4Dspace wrote:Thank you quickfur for your reply. Sorry I was not clear -- it's cause my understanding is evolving. I am new to topology.
I understand your example with threads in a piece of fabric and agree that 3D space can be distorted within 3D, without leaving its constrains. But. Think about it: if that were true with gravity in our world, then space would not appear to us perfectly flat, especially here on Earth. Spacetime is heavily curved around the planet, all around us. Yet we do not perceive this curvature in the refraction of light. (I'm aware that there is a lensing effect around stars, that's not what I'm talking about here.) What I'm saying is that space all around us on the planet is condensed and curved, yet we do not see it in all sorts of halos, mirages and lensings. This is possible ONLY if the distortion is in fact in the 4th spatial dimension, while the 3D is practically perfect.
The analogy here is with a thin 2D sheet of Flatland that is gently curved in 3D, appearing perfectly orderly and regular for the Flatlanders.
So, when I analyzed the situation in regard to curvature in (n+1) dimension, I saw that what curves is the surface plane from which the (n+1)th dimension arises. And to determine if it's curved, for 2D seen from 3D, I need to make orthogonal lines from 2 arbitrary points on the plane and see if they are parallel or not.
When I went from 3D to 4D, I noticed that the 3 planes that bind a cube from one side, all belong to the same plane from which the 4th dimension emerges (<-- as seen from the 4th D point of view).
I hope this is right -?
I want to learn to visualize this 4D thing well. Actually, I need only to see a 3D surface from the 4th D perspective, which is much easier. That's how I realized that the 3 faces of a cube lie on the same plane in 4D. - is that right? [...]
wendy wrote:When one says that space is curved, it is not bent in some higher space. All space, including euclidean, is curved. So if one supposes that 3d is curved in a 4d holding space, then the 4d holding space is curved in a 5d space, and so forth.
4Dspace wrote:quickfur, thank you so much for your explanation. And for your website, I am going to explore it now.
Regarding light distortions: when light passes through a lens or a prism, or simply uneven window glass (like they used to make in medieval times, 'cause they did not have the technology), the image such light delivers gets distorted. If spacetime is curved by gravity all around us, why don't we see similar distortions all around? <-- that's my line of reasoning.[...]
quickfur wrote:We do. Gravitational lensing is the prime example. I suppose your objection is that we don't see it more often -- but that's just a matter of degree. Most distortions are so tiny we don't notice it -- doesn't mean they aren't there. It has been shown, for example, that large nearby mountains have enough gravity to pull very long pendulums sideways. So the distortion is definitely there. It's just that when you couple light, which moves at extremely great speeds, and gravity, which is very weak as far as physical forces go, the effects are barely noticeable until you start talking about truly massive amounts, like entire galaxy clusters and millions of light years of distance. The spacetime around us does have all kinds of distortions, but they are slight enough that we don't readily notice them (e.g. the mountain bending a pendulum).
quickfur wrote:But this is really beside the point. The point is that space distortions don't have to require an additional dimension for them to be real. If you are confined to the surface of a piece of fabric, it's not so easy to tell, from your confined POV, whether the distortion is caused by curvature in 3D, or just stretching/compression within the fabric itself. In fact, it may not be possible to tell in some cases, without going outside of the fabric itself. You simply don't have enough information within the fabric itself to be able to prove conclusively that an additional dimension must be involved.
4Dspace wrote:[...] Light passing through the atmosphere of Earth, bouncing off surfaces of familiar objects before hitting our retinas, should be greatly distorted. Yet we see nothing. Here on Earth. Gravitational lensing around a galaxy or a star is an extreme case. We should see something on Earth with our own eyes. But we don't. Why?
You're saying this is because the distortion is so small, we don't notice it. But when glass has a minutest imperfection, we notice right away. We can tell very well when light is distorted ever so slightly. So, what's the difference?
I say, the difference between these two cases is in the 3d subspace (here I am referring to the fact that there are 4 3d subspaces in 4D). Gravity lives in 4D proper; i.e. it needs all 4 dimensions. Light (EM radiation) lives in 3d subspace of 4D. Prob'bly I am not using the right terms
[...]The point is that there can be different kinds of distortions. In our 2D analogy with a piece of fabric, yes it can be distorted in 2D (its threads having kinks), remaining flat in 3D. But it can also be curved in 3D, looking pretty orderly in 2D (its threads perfectly aligned). There is a difference. I want to know how to tell this difference.
[...] To hell with gravity. Forget it. The question is: How to tell if a 3D structure is distorted in 4D as opposed to the same 3D it is in?
4Dspace wrote:Actually, I just remembered, from computer graphics courses (yeah, me too, except that I ended up working with databases and as a sys analyst). To depict 3D we used 4x4 matrices, and that's because... something to do with the fact that you need n+1 dimension to fully describe n-dimensional object.
I think this has to do with.. an n-D object, no matter how distorted can always be contained in a Euclidean (n+1)D. There gotta be a theorem that states this -?
quickfur wrote: you need two additional dimensions to have a faithful embedding of the object into Euclidean space.
quickfur wrote:spatial distortion can be explained either way. There is really no way to tell the difference unless you have actual access to the additional dimension.
quickfur wrote:The difference is that the "minutest imperfection" to us is actually a gigantic mountain of faulty area at the atomic level, consisting of billions, nay, trillions, nay, quadrillions, nay, something on the order of 10^20 (that's 1 with 20 zeroes following it) atoms in size. That's when we macroscopic beings start noticing the effects of it.
wendy wrote:However, the curvature in space is not due to this, and by no where as large. Typically, you are looking at feet per light-years here.
wendy wrote:One should note that the hyperbolic geometry has a curvature less than euclidean space, and so a space of perfect euclidean curvature would be a curved thing in that kind of space.
Curvature is something that one does not have to suppose a larger space to do. It is partly suffice to construct lines perpendicular to a straight line. One then measures the length at some distance from the straight line. This will be longer, equal or shorter, as the space is negatively, zero or positively curved. If the case is that space is negatively curved, a euclidean construction would be as bent as a sphere. In fact, for these spaces, if one is significantly smaller than the radius of curvature (a measure which connects d(length)/length), then space will appear to be euclidean. Such is true, for example, on the surface of the earth, where one does not have to reckon curvature for the construction of a house, but does for large countries.
4Dspace wrote:Thank you very much for your replies!quickfur wrote: you need two additional dimensions to have a faithful embedding of the object into Euclidean space.
Awesome! Where do I read up on this? How this theorem is called. I wuv you!
So, it's +2, uh? Thank you thank you thank you.
[...]quickfur wrote:The difference is that the "minutest imperfection" to us is actually a gigantic mountain of faulty area at the atomic level, consisting of billions, nay, trillions, nay, quadrillions, nay, something on the order of 10^20 (that's 1 with 20 zeroes following it) atoms in size. That's when we macroscopic beings start noticing the effects of it.
You're saying that a spot with some messed up atoms on a piece of glass makes a greater distortion in otherwise straight ray of light than the dent the whole planet Earth makes in space? Here we differ. The distortion of space around the planet is so great that it requires phenomenal >11 km/s escape velocity to break through.
[...] And I have another question regarding the geometry/topology. I'm looking for a theorem that states that a symmetrical and highly ordered 3d structure in 4D is more rigid and stable than a 4d structure in 4D. I'm not sure how "stability" is determined. This gotta go in the same vein as such things as triangles being rigid in 2D or knots can be tied in 3D but not in 4D -? but I am not sure. [...]
quickfur wrote:It's +2 only in this particular case...
quickfur wrote:... It's all a difference of scale. ...
quickfur wrote: There's no special relationship with specific dimensions. ... In general, in n dimensions you need (n-1) legs pointing at the vertices of an (n-1)-simplex. Perhaps this is what you had in mind?
4Dspace wrote:[...]quickfur wrote:... It's all a difference of scale. ...
But it's the same light that passes through distortions, the same visible range, the same frequencies. It's the same scale for the main participant of the event: the light.
Today I spent some time googling trying to find out what's the value of spacetime curvature at ~1m off the surface of the Earth. After so many years, you'd expect that someone has already calculated this -? But I could not find it
Well, I tell you what I have in mind. I'm trying to deconstruct space. The one we live in. To me it's a fascinating subject and I hope you find it interesting too. I have a hunch that space is actually at least 4D. In fact, the rigidity of the 3D electromagnetic structure requires that an additional spatial dimension existed (for nuclei to move somewhere unobstructed). So, pursuing this line of reasoning, I am looking for some rules of topology that would make a 3D rigid structure naturally emerge out of a 4D structure in 4D, that is trying to assume the lowest energy state.
The most rigid and stable structure is actually 2D. The problem with it is that it is not bounded. => it needs to curve. This leads us to 4D <-- because the "real space" does not grow by drawing an orthogonal vector from a plane, it folds: 2d x 2d = 4D = 3D + 1D. This separation of 4D on 3D + 1D is what we got. 3D is occupied by the EM rigid structure and 1 empty D is where nuclei live, integrated into the 3D of EM via electron clouds.
So, what I'm looking is some clues, hidden in topology, as to why 3d structure in 4D precipitated out of a 4d structure in 4D. Any ideas?
Clue: all the structure "wants" is to assume the lowest energy state, to be as regularly distributed as possible. It can be modeled as nodes connected by edges. A node only "knows" what its 2 neighbors are doing and they are either pulling or pushing. The structure overall "wants" to equalize this pull-push among all of its nodes (yeah, it vibrates). That's the principle that drives the organization of this structure.
4Dspace wrote:Sorry, quickfur, you did not understand me. Forget physics. Think space. Think there is nothing in the world but the structure of space. When Universe is born, the first expression of its energy is the geometry of space. Space wants to be an even structure. And it quickly becomes. When it is perfectly balanced, that's what is called empty space. Each and every imperfection in its structure is something in space. And so, there is absolutely nothing in the world but topology describing the dynamic structure of space.
There is no physical forces in conventional understanding. Instead, there are distortions in the structure of space.
And so, you don't need to concern yourself with physics. The question to you is, why would a 4d structure in 4D precipitate into a 3d structure in 4D? Everything else is superfluous. I am not dealing with anything but structure of space itself. Nuclei and other trinkets are irrelevant here.
quickfur wrote:Where did the 4D structure come from in the first place?
quickfur wrote: And what do you mean by "precipitate"?
4Dspace wrote:quickfur wrote:Where did the 4D structure come from in the first place?
What does it matter? This is a given, like in chess: you have a position and the goal: white mate black in 3 moves. You don't ask how the position came about.
quickfur wrote: And what do you mean by "precipitate"?
By this I mean that, given a 4d structure in 4D (initially, the structure occupies all of the 4D = yes, initially the structure also defines the space), in order to get to the lowest energy state (it vibrates), it precipitates into a 3d structure.
The way I sort of see it, it collapses into 3d, because this configuration is more conducive to preservation of its energy. The 4 dimensions, from which it started though, remain. The 4th dimension becomes very different from the other 3. First, it appears "empty" in comparison to the other 3 where all the energies continue to run. This dimension sort of "helps" to absorb some of the "rattle" of the other 3. I know I sound phenomenally primitive, but... hey I don't know the terms yet. Where online I could read on the topology and functional spaces?
3D is the only dimension where the cross product is a binary operator. Generalizations to other dimensions are either not a binary operator, or do not have vector-valued results.
4Dspace wrote:Thank you quickfur! This quote from that link seems what I need:3D is the only dimension where the cross product is a binary operator. Generalizations to other dimensions are either not a binary operator, or do not have vector-valued results.
So, in 4D, the cross product is a plane.
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