I've been doing some thinking on the topic of extra spatial dimensions and I've worked out a measly theory of how to do part of the job of imagining a superdimension, eg a fourth spatial dimension. Please try to bear with me and don't be afraid to express your utmost contempt if you find this theory far too far fetched. Alright, let's start with an explanation involving the dimensions we know: how a two-dimensional object would see a 3D object. Let's create two characters, Larry and Daphne - Larry is a square living on a 2D plane while Daphne is a sphere living in a 3D space that encompasses the 2D plane that Larry happily lives on. Now, let's do some Platonian self-inquiry: Q: Can Larry see Daphne if Daphne intersects the 2D plane? What does Daphne look like to Larry? What if Daphne doesn't intersect with the 2D plane? A: Larry can only see Daphne if Daphne intersects with the 2D plane, at which time Larry perceives Daphne as a straight line - just as it perceives any other shape on the plane. When Daphne moves off the plane, it becomes invisible to Larry. Alright - that was easy - what this gives us is basic tool to work with: projection. Let's suppose Daphne doesn't have to intersect with Larry's plane, but Larry can, at any given time, see Daphne's orthogonal projection on the plane (that is, the projection is orthogonal to the plane and does not converge at Larry's viewpoint). Since Daphne is a superdimensional shape, there is no way for Larry to understand what Daphne actually looks like. However, it is very simple for Daphne to see what Larry looks like. Armed with this knowledge, let's create a hybrid of dimensions: let's combine what is known as 1.5D, 3D and 4D - in other words, let's try to create a projection of a 4D sphere onto a 3D "surface". Now, from the above it's rather simple to see that it's a lot easier to see the 3D surface from the point of view of a 4D object. In other words, it'd be far simpler to do the following substitution: divide four spatial dimensions as 1) left-right, 2) up-down, 3) to-fro (front and back), 4) yaw-pitch-roll. That is, in the imaginary fake 4D space, the first three dimesnions allow us to position ourselves in any direction and the fourth dimension allows us to move and look around (defines directionality) - this is just a substitution, so don't dwell on it. Q: Now, suppose you were to place a 3D object (say, Daphne, who only has positional information) into this fake 4D world - what would it give you? A: You'd get an ordinary 3D world with another 3D object located at a point in these three dimensions. The only difference would be that you could actually move around and look at the object from every direction whereas the object wouldn't be able to do the same. You could move away from the object whereas the object couldn't because it is not directional. This would give you a miniscule (albeit fuzzy) understanding of how one might imagine a 4D environment as it sees a 3D object. Let's move on. Q: Now, it's very difficult to see the fourth dimesnion that is around you from the POV of the 3D object because you're not equipped with the appropriate abilities. However, at any given time you can see the projection of the extra dimesnion that is cast on the object as a reference. What would this projection look like? A: let's use some graphics terms: billboarding and skybox. To explain, since you (the object) lack directionality, you can only see the projection of the extra dimension based on how you're positioned in it. However, you have no control over the positioning. Furthermore, you lack any ability to move around in this dimension, which implies that you're, at any given time, at a constant distance from any point in the extra dimension (that is, it is not the case that all points in the extra dimension are equidistant to you, but their distance never changes). Since a visual understanding of this is most likely impossible because there is no reference which one could compare it to, the only way to see this is intuitively. However, it seems to me that a billboarded skybox is pretty much the closest visual reference to the projection of an extra dimension that one could come up with. To those who don't know what these terms are, try the following: Billbording is a method of having a mesh face the camera at any given time - run any of your favorite 3D games and try to "walk around" a corona (the glare) of a light source. You'll notice that you can't because it's always facing you. A skybox is essentially a cube that exists in the same 3D space as you do, but is always centered on you. That is, it is a 4D box projected into 3D - you can move and look around in it, but you can't move away from the center of the box without becoming non-equidistant from all the sides of the box (if you could, you'd be moving in the fourth dimension). Regardless, you can move around in the game world with ease and everything looks like coold 3D. Load up some game like UT and clip outside of an open-space level to see the skybox in action. Q: Why is it so much more difficult to imagine what the actual extra dimesnion looks like? A: Imagine this (note that I'm not providing any mathematical proof, only a simple example that I came up with): when going back to Larry and Daphne, how can Daphne's projection be created on Larry's "home plane"? There are two ways: 1) Use the Daphne's silhouette directly and generate the circle on the plane. The silhouette for a sphere, streteched from Daphne to its projection, would be a cylinder. 2) Take the silhouette and split it into points. In three dimensions it is easy that, knowing the distance of Daphne from the plane and the distance of the projection from Larry, we can, credit goes to Pythagoras, know how far Daphne is from Larry. I personally can't think of a way to know this kind of a relation between 3D and 4D. That's the caveat. Alright - bash me, but please, constructively! edit: fixed topic name - I mean this stuff seriously

Mmkay - a bump... Just in case anyone would even bother to read it.

If it makes you feel any better, I read the first part about 2D/3D interaction, thought "didn't Edwin Abbot already cover this in Flatland?", and skimmed the second part. Good luck getting anything useful out of that.

Dang - I guess this is where you say "welcome to 120 years ago"... Ah well...

lol, i read it all. but i have a couple of issues.

the 4th dimension is time.(or as Einstein said the 4th and 5th dimension is Ifth and Oofth.

We know how to move in time(relative to velocity)

We also know that up until the critical point "C" , an object moving in Time is observable.

IIRC he felt the 5th was gravity, or rather gravity was the tell-tale sign of the 5th dimensions influence.

IMHO, when you limit your analysis to spatial organization, you limit the scope of your ability to correctly attempt an understanding.

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What does Daphne look like to Larry?

Depends, is she naked?

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the 4th dimension is time.(or as Einstein said the 4th and 5th dimension is Ifth and Oofth.

That's probably why the OP said 4th spatial dimension.

Now, here's something to think about:

There is no such thing as 1 dimension or 2 dimensions in real space. It's a human concept used to simplify things for calculations and the like. How can you possibly draw a 2-D line? The ink you use has height -- it's 3-D. You can't create something in the real world that has a size of 0 in any given dimension.

Quote:Original post by Dreddnafious Maelstrom
IIRC he felt the 5th was gravity, or rather gravity was the tell-tale sign of the 5th dimensions influence.

Almost. Space time curvature is believed to be an additional dimension. Gravity, being a force, can influence this curvature by distorting it.

Crispy: there is one very important point you should consider when talking about "spatial" dimensions: the definition of the term "spatial". Spatial refers to dimensional coordinates perceived as space extends. There are only three of them, there can't be a fourth, simply because we wouldn't perceive it as a spatial extend. Basically, in your example, Daphne doesn't have an additional spatial dimension to Larry, because Larry lacks the ability to perceive it as a spatial extend. However, Daphne could extend it's 3rd dimension within the time dimension of Larry.

And that's the whole trick with projections: Larry would see Daphne as a projection of his 2D surface space along his time dimension, not along a 3rd spatial one.

As Dredd said, for our world the 4th dimension is time. A 4D object would live within our 3 dimension and time. Imagine a tesseract (a 4D cube). Geometrically, a cube corner is defined as a point in space, where d edges meet at 90° angle to each other (where d is the number of dimensions). In other words, each dimensional major axis is orthogonal to each other. For a 4D cube, the fourth axis is time. We would perceive a 4D cube as a regular 3D cube, where each edge meets at 90° within the time dimension. Of course, being 3D beings, we cannot perceive such a 4D cube directly, because we are not able to freely observe time outside of its linear constraint.

Space curvature could be seen as a semi-spatial dimension, in the sense that it directly modifies the structure of space, ie. it can bend and deform it.

Quote:Original post by Etnu
You can't create something in the real world that has a size of 0 in any given dimension.

This really does depend on what you mean by "something". [grin]