Actually seeing 4D

[Wilco]

Thinking about a "possible" 4th dimension when reading about tetraspace I thought of the following:
When looking to a specially created 2D image using the "look crossed" method you can see three dimensions. If you encounter several correctly placed repeated elements, it is possible for our mind to see "depth" in the image.
Now when looking a two or more 3D objects with the same characteristics can it be possible to see a fourth dimension?
I think the difficulty lies in the problem that it's impossible to see a 3D object in whole at one time because every object always has a "back" side which you can't see no matter from which direction you look at it. This problem doesn't exists with 2D.
And still remains the question if the 4th dimension exists and more importantly if a human can see/experience it and comprehense it.

[pat]

One of the things that I'd like to try to work out some time is a 3-D stereogram sculpture. I'm imagining several wavy cylinders. One would stand outside of a circle of these cylinders. The wavy edges of the cylinders would form a stereographic pattern of a projection of a hypercube. But, as you walked around the circle of cylinders, the pattern of edges would be shifting in such a way that you were walking around the hypercube in a way involving a fourth dimension. This wouldn't let you really see in 4-D, but it would sort of let you physically rotate a hypercube.

Somewhat better may be to make small cylinders mounted on a "lazy susan" (rotatable platform) so that you can rotate the hypercube instead of walking around it.

[Geosphere]

No, you cannot 'see' the fourth dimension. No way, no how.

You body is physically equipped to see 3D. Not 4. Stereogram pictures have no 'magic', and are simply two planes of vision scrambled together. All light that reaches your eye is encountered as a 2D plane. Your brain then describes the discrepencies of the two images as 3D. Whether the two planes encountered overlap or not is merely a cool physics trick.

[jinydu]

Actually, there are already many webpages on the Internet that allow you to rotate "hypercubes". Unfortunately, they just look like a jumble of lines to me.

Try searching: hypercube +view +rotate, or something like that.

[pat]

Yes, I know there are lots of pages that let you rotate hypercubes. But, they all suffer from the fact that rotating something with your hands is much more intuitive than rotating something with your mouse.

[PWrong]

I built a 3D projection of a hypercube out of toothpicks and bluetack. It's not stereoscopic, but it's a start.

I was thinking about the shape our eyes are, and whether they could theoretically be different to allow us to visualise things better. For instance, what if we evolve so that our eyeballs gradually stretch out and became more flat, then eventually become concave rather than convex, and finally they curve so far that they close off entirely, so the eyeball is enclosed within the head, we end up seeing what ever is inside the hole. Then we could put a stereoscopic hypercube inside and look at it without having to rotate! Ok, it's not a very practical solution. Completely useless.

But ridiculous ideas like that show that it's theoretically possible. With a stimulus that approximates 4D, the brain will eventually learn to think in 4D. All we need is a good enough stimulus, a 4D body, a good training program, and a lifespan of a few hundred years.

[RQ]

Extending a dimension up from a piece of paper that has a 3D drawing in it does not make it a 4th dimension, because the paper is just a representation, so it's not gonna be 4D in 3D which is absolutely absurd, however he might be onto something about visualization.

[PWrong]

I never said it was really 4D. We'd use a stereoscopic hologram like Pat said, or alternatively just send 4D data directly into the brain from a computer.

[Solid-G]

Hello I,m new to the forum and my typing is no the best, however I endevor to make up for my communication flaw by the ideas and thoughts that I may offer. I saw the line of posts and have had interest in this area of physics for some time. I want to put forward an idea that may offer a workaround the the obvious difficulties in the efforts to gleam useable understanding from making observation of what is loosely refered to as 4D space.
I suggest that while it is not possible for us as 3d dimensional beings to observe 4d space and events directly. It is possible to extend our perception into a piece of 4D space by redefining the perameters of our vision, such that we may map out a one to one slice of the domain in which we may make rational sence of the enviorment and thus retrieve some insight applicable to our 3D domain.
I've arrived at a representatonal solution that has yielded some interesting result. I'm interested in discussing this subject. I want to bounc a few ideas out there and hear some responce.
My investigations into this have lead to the discovery of what I feel is a flaw in the logic of higher dimensional representation which inhibits us from visualizing this enviorment with any comprehention.

Contact me to exchange data and discuss.

[quickfur]

I think the difficulty lies in the problem that it's impossible to see a 3D object in whole at one time because every object always has a "back" side which you can't see no matter from which direction you look at it. This problem doesn't exists with 2D.

For a 3D person, yes, this problem doesn't exist in 2D. But if you were a 2D person, you'd have exactly the same problem.
In fact, if you were a 4D person, you could only ever see 4 cells of the hypercube at a time for the same reasons...

And still remains the question if the 4th dimension exists and more importantly if a human can see/experience it and comprehense it.

Well, see it <b>physically</b>, probably not. But comprehend it, I think yes! And I mean comprehend not just in an abstract way, but actually <i>visualize</i> it as a true 4D being would see it. I'm actually writing up a webpage describing how this might work; I'll post it here when it's done. The basic idea is to look at 4D not from an abstract mathematical view, but from the perspective of what a 4D person would actually see in her 3D retina.

Our eyes only have 2D retinas, yet we are quite capable of reconstructing the 3rd dimension from these 2D images. Although our eyes will never see "true 3D" (in the sense of seeing every point in 3D space simultaneously), we can, and do, visualize entire 3D objects in our mind. So all we need to do is to find out what a 4D person sees in her 3D retina, and learn how to infer the 4th dimension from it. Even if we wouldn't fully comprehend the implications of 4D this way, we'd get a very good idea of what 4D looks like to a 4D person.

[trill]

I think it's worthwhile to mention that our brain develops throughout early life to accomodate to see three dimensions (stereovision). The brain takes images seen by each eye and calculates what the three-dimensional vision would look like. In fact, in cases where patients were blind from birth but after surgery later in life were able to see, they were not able to perceive three dimensional space. Instead, they would see everything in two dimensions with shadows just seeming to be darker two-dimensional areas.

With that in mind, I don't think an adult who had grown up in a three-dimensional world would be able to perceive four dimensions.

Regardless, I don't think that even a pesron (speaking of a human like us, not a non-human four-dimensional being) who had grown up in a four-dimensional world from birth would be able to see or perceive four dimensions visually because with our three-dimensional visual system, we would only be able to see a three-dimensional slice of a four dimensional universe.

[quickfur]

(...) With that in mind, I don't think an adult who had grown up in a three-dimensional world would be able to perceive four dimensions.

Regardless, I don't think that even a pesron (speaking of a human like us, not a non-human four-dimensional being) who had grown up in a four-dimensional world from birth would be able to see or perceive four dimensions visually because with our three-dimensional visual system, we would only be able to see a three-dimensional slice of a four dimensional universe.

Like I said, we would never be able to physically perceive 4D. Nevertheless, just as our brains learned to infer 3D from 2D images in our retina, I believe it should be possible (if only to a limited extent) to train our mind to infer 4D from 3D models we construct mentally.

[trill]

interesting idea...I come with a background of neurology and I honestly wonder whether our brains would be capable of perceiving more than three dimensions. Maybe so? It's interesting to think about! I personally have looked at two-dimensional depictions of a hypercube (such as in applets where you can rotate it) and tried to perceive what it would look like--haven't had much success

[quickfur]

interesting idea...I come with a background of neurology and I honestly wonder whether our brains would be capable of perceiving more than three dimensions. Maybe so? It's interesting to think about! I personally have looked at two-dimensional depictions of a hypercube (such as in applets where you can rotate it) and tried to perceive what it would look like--haven't had much success

The problem with looking at 2D projections is that it's, well, only 2D. I doubt our mind is very good at inferring 2 additional dimensions from a 2D image. However, if we could "look at" a 3D projection of a hypercube, we might be able to take the step into 4D.

Of course, this is not the same as looking at the 2D projection of the 3D "image" of the hypercube, since we'd only see its surface. However, if we examined dissections of this 3D projection (such as individually seeing the projected hypercube cells), our mind could easily reconstruct the full 3D image mentally. From there, we just have to take the next step: to infer 4D depth.

I certainly believe this is possible, even if only to a limited extent. In fact, I've written up a webpage that describes this approach to 4D visualization. I still badly need good diagrams to illustrate what I'm trying to describe; I'm still working on that. Nevertheless, after I went through the process of writing up these descriptions in the first place, I find that I can better visualize 4D objects now (although still not anywhere near facile).

For example, I can reconstruct in my mind the 3D image of a rotating hypercone. The "tip" of the hypercone lies in the middle of its spherical projection when viewed point-on. (The sphere is where the nappe is attached to the "bottom".) As you rotate the tip away, it slowly moves towards the boundary of the sphere, and the sphere itself compresses into a flat ellipsoid. When the tip "emerges" from the sphere, you get a 3D cone attached to a sliced ellipsoid. When your view angle reaches 90 degrees, the ellipsoid has flattened into a circle, and you see a 3D cone. If you keep rotating it, the process reverses; but this time, the tip is on the "other side" in the 4th direction, so the ellipsoid is "eating into" the conical nappe as you continue turning it. Once the tip passes below the ellipsoid, it vanishes, because now you're looking at the bottom of the hypercone. When the tip is pointing 180 degrees away from you, the projection is again a sphere (but without the "tip" in its center).

(I left out the description of the shapes of the perceived shading of the different parts of the hypercone---I'm saving that for when I add it to my webpage. :-))

[RQ]

Anything can happen in natural selection.

Perhaps a species could have evolved into seeing 4D objects, but died out of course. Perhaps our brains could be "trained" to see 4D, but I guess it depends on whether it is possible without electronic technology inserted in our brains.

[quickfur]

Anything can happen in natural selection.

Perhaps a species could have evolved into seeing 4D objects, but died out of course. Perhaps our brains could be "trained" to see 4D, but I guess it depends on whether it is possible without electronic technology inserted in our brains.

I'm not sure I understand why electronic technology is necessary. I think it should be possible, using our brains' familiarity with 3D objects, to be trained to infer 4D depth from 3D projections of 4D objects. Of course, it would be a lot harder than when we learned to see in 3D when we were young, 'cos we don't have visual stimulus for 4D objects. I guess that's where electronic technology would be nice. But I'm not sure it's necessary.

[A Drug Against War]

Visualising 4-d is one of the hardest things for the brain to imagine, I only started an interest in the 4th spatial dimension a few weeks ago. I try to think your experience would be like this:

Sit in a chair preferably when you’re tired or sleepy, sit upright, now imagine the uprightness is flat, your actually laying down, it feels right, your sleeping away. Then you hear something, so you get up, this feeling would feel like from the moment you stood up for a split second you feel like your in a room, but the whole room and you are facing gravity face on, but you don’t fall, you are free to move in that direction freely. Note thou I have a hard time in imaging a 4-d body, that’s just too much for the brain, lol.

[quickfur]

Visualising 4-d is one of the hardest things for the brain to imagine, I only started an interest in the 4th spatial dimension a few weeks ago. I try to think your experience would be like this: {...}

Well, I think that trying to imagine our 3D world interacting with a 4D one is the wrong way to approach the matter. Just like a creature in flatland would be completely confused if a 3D being suddenly started to interact with it, it's not surprising that it's very hard to imagine what we, as 3D beings, would feel in a 4D world.

I think a better approach is to begin in 4D from the start. Instead of trying to imagine what it's like for a 4D being to interact with our 3D world, we start in a 4D world itself, and imagine we have eyes that can see 4D. The key here is that if our 4D eyes are anything like our 3D ones, the retina would be 1 dimension less: i.e. a 3D retina. We don't have too much trouble imagining 3D images, so that's a good starting point. We can start exploring 4D by seeing what goes on in our hypothetical 3D retina as we look around. In other words, we look at the 3D projections of 4D objects, how they change when viewed from different angles, etc..

I summarized my ideas in another topic, although I haven't had the time to do very much with it lately.

(Now if you've ever seen those 4D wireframe Java applets, you might wonder how on earth that could help you visualize 4D... part of the problem IMHO is that they don't have hidden surface removal. It's like living in a world where everything is transparent... very confusing. Worse if you don't have direct physical access to the world. Now if somebody would only write an applet that does hidden-surface clipping...)

[pat]

(Now if you've ever seen those 4D wireframe Java applets, you might wonder how on earth that could help you visualize 4D... part of the problem IMHO is that they don't have hidden surface removal. It's like living in a world where everything is transparent... very confusing. Worse if you don't have direct physical access to the world. Now if somebody would only write an applet that does hidden-surface clipping...)

So, do you mean 4-D hidden line removal or hidden line removal on the 3-D rendition?

By my estimation, hidden line removal in the 4-D sense would leave a cube looking still pretty confusing:


And, hidden line removal in the 3-D rendition would obliterate things you really should be able to see.

[quickfur]

So, do you mean 4-D hidden line removal or hidden line removal on the 3-D rendition?

By my estimation, hidden line removal in the 4-D sense would leave a cube looking still pretty confusing:

Um, are you sure the correct lines are removed? According to my derivations you should see the wireframe of 3 attached distorted cubes. Partially hidden lines would, of course, need to be taken into account as well. Removing the entire line just because it is partially hidden does make it very confusing as well.

I guess I should've said clipping rather than removal. What I meant was that the 3D volumes resulting from the projection should be clipped w.r.t. to 3D volumes that lie in front of it in the 4th direction. Dealing with lines alone really doesn't account for all possible cases, since you'd want to render the outline of the overlapping regions rather than simply clipping the lines abruptly. (I guess this is why I haven't seen Java applets that do this, since it's not trivial to implement.)

And, hidden line removal in the 3-D rendition would obliterate things you really should be able to see.

Yeah, you don't want to do this. This is like trying to visualize a cube given a black hexagonal shadow with no visible internal structure.

[pat]

Um, are you sure the correct lines are removed? According to my derivations you should see the wireframe of 3 attached distorted cubes.

Well, it depends on exactly what overlaps happen in the 4-D direction. But, I'm still confused as to what exactly you think the best process would be for doing things. Here's my best guess (but I have others):

For each one-dimensional face (i.e. line) consider only the portion(s) of the line you would be able to see if you could see four-dimensionally. Project that portion(s) down to 2-d (or 3-d, then 2-d) to display. Display.

[quickfur]

{...} Well, it depends on exactly what overlaps happen in the 4-D direction. But, I'm still confused as to what exactly you think the best process would be for doing things. Here's my best guess (but I have others):
{...}

I guess I should explain exactly what I mean by hidden surface removal. This won't have precise details on how to actually do the computations, but at least it gets the idea across. You can probably find a way to implement this efficiently.

(1) Take any 4D object, say a polychoron for simplicity's sake. This object will be bounded by 3D volumes, e.g. the 8 cubes in a tetracube. Call these volumes V₁, V₂, ... V_n.

(2) Project each of V₁ ... V_n into 3D. Each volume V_i would map onto some 3D volume, W_i, in the projected space. The projected volumes may overlap with each other.

(3) Given two projected volumes W_i and W_j, if V_i is closer along the line of projection than V_j, do a CSG subtraction of W_i from W_j. (I.e., 3D clipping.) Do this for all possible pairs of projected volumes W_i and W_j. Call the resulting clipped volumes X₁...X_n.

(4) Compute the edges that bound X₁...X_n, and render them. (These may or may not have a direct correspondence with the edges of V₁...V_n).

I hope this makes sense.

The above of course only works for polychora, but it can be generalized to arbitrary 4D objects if we do voxel-level Z-buffer (W-buffer?) clipping. I don't know if there's a memory-efficient way of implementing this, though.

[pat]

(3) Given two projected volumes W_i and W_j, if V_i is closer along the line of projection than V_j, do a CSG subtraction of W_i from W_j.

When you say "closer along the line of projection", that's where I think you're doing some flattening that you don't mean to do. It is very possible that one could view a hypercube from such an angle that the cubical facet that is closest along the line of projection has points that are farther away along the line of projection than some (but not all) of the points in the cubical facet opposite this one. So, it's vague... do you mean the facet with the "closer facet-center" or the facet with the "closest vertex" or ...?

Let me describe another way to go about this, and see what you think.

Pick a point interior to a convex polytope and call it the origin. Now, for every facet on the polytope, think about all of the rays which originate at the origin and go through the facet. For many facets (at least 1/2 of them if the polytope is regular), none of those rays will ever intersect the viewer's retina (even if we pretend the viewer's retina is an infinite hyperplane). It is my contention, that those are entirely hidden facets and should not be drawn at all. Any possible way one could view those facets would have to go through a different facet before getting to the facet in question. With any facets that one has left, project those facets down to 3-d and render the CSG union of those facets.

But, of course, I think this should actually be done from a point outside of the facet... from the viewpoint. Take a ray from the viewpoint to each vertex of the polytope. Discard any vertexes if the ray had to pass through a facet to get to the vertex. Render the remaining vertexes connected in the way they are connected. I believe, for convex polytopes, only checking the rays to the vertexes will be enough. Consider, there is no way to orient a cube such that it obscures part-of, but not all-of, any edge. The same goes for any 3-d, convex polyhedron. The same also goes for any 2-d, convex polygon. The same goes for n-d, convex polytopes.

Of course, you're probably going to want to see more than one polytope at a time, aren't you.

[monkeymeister]

Because we are grounded in the 3rd dimension it is impossible for us to perceive with our senses past it.

Every object within our dimension is part of a higher dimension.
its easier for us to think about in in lesser dimensions...

If we lived in a 2 dimensional world then we would be only seeing one plane of the 3rd dimension. As we are living in the 3rd dimension we only see part of the 4th dimension. Because of the physical laws that govern this dimension we will never be able to see all sides of a 4th dimensional object just like a citizen of flat land would not be able to see more than one side of a plane at a time.

[quickfur]

(3) Given two projected volumes W_i and W_j, if V_i is closer along the line of projection than V_j, do a CSG subtraction of W_i from W_j.

When you say "closer along the line of projection", that's where I think you're doing some flattening that you don't mean to do. It is very possible that one could view a hypercube from such an angle that the cubical facet that is closest along the line of projection has points that are farther away along the line of projection than some (but not all) of the points in the cubical facet opposite this one. So, it's vague... do you mean the facet with the "closer facet-center" or the facet with the "closest vertex" or ...?

Hmm. You're right, my statement was ambiguous. Perhaps, if we assume only convex polytopes, then it should be possible to partition the cells into "front-facing" and "back-facing" (i.e., take the vector from the center of the polytope to the center of the cell; if the angle of that vector with the eye-to-polytope-center vector is greater than 90, then it's back-facing, otherwise it's front-facing). Then you can just render all front-facing cells, since the assumption of convexity guarantees that none of them would obscure each other.

Let me describe another way to go about this, and see what you think.

Pick a point interior to a convex polytope and call it the origin. Now, for every facet on the polytope, think about all of the rays which originate at the origin and go through the facet. For many facets (at least 1/2 of them if the polytope is regular), none of those rays will ever intersect the viewer's retina (even if we pretend the viewer's retina is an infinite hyperplane). It is my contention, that those are entirely hidden facets and should not be drawn at all. Any possible way one could view those facets would have to go through a different facet before getting to the facet in question. With any facets that one has left, project those facets down to 3-d and render the CSG union of those facets.

Yeah, that should work as long as the polytope is convex.

But, of course, I think this should actually be done from a point outside of the facet... from the viewpoint. Take a ray from the viewpoint to each vertex of the polytope. Discard any vertexes if the ray had to pass through a facet to get to the vertex. Render the remaining vertexes connected in the way they are connected. I believe, for convex polytopes, only checking the rays to the vertexes will be enough. Consider, there is no way to orient a cube such that it obscures part-of, but not all-of, any edge. The same goes for any 3-d, convex polyhedron. The same also goes for any 2-d, convex polygon. The same goes for n-d, convex polytopes.

Yeah, that should work. Though one should be careful of boundary cases, such as when a cell is exactly 90 degrees from the viewpoint-to-vertex ray. It seems that it should work, although I'm not 100% sure there's no catch there.

Of course, you're probably going to want to see more than one polytope at a time, aren't you.

Well, let's at least start with one polytope at a time, and see if it can be improved from there. :-) Then if we can extend this algorithm to work with non-convex polytopes, I suspect it would also be able to handle multiple polytopes.

Now the other thing about 4D visualization is that a hypothetical 4D person would notice mostly the interior of the 3D projection of a 4D scene, rather than the shape of its boundaries. I.e., if the 4D person looks at a hypercube vertex-on, he would take more notice of the tetrahedral distribution of the projected cells, rather than the fact that the outer boundary of the projection is a rhombic dodecahedron. But I've no idea how to visually draw the attention of the viewer to this fact.

[pat]

Okay, here's an applet that displays a hypercube with hidden line removal. I am fairly confident (though I will have to do some further checking to be proof-positive) that every line shown would be simultaneously visible to someone with a realmar retina. The caveat is that they wouldn't see things scrunched down to overlapping like we have to.

Unsurprisingly, the phpBB won't let me include an applet directly, so you'll have to look here: http://www.nklein.com/products/hsr.

Hopefully, I'll get the source code up there sometime soon and get some more polytopes into the applet.

[quickfur]

Okay, here's an applet that displays a hypercube with hidden line removal. I am fairly confident (though I will have to do some further checking to be proof-positive) that every line shown would be simultaneously visible to someone with a realmar retina. The caveat is that they wouldn't see things scrunched down to overlapping like we have to.

Yeah, but that shouldn't be too bad if we use shading for 3D depth. (That would preclude using shading for 4D depth... But I don't think that makes much difference, really, since our brains aren't familiar enough with 4D to make much use of that info anyway.)

Unsurprisingly, the phpBB won't let me include an applet directly, so you'll have to look here: http://www.nklein.com/product/hsr.

Hmm, I can't seem to access that URL. I get redirected to a 404 page.

Hopefully, I'll get the source code up there sometime soon and get some more polytopes into the applet.

Cool.

[pat]

Oops... I got the URL wrong. I fixed it above. But, for completeness, it's: http://www.nklein.com/products/hsr.

[quickfur]

Oops... I got the URL wrong. I fixed it above. But, for completeness, it's: http://www.nklein.com/products/hsr.

Cool!! That's awesome. There seems to be some odd boundary condition where I somehow managed to get a concave projection, but it seems to have something to do with clipping to some volume (when I use the 3rd mouse button to move the hypercube closer). Doesn't happen if I let it stay at the distance the applet starts out with. Other than that, it looks correct to me. You can only see at most 4 cells at a time. Sometimes you can see 3 cells, sometimes 2, and occasionally 1. Wonderful!! Best 4D polytope applet yet. Can't wait to see the other polytopes in action. :-)

It would be nice to have some way to tell 3D depth in the image, though. In some angles, when a cell comes into view, it's sometimes hard to see whether it's appearing in front of or behind the other visible cells. The "tumbling cube" effect can be quite bad as well, although rotating X into Y slightly does show enough parallax to resolve the ambiguity.

Maybe one way to solve this is if you could rotate the 3D projection independently of rotating the tetracube itself. That way you can examine the 3D shape of the projection without changing the 4D view angle, and get a better idea of the exact shape of the realmar projection without having it change on you in the meantime. Is this relatively easy to implement? Perhaps we can use the 3rd mouse button for this... I don't see a pressing need to be able to move the tetracube away/towards the viewer, since it's essentially just a scaling factor.

[pat]

Cool!! That's awesome. There seems to be some odd boundary condition where I somehow managed to get a concave projection, but it seems to have something to do with clipping to some volume (when I use the 3rd mouse button to move the hypercube closer).

Yes, I let you get close enough that sometimes a vertex could get behind you. In that case, I'm not 100% sure what will happen with the display. I suppose that I should make that more explicit.

As for your other comments, hopefully I will get to addressing them tonight along with adding in a tet.