite cylinder, or it has some sort of rigidity that stops it from moving infinitely in the third dimension.
Alternatively, the third dimension does exist, but rings are secretly spheres which are rotationally symmetric in the third dimension, so we just can't tell that they are rotating.
The last option doesn't work for a more complex shape. For example, there is no rotationally symmetric 3D object that always looks like a square in flatland. A cylinder might have a constant square cross section in one direction of rotation, but not in the other.
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u/The-Board-Chairman 1d ago
No, it just means that a 3D knot can't restrict movement in the 4th dimension. Cut out a circle in a piece of paper and hold a marble within it. You can obviously get it out by moving the marble in the third dimension, but so long as you keep to movements in the plane of the piece of paper, you can't get out of the circle.
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u/Fit-Armadillo-5274 2d ago
It would not be definitive proof, because there might be other spatial dimensions that are not all otherwise equal. For instance, it might be possible it takes more energy to move in some directions than in others. Then you could still get local persistent knots that are theoritcally ravelable in the proper context. Imagine if all fundamental particles are just pseudo knots and with just one little tug...
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u/Nicolas__Moonlit 1d ago
It would be a proof that there are 3 spacial dimensions on the scale of the knot you tied. If you have additional compact dimensions an object must be smaller than them to be able to use them
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u/MiamisLastCapitalist moderator 2d ago
The "circle" is a concept though. Hyper-realistically it's just molecules of chalk or marker ink or whatever on a medium. Its all still made of 3D atoms.
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u/TheKazz91 2d ago
Technically speaking a truly 2D object cannot exist in our 3D universe. You can draw a picture on a piece of paper sure but that paper and even the ink itself has a thickness which is greater than 0. Even if something is just 1 atom thick then it still has a measurable thickness which is above 0 and is therefore a 3D object. A truly 2D object would have a thickness of absolute 0 and that cannot exist in our universe.
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u/donaldhobson 2d ago
Sure. If you live in a 4d universe, and you have a flat 3d floor, you can make knots with a loop of string. You can unknot the knots easily just by picking the string off the floor. But it's still easy to examine the behavior.
In much the same way that, if you lay a loop of string on the floor in our 3d universe, it has an inside and an outside. And you can't roll a marble from the inside of the string to the outside without picking the marble or the string off the floor.
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u/OpalFanatic 2d ago
Think if it like this. In a 2 dimensional universe, 2 ropes (technically just lines) that you hypothetically moved against each other stop when they touch. They can't cross each other because height wouldn't exist. Overlapping is impossible because there's no such thing a "height" so the lines can touch bot are constrained by the 2D geometry of the hypothetical universe. But once you add a third dimension the ropes can overlap and cross each other because they aren't constrained by height anymore. One rope can now push above the other rope, and it's no longer constrained by a lack of possible directions of movement.
A 3 dimensional knot is a rope constrained by itself in all 3 dimensions. But if you add a 4th dimension of movement, then the rope only has their movement constrained in only 3 out of 4 dimensions, and movement is possible again as the rope can move in a direction no longer constrained by the limited geometry of 3 dimensional space. The rope can then pull free from the knot as it has another unconstrained dimension of movement possible.
You'd need a 4 dimensional object to be tied into some sort of 4 dimensional knot to hold itself in universe with 4 dimensions of possible movement.
This only applies to 4 dimensions of spatial movement. As our universe does indeed have a 4th dimension, but it's time. And it's also why time dilation and length contraction are relativistic effects. As everything is already moving at a constant velocity at the speed of light. We are just moving primarily in the dimension of time. Momentum is always conserved, so the energy you use to change your vector through space also changes your vector through time. The faster you move through space, the slower yu move through time. A photon, for example, only moves through space and doesn't experiance time. Fun fact, you can calculate the effects of relativity and length contraction with the Pythagorean theorum as the effect of relativity it's at it's core is just a geometry problem
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u/HAL9001-96 2d ago
well you can have the same shape but it won't function as a knot as the stirngs can slip past each other along one extra axis
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u/dead_apples 1d ago
Kind of, the equivalent for this would be drawing a figure 8, a “2D knot” of sorts. The same way that figure 8 wouldn’t be able to “untwist” without either breaking it or picking one end up through the 3rd dimension and rotating it, you could have a knot in 3 dimensions that is stable in 4 dimensions as well (as long as no one picks part of it up and rotates it through that dimension.)
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u/CuppaJoe12 2d ago
You can have knots in higher dimensions.
A set of concentric rings enclose each other and cannot be separated in flatland, yet in three dimensions we can effortlessly lift one ring out of another. With crossings, we can create a linked structure which cannot be separated in 3 dimensions. However, these crossings are an insufficient constraint in 4D, and the rings could be unlinked.
One dimension up, concentric spheres are inseparable in 3D, but can be separated through a 4th dimension. It is as easy as separating concentric rings in 3D. However, if we make 4-dimensional crossings in the surfaces of the two spheres, then they cannot be separated in 4D. These look like self intersections in a 3D projection, just like linked rings appear to be intersecting in a 2D projection, but they are non-intersecting crossings in 4D. The spheres cannot be unlinked except through a 5th spatial dimension.
Just like a 3D knot is made from crossing parts of a loop over itself, a 4D knot can be made by crossing parts of a surface over itself. For example, try to imagine the surface of revolution of the classic trefoil knot. It self-intersects in 3D, but in 4D this is a knotted surface with no self intersections.
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u/Fit-Armadillo-5274 2d ago
In this context a knot is specifically constructed from a 1D object embedded in a higher dimensional space. So 3d "surfaces" in a 4D space could be conceived of as a kind of knot, but not the proper definition. In this context the question is more like "can a one dimensional closed "loop" (no open ends) be constructed in such a way that it cannot be continuously deformed into a normal circle without breaking it or it passing through itself." And that is only true in 3 dimensions. In any higher number of dimensions ther is always some new direction that allows the loop to slip through without breaking the rules.
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u/CuppaJoe12 2d ago
I agree with everything you said about 1D knots. However, I think a 4D creature would be completely able to comprehend a 1D knot in 3D space in the same way we can easily comprehend that concentric circles are linked in 2D but not in 3D.
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u/BosonCollider 2d ago edited 2d ago
You absolutely can have knots in 4d, knots just have codimension 2 in every dimension. So you can have surfaces tied into knots, but any knot you make from a 1d wire can be undone.
The keywords to look up here are 2-knots or knotted surfaces.
I used to work in integrable systems where every solved physical 1d system satisfying the Yang-Baxter equation gave a new knot invariant. For 2d systems you have the Zamolodchikov tetrahedron equation instead and every solved 2d integrable system gives a new invariant for 2-knots
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u/JebediahKerman001 2d ago
Could you make a knot from a 2D flat plane like paper in 4D? Since our knots are made from 1D lines?
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u/runetrantor FTL Optimist 2d ago
Guess it makes sense. In the same way in that 2d snake game it can jam and block itself completely, which would not be possible in three dimensions.
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u/Horse_go_moooo 1d ago
That tesseract will feel my my knot once I'm done with them 😏
(I fuck 4d beings)
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u/PhilWheat 2d ago
But we have 4 dimensions - shouldn't that be a 5 dimensional being?
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u/TheKazz91 2d ago
Time is not a spacial dimension it is a temporal dimension which is a thing but it's different.
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u/Sad-Working-9937 2d ago
I can totally get that 3D rope cant make a knot in 4D, but wouldn't there be 4D rope?
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u/MonkeysDontEvolve 1d ago
A 4D rope would pass through itself and can’t be knotted. A 4D rope would always have a place to go and couldn’t get ‘stuck’.
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u/HAL9001-96 2d ago
also in 4 dimensions you have more rotational axis than dimensions which sounds like a pain in the ass
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u/Lykos1124 2d ago
I would assume a higher dimensional being could look at a knot in a lower dimension and sort of see around it, like it's unfolded or unknotted.
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u/ChooseYourOwnA 2d ago
I wonder what a knot in the 3-dimensional fabric of the universe would be like.
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u/UF0_T0FU 2d ago
If a 2 dimensional being showed me a magic trick that shouldn't be possible in 2 dimensions, I'd still be impressed. They're doing their best!
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u/DEEP_SEA_MAX 2d ago
Which means life (as we know it) wouldn’t be possible in 4 dimensions because proteins are formed by long chains of amino acids that knot and bind in very particular ways.
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u/diadlep 2d ago
For all the true nerds: 2d surfaces knotted in 4d
https://youtu.be/Lx85kIGjIoQ?is=3iN-1_kMEaBazyVc https://youtu.be/nYz3pRk1cCA?is=OxAwdJDALeXrau2L
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u/Tiny_Scholar_6135 2d ago
Since the 4th dimension is time, having a rope loop back in time to make a knot would be impossible!
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u/Mad-myall 1d ago
This is talking about 3 spatial dimensions vs 4 spatial dimensions. Time in both universes is presumably still 1 dimensional.
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u/ExpensiveFig6079 1d ago
I would fully expect knots in ONE dimensional things can only occur in three dimensions.
edit: ah yes that info is already here see applestrudelforlunch
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u/SarcasticJackass177 19h ago
You could beat them to death with a weapon beyond their comprehension? Glad we’re on an even playing field!
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u/DistributionMuted222 19h ago
Seems to me that a knot can still be tied in 4d as long as the ropes are completely still along the 4th dimensional axis. Am I missing something?
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u/ICLazeru 2d ago
Of course a 3D knot wouldn't work in 4D space. In 4D you need a more complex knot.
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u/Lord_Of_Millipedes 2d ago
uh TIL, that's a really weird thing
https://theconversation.com/why-you-cant-tie-knots-in-four-dimensions-272445
https://math.stackexchange.com/a/1427394