Dimensional Baby Steps

[4Dspace]

Hi 4DSpace. We really aren't looking to argue with you or your silent team of supporters. All we are simply looking to do is study a particular idea within a particular framework and we are getting the deep impression that our framework does not gel with yours at all and that yours belongs in a different area of this forum. There is nothing personal about it for me. I certainly don't speak for all here but I regard our framework, and this area of the forum, as purely fictitious and solely interesting from a geometrical point of view. I would also like to think that it also gives me some interesting insights into what I regard our 3D geometrical terminology and thinking to be; or at least throws a different perspective on it.

This is the forum for discussing the geometry in 4D and higher. You and quickfur only discuss mythical beings, not real properties of real N-spaces. You sound like a bunch of kids. Not only that, you are unfamiliar, apparently, with the most basic concepts of geometry and => "vision" which implies, first of all a POV. As this discussion revealed, quickfur was not clear, what a POV was and how it related to geometry, so what "vision" were you talking about here?

As for the newcomers to the discussion, clearly, they can't tell whether what I'm saying is true or not. They apparently think I'm the troublemaker here, because they continue to appeal to quickfur to tell them more about his "4Der vision", because he represents the authority for them. That he is a blind authority, speaking about vision devoid of a POV does not strike them as odd.

So, you gongahgah should not try to misrepresent the situation here either.

If this section of the forum was for dimwits, you should have put a sign to this effect.

[gonegahgah]

Our apologies. This section of the forum is for dimwits. This really is not the place for you 4DSpace.
So I would encourage you to seek elsewhere...

[4Dspace]

And I rather say that dimwits should leave their discussions of mythical beings and those beings vision out of geometry of higher dimensions.

I'll tell you gonegahgah how people with Asperger differ from the rest of you "normal" people. And, luckily, there is more and more of us are being born. To replace the dimwits that took over the boards here and elsewhere. We, Asperger folks, are not concerned with your pecking orders, which is really just an atavism humans retained from the times they were baboons. And for a baboon, the pecking order and where and how you fit in it, is the most important thing. Also, for a baboon, what is true is not what actually is true, but what is the main baboon is saying. And the pecking order demands that whatever the main baboon was saying was the truth, truth and nothing but the truth. It's because your relationship with other baboons is what matters the most. To hell with truth.

But enough already!

[gonegahgah]

Yes, that was what I was thinking. I somehow imagined the 2D beings to have to be opposite the objects and therefore I thought they had to rotate in the direction of 3D. But now that I have seen your drawings, I'm quite sure it's nonsense what I thought. Because in your drawings they are on the same 2D plane. I hope there will be a sequel to the nice drawings with 4D examples ? Maybe then I can understand better what you wrote about the 4Ders that can't see some edges of a cube (and/or a hypercube?) and how this goes together with quickfurs model of 4D seeing, where they can see every little bit at once. Somehow it sounds both true but somehow also a little contradictory, in any case, quite difficult to comprehend.

I have some more pictures in mind so I'll add those as I can.

Step 4. The 3Der speculating on a 4Der's view:



I've combined a few pictures here.
On the left is one of us looking at the front of a big cube. The cube is made of red wood, which we can't see, and is painted green on the outside.
The second picture shows the image on our retina (I'm ignoring the whole upside-down thing that our eyes do).
I've also shown the cones as full colour cones this time which would really contain a mixed pattern of red, green, and blue cones (with b&w rods as well).
The third picture is us pondering what a 4Der will see.
We find it difficult to understand how a 4Der can see when they are supposed to be able to see things that are, to us, behind something that is in front.

Step 5. Comparing the 2Der and the 3Der speculating:



In this picture we have the same dilemma occurring for both the 2Der and the 3Der.
The 2Der wonders how our eyes can possibly see the inside of the square when our eye cones must be in columns, like theirs, but multiple columns that sit behind each other.
Surely only the front retinal cones would be able to see because they are at the front and the ones behind couldn't see because they are stuck behind the front ones.
We 3Ders know the truth that each column of cones is not behind each other but are actually all side by side so we can see the inside of their square with no problem.
It is a hard concept for a 2Der to understand how something can be side-by-side; and not just in front of or behind.

But, we, in turn, wonder how the 4Der can possibly see the inside of our cube when their eye cones must be in an array, like ours, but multiple arrays sitting behind each other.
Surely only the front retinal cones would be able to see because they are at the front and the ones behind couldn't see because they are stuck behind the front ones.
Does the 4Der think that it makes perfect sense? Can they see inside our cubes in a similar manner to the way we can see inside the squares of the 2Der?

I hope this helps to highlight the dilemma that we tend to face when conceptualising what a 4Der would see and how a 2Der faces a similar dilemma when thinking about how we see. But we already have a sideways; so the difficulty for us is to conceptualise how the 4Der can have a whole 359deg of sideways other than our single left-right sideways that we know.

[ac2000]

Thanks for the nice drawings, gonegahgah.
So far that looks very plausible to me. The approach seems to me to be very much like the one of C.H. Hinton in his book "The fourth dimension" (1904) (although I have read only the first chapter or so, and it's much more complicated there. Just in case anyone is interested, here are the links to a description of the making of the colour cubes he used for visualization and an edition of the book, which is in the public domain by now):
http://www.scribd.com/doc/3046062/Hinton-Cubes
http://hermetic.com/93beast.fea.st/files/section1/hinton/Hinton%20-%20The%20Fourth%20Dimension.pdf

What I still don't quite get is, which part of inner edges could possibly be obstructed for a 4Der when he looks at the 3Dcube. Because when the "slices" of the cube are infinitely thin, then they wouldn't be thick enough to obstruct anything from view, wouldn't they? But maybe I can still figure this out.

[quickfur]

[...] But, I do really like your example. It is about as close to true 2D that we will ever get. Good one!

Thank you .

Yes, that was a very clever idea. I like it too.

Thanks for the nice drawings, gonegahgah.
So far that looks very plausible to me. The approach seems to me to be very much like the one of C.H. Hinton in his book "The fourth dimension" (1904) (although I have read only the first chapter or so, and it's much more complicated there. Just in case anyone is interested, here are the links to a description of the making of the colour cubes he used for visualization and an edition of the book, which is in the public domain by now):
http://www.scribd.com/doc/3046062/Hinton-Cubes
http://hermetic.com/93beast.fea.st/files/section1/hinton/Hinton%20-%20The%20Fourth%20Dimension.pdf

What I still don't quite get is, which part of inner edges could possibly be obstructed for a 4Der when he looks at the 3Dcube. Because when the "slices" of the cube are infinitely thin, then they wouldn't be thick enough to obstruct anything from view, wouldn't they? But maybe I can still figure this out.

If you're talking about mathematical cubes which are infinitely thin, then, no, nothing will be obscured from view.

But, as I mentioned in an earlier post, if we assume that the cube actually has non-zero 4D thickness (i.e., it's a tesseract in disguise), then the "inner faces" (or rather, the inward-facing part of the faces) will not be visible.

Consider the 2Der's case, for example. If the square that the 2Der is looking at is actually a very thin cube, then we may regard it as being made of cylindrical atoms packed together to form a square. The lids of these cylinders face outwards, in the positive and negative axis in the 3rd direction, whereas their curved sides face the directions in the 2D plane.

Now suppose the lids of these cylinders are blue, and their curved sides are green. The 2Der, confined the 2D plane, will always only see the curved sides of the cylinders, so to them, the square is green. They can cut it up into smaller pieces, and it will still appear green. But the 3Der, looking from the 3rd direction, sees not the curved sides of the cylinders but their lids, which are red. So they always appear blue.

Now let's say the outer edges of the square are made of cylinders which are painted red on the outward-facing parts of their curved sides, and yellow on the inward-facing parts of their curved sides. For simplicity, let's say the edges of the square are just its 1-atom-thick perimeter. So we can say that we've painted the "outside" of the edges red, and the "inside" of the edges yellow. But the parts of the cylinders that are painted yellow are touching the cylinders inside the square (which haven't been painted, so they are green on the sides with red lids).

When the 2Der looks at the square now, he will see what he thinks is a red square. Of course, if he cuts it open, he will discover that it is green inside; and if he peels away the outermost 1-atom-thick layer, he will see that the inward-facing sides of the edges are yellow.

A 3Der looking at the situation sees only a blue square -- because he sees the blue lids of the cylinders. If he were to shift his view to the plane that the 2D being lies in, he would see the red-painted outward-facing part of the square's edges. But now he cannot see the yellow at all; not from the plane that the 2D being lies in, because those yellow-painted parts of the cylinders are obscured from that POV, neither from outside the plane, from a 3D POV, because the yellow-painted parts of the cylinders are touching the cylinders inside the square, and the sides of the cylinders are obscured by neighbouring cylinders, so only the cylinders' lids are visible.

Now in the 4D case, a similar situation can be setup. Suppose what we think are spherical atoms that make up the cube are actually thin 4D spherical cylinders (the extrusion of a sphere into 4D). This shape as two "lids" which are spherical, analogous to the 3D cylinder's lids, and a curved side, which is curves in a spherical manner analogous to the 3D cylinder's curved side. From our 3D point of view, we only ever see the curved side, which appears to us as a sphere (the cross-sections of the spherical cylinder parallel to its lids are spheres). The cube, then, is made of these spherical cylinders. Let us suppose that this spherical side is painted green, but the lids are painted blue. Then, a 4Der looking at the cube from the 4th direction will see the blue lids, so he perceives a blue cube. From our point of view, though, we see a green cube. If the 4Der would shift his POV to our 3D hyperplane, he would also see the green sides of the spherical cylinders that make up the cube, though normally, from a 4D pov, the cube appears blue.

Now suppose we peel off the outer 1-atom-thick layer of the cube and paint the inward-facing side yellow, and the outward-facing side red (if you like, think of the result as a 1-atom thick layer in which the atoms are painted half red, half yellow), then we glue back this perimeter onto the cube. What we see now is a red cube, which, if we cut it open, is green inside. The yellow would be visible if we peel away the perimeter layer, of course. But if we leave it unpeeled, we won't see any yellow. And if we don't cut the cube, we don't see any green.

Now, what we thought are spherical atoms painted red on one half and yellow on the other half are actually spherical cylinders in 4D; so what we have really done is that we've painted half the curved side of the spherical cylinder red, and the other half yellow. The blue lids remain blue -- they face outwards along the 4th direction so we have no way of accessing them.

When the 4Der looks at the cube now, he still sees a blue cube as before (he sees the blue lids of the spherical cylinders). He can see the green sides if he moves into our 3D hyperplane. But now he can't see the yellow -- not from our 3D hyperplane, because that part of the spherical cylinders are obscured from that direction, neither from the 4D pov, because the yellow is on the inward-facing part of the spherical cylinders' curved sides, which is being obscured by neighbouring cylinders (in much the same way as 3D cylinders closely packed will obscure each others' curved sides). Thus, the 4Der can't see any yellow at all.

That is to say, the 4Der sees the "inside" of the cube in the sense that he sees the blue lids of every atom inside the cube, but he doesn't see the curved sides of those atoms because they're obscuring each other. We, of course, have a drastically different view of things, since we're always looking at the curved sides of these atoms, and can never see their blue lids. We can't see inside the cube because the atoms are obscuring each other from our pov. When we're told that the 4Der can see "inside" the cube, we imagine that they must have xray vision, because that's the only way you can see the obscured curved sides of the atoms inside the cube; however, this is a misconception, because the 4Der's can't see the obscured curved sides of the atoms either. What they can see is the blue lids of these atoms, which we can't see (and have no idea about).

[gonegahgah]

Thanks for the link to that book AC2000. My approach won't be like CH Hinton's. His book proposes his multiple combinations of results from a primary number of available viewing rotations which give that many patterns that could be produced. It's kind of like having a Rubik's Cube and seeing how many colour orientations we can directly see it in. ie. Red-Green-Yellow, or Blue-Red-Yellow, or Orange-Green-Yellow, etc., but never, for example, Red-White-Orange, or White-Orange-Yellow, etc. For the Rubik's Cube this is because we 3Ders can only see three faces at once; and can't see the opposite faces together. He shows the result that he has produced at the start and then proceeds through the rest of the book to explain how it fits together.

I want to more directly help us understand where the 4Der is seeing from; not how many combinations they will see. From the brief read of CH Hinton's book I think it would still leave the reader thinking that the 4Der sees cubes from our outside point-of-view; which I'm hoping to show otherwise.

[Hugh]

The existence of a 3rd orthogonal direction allows a 3D viewpoint to see the same 2D square from another orthogonal direction.

Perhaps the existence of a 4th orthogonal direction allows a 4D viewpoint to see the same 3D cube from another orthogonal direction.

[gonegahgah]

The existence of a 3rd orthogonal direction allows a 3D viewpoint to see the same 2D square from another orthogonal direction.
Perhaps the existence of a 4th orthogonal direction allows a 4D viewpoint to see the same 3D cube from another orthogonal direction.

That's exactly where I am heading towards Hugh. You've been thinking ahead; very good.

Though, I will keep taking little steps diagrammatically to help piece it together.
Sorry, it will be taking several more steps to depict.

Step 6. Looking at things from 2Ders Point-of-View:



It will be worthwhile to explore the dilemma faced by the 2Der in trying to understand where we are when we are looking at their square.
I've tilted it on it's point so that two 2-faces are in view to the 2Der.

Let me explain a little about faces as I am expressing it. When you look at someone you see their face. The face falls on our retina to form a flat image. (It is the combination of the two flat images from our two separate eyes that gives us our stereoscopic effect). To us a face is an image that can be shown on a flat canvas.

A 2Der looks at another 2Ders face and that face only has up and down, and no sideways. Their 2Der friend's face falls on the back of their eye to form a flat line.
To them a face is an image that can be shown on their flat canvas.

This is why I keep mixing terms and putting the dimension in front, an idea I picked up from QuickFur.
So I refer to 2-volume for a 2Der's containers (eg. cup) and 3-volume for our containers.
Or, in the instance above I refer to a 2-face to refer to faces as a 2Der thinks of them - which we think of as edges. To them a 'face' is what we think a line.
Afterall that is what they face (pun intended but meaning also implied).

Back to the picture... The image on the left shows the 2Der looking at their square. We can see the action from side on.
The image on the right then attempts to depict what the 2Der is seeing; though I have made the material a little transparent so they can see the shadow off the back edges.
I have had to give it a little width - which the 2Der doesn't have - otherwise we wouldn't actually be able to see what they are seeing.
I've put a little line for the 2Der as well but obviously they won't be able to see themselves from the back.

[Hugh]

The existence of a 3rd orthogonal direction allows a 3D viewpoint to see the same 2D square from another orthogonal direction.
Perhaps the existence of a 4th orthogonal direction allows a 4D viewpoint to see the same 3D cube from another orthogonal direction.

That's exactly where I am heading towards Hugh. You've been thinking ahead; very good.

Though, I will keep taking little steps diagrammatically to help piece it together.

Thanks gonegahgah!

[quickfur]

[...] Consider the 2Der's case, for example. If the square that the 2Der is looking at is actually a very thin cube, then we may regard it as being made of cylindrical atoms packed together to form a square. The lids of these cylinders face outwards, in the positive and negative axis in the 3rd direction, whereas their curved sides face the directions in the 2D plane.

I made some povray models to illustrate what I mean. Here's a scene showing a 2Der and a 3Der both looking at a square from their respective viewpoints:


To illustrate the effect of 2D atoms being cylinders in disguise, I've exaggerated the size of the atoms in the square. The following image shows a close-up view of the atoms in the square:



As you can see, the lids of these cylinders are blue, but their curved sides are red. So how does this square appear to the 2Der?



The 2Der's eye can only hold a 1D line. Technically, that line should be infinitely thin, but like gonegahgah, I've added some width to it so that we can actually see it. Again, the size of the atoms are exaggerated for the purposes of this illustration, so you see a "bumpy" line; but with normal microscopic atoms, this would just be a solid red line. Note that the 2Der never sees any blue, because the blue can only be seen from 3D.

Now suppose we tell the 2Der that we can see the entire square -- every atom that's in the square. The 2Der will be surprised, and think that we have "xray vision" of some sort -- because as far as the 2Der knows, that's the only way we can see through that solid red wall of the outer atoms of the square. However, we can't see through the red part of the atoms; if we were to look at the square from the same viewpoint as the 2Der, we'd see this:



In other words, what we see from the 2Der's viewpoint isn't too much different from what they see (except that we see the width of the cylinders, they don't). As long as our viewpoint lies in the 2D plane, we cannot see anything more than what the 2Der can see. Our sight is under the same restriction as theirs.

However, if we were to look at the square from our 3D point of view, we see this:



Again, the size of the atoms are exaggerated for the sake of this illustration; with normal microscopic atoms, the above image would simply be a featureless blue square. Now we can see every atom in the square, no problem. We can also see all four edges of the square at the same time. To the 2Der, this may seem "magical", or require incredible feats of omniscience or omnipresence. The 2Der thinks so, because when we say that we can see "every atom in the square", they are thinking in terms of seeing every atom in the square from their viewpoint, where they see the red curved sides of the cylinders. However, we do not see the atoms in the square like this at all! What we are seeing is the blue sides of the cylinders, not the red curved sides. The red curved sides of the atoms inside the square are just as obscured from our view, as they are obscured from the 2Der's view.

When we see every atom in the square, we're seeing a completely different aspect of the 2D atoms; we're seeing a side that is completely inaccessible to the 2Der. We don't have any xray vision at all; the atoms are still as solid and as opaque as they are from the 2Der's point of view. In fact, as is obvious from the last image above, these atoms obscure any objects that may lie behind them, so they still block our vision, just as they did when we look at them from the 2Der's point of view. The only difference is that when we were looking from the 2D plane, these atoms are obscuring each other; the atoms on the edge of the square are blocking the view of the atoms inside the square. But when we look from our 3D point of view, even though the atoms haven't turned transparent or anything of the sort, they no longer obscure each other. Why? Because from our 3D vantage point, every atom now lies along a distinct line-of-sight -- they still obscure whatever's behind them, but none of them are behind each other so all of them are visible.

Consider now the analogous 4D case. When the 4Der looks at a cube from their 4D viewpoint, they can see every atom in the cube. But it's not how we imagine it -- xray vision, or omnipresence, or whatever. We think xray vision or omnipresence is necessary for this feat because we're thinking in terms of the "curved sides" of the atoms that we're familiar with. (We are not familiar with any other faces of the atoms, since we can't see those faces.) We think that these curved sides, which appear to us as spheres, are all the sides the atoms have. But actually these atoms are not mere spheres; they are spherical cylinders in 4D. They have a completely different aspect, their "blue lids", which we cannot see or access. The 4Der can see all the "blue lids" of every atom in the cube -- so in a sense, they are seeing every part of the cube, all 6 faces, and the entire inside content of the cube. But in another sense, they aren't seeing the inside of the cube -- they cannot see the "curved sides" of the atoms, just as we cannot see them, because the surrounding atoms are blocking the view. Neither can they see inside the atoms (the inside of the cylinders); they only see the "lids" of the atoms.

Should the 4Der move their viewpoint into our 3D hyperplane, then they would be subject to the same limitations we are. They would only see the curved sides of the atoms -- the sides that we see -- and they can't see inside the cube at all, because the atoms on the surface of the cube are blocking the view of the atoms inside. But when they lift their viewpoint out into 4D space, they begin to see the blue sides of these atoms. From their new point of view, now every atom in the cube is visible to them, because now the atoms all lie on different lines of sight, and so they no longer occlude each other. But this does not mean the 4Der can see the "curved sides" of all the atoms, as we might imagine; they can't. What they see is a wholly new aspect of the atoms that is completely inaccessible to us. So while the 4Der can see every atom in the cube, they only see a part of every atom -- the hyperface that's facing them. They cannot see the "curved sides" of the atoms, and neither can they see the other blue side of the atoms (the other hyperface of the atoms).

[gonegahgah]

Cool pictures quickfur.

[quickfur]

Cool pictures quickfur.

Thanks! I owe you the idea of comparing the different POVs of different dimensions in this way. I also owe 4DSpace for making me think more thoroughly about dimensional terminology, that eventually led me to realize the far-reaching consequences of objects needing non-zero n-bulk in order to exist in n-space, and what that means for 4D (and higher) visualization. Now I understand why people struggle so much with the idea that 4D (and higher) beings can see every part of 3D space, attributing it to some kind of "magical" x-ray vision or omnipresence or some such. They are thinking in terms of the "sides" of atoms that are visible to us, imagining that somehow the 4D being's sight is magically bypassing the occlusion of atoms in the back by the atoms in the front. But what's really happening is that these atoms have "sides" that are wholly unknown to us -- sides that face directions outside of 3D space. These sides are oriented such that they do not occlude each other, and that is how the 4D being can see all of them at the same time. The occluded sides are still just as occluded as before. While it is true that the 4Der sees every atom in the object, they only see a part of these atoms -- only the sides that are facing them. The sides that are obscured to us are just as obscured to the 4Der.

[ac2000]

However, we do not see the atoms in the square like this at all! What we are seeing is the blue sides of the cylinders, not the red curved sides. The red curved sides of the atoms inside the square are just as obscured from our view, as they are obscured from the 2Der's view.

Hmm, these are really interesting new aspects, but somehow I'm not completely convinced.

First of all: We could see the red and yellow curved sides of the cylinders, if we would pan our POV slightly to the left or right, couldn't we?

But what I think is a bigger problem for analogy:
The curved sides are a result of the thin "helper"-Dimension, that we just assumed, so we wouldn't have to deal with zero thickness, right?
But if one starts to paint the sides and thinks about them being obscured by the sides of neigbouring cylinders, then one is dealing in fact with the 3rd dimension and not with the 2nd anymore. And that neigbouring things (whether cylinders or atoms) obscure each other in the 3rd dimension for 3d beings is quite normal.
I thought, the question was rather, whether something in the 2nd dimension could be obscured for the 3d being or something from the 3rd for the 4d being.

The greatest trouble with these analogies seems to be that we have to assume a non-zero thickness (which I called "helper"-dimension above) that extends into the next dimension, otherwise nothing would be visible and, e.g. the 2D square, would simply not exist. But as soon as we assume this non-zero thickness, it leads to all kinds of other assumptions and somehow everything gets jumbled up -- and I get the feeling that I think in circles .

What about one of the examples in the earlier posts: when we paint a 3d cube on each side with a number (I wouldn't mind a rather thick layer of paint ) and would also allow an additional extension into the 4th dimension (a "helper"-Dimension), so it could be seen at all be the 4d being:
would you still think that a 4d being in 4D space could see something of all six numbers at the same time? Or do you think they could not, because of the analogy with the neigbouring cylinders/atoms that obscure each other? And if not, what has happened to their 3D retina?

[Hugh]

The greatest trouble with these analogies seems to be that we have to assume a non-zero thickness (which I called "helper"-dimension above) that extends into the next dimension, otherwise nothing would be visible and, e.g. the 2D square, would simply not exist. But as soon as we assume this non-zero thickness, it leads to all kinds of other assumptions and somehow everything gets jumbled up

I agree. Why give thickness to a 1D line when in reality there would be none?

The 2D being can't see a 1D line any more than we can.

Looking at a dimension edge on makes it vanish from view.

[ac2000]

I agree. Why give thickness to a 1D line when in reality there would be none?

Well the first reason, I guess, is that without that thickness it would not be possible (at least for me) to imagine it, and without this it wouldn't be possible to think about analogies from 2D to 3D, and without this I probably could not think much about the 4th dimension at all, because I'm really really bad at maths .

And the second reason might be that, if there was a 4d space the creatures might say the same thing: why should we give thickness to a 3d er when in reality they have none.

[Hugh]

And the second reason might be that, if there was a 4d space the creatures might say the same thing: why should we give thickness to a 3d er when in reality they have none.

No one could deny that 2D creatures have area, and 4D creatures couldn't deny that 3D creatures have volume.

[gonegahgah]

What about one of the examples in the earlier posts: when we paint a 3d cube on each side with a number (I wouldn't mind a rather thick layer of paint ) and would also allow an additional extension into the 4th dimension (a "helper"-Dimension), so it could be seen at all be the 4d being:
would you still think that a 4d being in 4D space could see something of all six numbers at the same time? Or do you think they could not, because of the analogy with the neigbouring cylinders/atoms that obscure each other? And if not, what has happened to their 3D retina?

Hi AC2000, this is an interesting question and it reveals an interesting answer...

Again, I'll use the 2Der to help us. Let's say a 2Der paints his equivalent for the numbers 1-4 one on each of the four 2-faces (what we would call edges) that surround his square. What would we see?

The first trick is that generally we don't look at the square via it's edges; like the 2Der does. So, if the paint is not very thick, then we would probably not even notice that the 2Der had written anything at all. We would have to know to look at the square edges side on in this case to see the markings. Then we could only see two of the numbers at once, just like the 2Der, as we would be placing ourselves in the 2Der's line of view.

In this situation, as we're all saying, the paint and square would be shown as smearing out of the 2Der's dimensions - purely for our convenience as the 2Der doesn't need the third dimension. We are afterall just imagining these things so we are not trying to make 2Ders or 4Ders real. Instead, we are just trying to allow ourselves to see what they would see. Smearing a 2Der's universe sideways into ours is purely for our convenience; and not theirs.

If you did use super thick paint then we would start to notice the disfiguring of the edges from front on. Depending on how big the numbers were and how thick the paint, the square might even stop looking like a square; but that's a lot of 2Der paint. If it were super thick you could even read these from our normal viewing direction; because the edges would be bumpy in the same general form of their numbers. If you knew what you were looking at, and how to translate them to our numbers, then you could work out which edge had which number. And, in this case we could see all four numbers at once; as all the edges would be bumpy.

The same applies to the 4Der. If you paint the numbers thinly onto the faces of our cube then they probably wouldn't even notice them unless they decided to look at the cube 4-edge on. Then they could only see, at most, three numbers at once because they would be in one of our point-of-views. Again, if the numbers were painted very, very thick, they would notice them from their usual viewing direction; and could even possibly translate them if they knew how to. In this later situation they would be able to see all six numbers at once.

Hopefully the next few pictures will make this all clearer too.

[gonegahgah]

Thanks! I owe you the idea of comparing the different POVs of different dimensions in this way.

You can give me a brief mention in your book
Which is a great idea if it can be done; a book that is. Hopefully we can generate more subject matter for your story in these posts.
I don't know if a 4Der story has ever been written before. I would certainly buy it.

I also owe 4DSpace for making me think more thoroughly about dimensional terminology, that eventually led me to realize the far-reaching consequences of objects needing non-zero n-bulk in order to exist in n-space, and what that means for 4D (and higher) visualization. Now I understand why people struggle so much with the idea that 4D (and higher) beings can see every part of 3D space, attributing it to some kind of "magical" x-ray vision or omnipresence or some such. They are thinking in terms of the "sides" of atoms that are visible to us, imagining that somehow the 4D being's sight is magically bypassing the occlusion of atoms in the back by the atoms in the front. But what's really happening is that these atoms have "sides" that are wholly unknown to us -- sides that face directions outside of 3D space. These sides are oriented such that they do not occlude each other, and that is how the 4D being can see all of them at the same time. The occluded sides are still just as occluded as before. While it is true that the 4Der sees every atom in the object, they only see a part of these atoms -- only the sides that are facing them. The sides that are obscured to us are just as obscured to the 4Der.

I concur fully with you on this QuickFur.

[quickfur]

However, we do not see the atoms in the square like this at all! What we are seeing is the blue sides of the cylinders, not the red curved sides. The red curved sides of the atoms inside the square are just as obscured from our view, as they are obscured from the 2Der's view.

Hmm, these are really interesting new aspects, but somehow I'm not completely convinced.

First of all: We could see the red and yellow curved sides of the cylinders, if we would pan our POV slightly to the left or right, couldn't we?

Yes we can. Or at least, we can when we're talking about macroscopic exaggerated-sized atoms, as I have used for illustrative purposes. If our own 3D atoms were big enough, we could in theory see "through" solid rock simply by looking through the gaps between the atoms (at least, we could see quite some distance into the rock, if not the whole way through, by looking along a line that passes through a large number of said gaps). But atoms being microscopic, and even smaller than visible wavelengths of light, means that these gaps don't really factor into what we see. So I didn't bring up this issue in my previous post -- I felt it would distract from the main point of the analogy.

But what I think is a bigger problem for analogy:
The curved sides are a result of the thin "helper"-Dimension, that we just assumed, so we wouldn't have to deal with zero thickness, right?

Correct.

But if one starts to paint the sides and thinks about them being obscured by the sides of neigbouring cylinders, then one is dealing in fact with the 3rd dimension and not with the 2nd anymore. And that neigbouring things (whether cylinders or atoms) obscure each other in the 3rd dimension for 3d beings is quite normal.
I thought, the question was rather, whether something in the 2nd dimension could be obscured for the 3d being or something from the 3rd for the 4d being.

OK, the whole deal with "painting" the sides is really just a way of speaking about how these curved sides of the cylinders occlude each other. Obviously it's impossible to "paint" an atom; paint itself is made of atoms, so to actually paint the "inside" of the square's edges would mean that additional layers of atoms (atoms that constitute the paint) would have to be inserted between the 1-atom-thick perimeter of the square and the 2-bulk (i.e. inner area) of the square. Such additional layers of atoms would certainly be visible from the 3D perspective, since they also would have the "blue lids" which are plainly visible to us.

Further, as you point out, this "helper dimension" really is just a device to avoid having to deal with zero thickness. The curved sides of the cylinders are practically just circles of arbitrarily small width. The 2Der's vision is merely a 1D line, and so even zero-thickness line segments fill up his vision. The thickness shown in the images that I made is greatly exaggerated so that it's clearly visible; I could have made it so small that it's effectively invisible (i.e. microscopic), but then the resulting diagrams would be unhelpful (the red parts would be so thin they're invisible).

The whole thrust of this illustration, though, is to draw the distinction between what parts of the atom the 2Der sees, vs. what parts of the atom we see. Even though we're looking at the same atoms, the 2Der sees only the circular boundaries of them, boundaries which fill up some length in his 1D retina; but we, looking from 3D, mainly see the area covered by the atom (the "lid" of the cylinder), which fill up some area in our 2D retina. Do we see the circular boundaries of these atoms? Yes and no. Yes, because it's obvious that the area we see has a boundary, and that boundary is a circle. No, because what we see as the atom's boundary (a circle) is fundamentally different from what the 2Der sees as the atom's boundary (a line segment).

The 2Der has trouble understanding how we can possibly see the entire boundary of the atom simultaneously -- attributing it to xray vision or omnipresence --, because he is thinking in terms of line segments. To him, to see the entire boundary of the atom requires that one looks at it at least from the front and the back (each time seeing a line segment corresponding to half of the circular boundary). However, what we 3Ders see is fundamentally different. We aren't looking at the atom edge-on at all. We're looking at it from a fundamentally different direction that enables us to see the area that the atom covers. We don't see line segments at all. We see area, and we realize that this area has a boundary. Such a view is very hard for the 2Der to grasp, because he has no experience of it whatsoever.

The greatest trouble with these analogies seems to be that we have to assume a non-zero thickness (which I called "helper"-dimension above) that extends into the next dimension, otherwise nothing would be visible and, e.g. the 2D square, would simply not exist. But as soon as we assume this non-zero thickness, it leads to all kinds of other assumptions and somehow everything gets jumbled up -- and I get the feeling that I think in circles .

What about one of the examples in the earlier posts: when we paint a 3d cube on each side with a number (I wouldn't mind a rather thick layer of paint ) and would also allow an additional extension into the 4th dimension (a "helper"-Dimension), so it could be seen at all be the 4d being:
would you still think that a 4d being in 4D space could see something of all six numbers at the same time? Or do you think they could not, because of the analogy with the neigbouring cylinders/atoms that obscure each other? And if not, what has happened to their 3D retina?

For this part, I agree with gonegahgah's response.

Again, the key point here is that what we see is fundamentally different from what the 2Der sees, and similarly what the 4Der sees is fundamentally different from what we see. When we look at a sphere, for example, we see only half of its surface, the near half, that's facing us. If we want to see the other half of its surface, we have to be standing on the other side of it. So when the 4Der tells us he can see the entire spherical boundary of the atom as well as what's "inside" the atom, we are baffled. How can one possibly look from two different viewpoints simultaneously? How can one see "through" an opaque surface? But what the 4Der is seeing is a whole new "side" of the sphere that we don't have access to. The 4Der sees the volume that the sphere covers, and perceives that the boundary of this volume is a sphere. The 4Der does not see the surface of the atom the same way we see it. We see the surface projected onto a circular area in our eye, and we imagine that the 4Der somehow is seeing two circular areas (the two halves of the sphere's surface, projected into two circular areas) simultaneously. But the 4Der doesn't see such a thing at all. He sees the volume (the "hyper-area", if you like) of the sphere, which we can't see.

[quickfur]

Thanks! I owe you the idea of comparing the different POVs of different dimensions in this way.

You can give me a brief mention in your book
Which is a great idea if it can be done; a book that is. Hopefully we can generate more subject matter for your story in these posts.

Book? I'm a writing a book now? And there's a story? Heh. It was just a story idea. I was just thinking more of an after-the-fact story that presumes the reader already has a little understanding of 4D, that starts out sounding like it's just another version of Abbott's "Flatland", when suddenly it turns out to have been talking about a 4D being dreaming about a "flat" 3D world all along.

I don't know if a 4Der story has ever been written before. I would certainly buy it.
[...]

Oh, there has been. Rudy Rucker has written a novel called "Spaceland", about an encounter with 4D beings that visit our world. I haven't read it myself, though I have read the summary and reviews. From what I can tell, it pretty much presents the same understanding of 4D as we have here. (There is purportedly even a brief episode near the end where the protagonist goes one step higher, into 5D.)

Rudy Rucker is a professional writer... I'm not even close to being called a writer. So I don't think my story will ever attain to what he has.

[4Dspace]

All this is so sad. I feel sorry for the visitors to this site, as they are being mislead by the resident troll here. Yes, I mean the quickfur. Why wendy, "who sees up to 8D" does not interfere into this bacchanalia of dimwittedness and at least help this situation of blind leading the blind?

If we assume that the cube actually has non-zero 4D thickness (ie, it's a tesseract in disguise), then the "inner faces" (or rather, the inward-facing part of the faces) will not be visible.

This is wrong.

From the brief read of CH Hinton's book I think it would still leave the reader thinking that the 4Der sees cubes from our outside point-of-view; which I'm hoping to show otherwise.

How? by placing the POV inside the cube? If you used analytic geometry instead of lame analogies, you would see very quickly that in 4D, a ray from a POV outside the cube cuts through the middle of the cube, thus allowing to see the "inner side" of its far faces. The front faces are seen from outside.

Consider now the analogous 4D case. When the 4Der looks at a cube from their 4D viewpoint, they can see every atom in the cube. But it's not how we imagine it - xray vision, or omnipresence, or whatever. We think xray vision or omnipresence is necessary for this feat because we're thinking in terms of the "curved sides" of the atoms that we're familiar with. (We are not familiar with any other faces of the atoms, since we can't see those faces.) We think that these curved sides, which appear to us as spheres, are all the sides the atoms have. But actually these atoms are not mere spheres; they are spherical cylinders in 4D. They have a completely different aspect, their "blue lids", which we cannot see or access. The 4Der can see all the "blue lids" of every atom in the cube - so in a sense, they are seeing every part of the cube, all six faces, and the entire inside content of the cube. But in another sense, they aren't seeing the inside of the cube - they cannot see the "curved sides" of the atoms, just as we cannot see them, because the surrounding atoms are blocking the view. Neither can they see inside the atoms (the inside of the cylinders); they only see the "lids" of the atoms.

This is so pathetic! Why are you using lame analogies when there are tools of analytic geometry that allow you to "see" what is seen of what sort of object, projected to a given POV, in whichever dimension?

You said you want to talk geometry here, but you're are inventing mythical atoms in mythical lands, populated by mythical beings with weird types of visions. This is not geometry.

Again, If wendy sees up to 8D, why does not she interfere into this blind leading the blind situation? Why the owners of the site allow this troll to continue to spew onto unsuspecting visitors his nonsense? Yes, what he is saying is pure nonsense. And it is easily checked with proper geometry tools.

Again, I'll use the 2Der to help us. Let's say a 2Der paints his equivalent for the numbers 1-4 one on each of the four 2-faces (what we would call edges) that surround his square. What would we see?

First, the answer to this simple question was already given above. What of it you, ac2000, did not understand?

Second, again, instead of geometrical tools, you use... strange, to say the least, analogies. Which keep confusing you. Use compass and ruler and a bit of math. As Euclid did. And everything will be clear.

If you did use super thick paint then we would start to notice the disfiguring of the edges from front on. Depending on how big the numbers were and how thick the paint, the square might even stop looking like a square; but that's a lot of 2Der paint. If it were super thick you could even read these from our normal viewing direction; because the edges would be bumpy in the same general form of their numbers. If you knew what you were looking at, and how to translate them to our numbers, then you could work out which edge had which number. And, in this case we could see all four numbers at once; as all the edges would be bumpy.

The same applies to the 4Der. If you paint the numbers thinly onto the faces of our cube then they probably wouldn't even notice them unless they decided to look at the cube four-edge on. Then they could only see, at most, three numbers at once because they would be in one of our point-of-views. Again, if the numbers were painted very, very thick, they would notice them from their usual viewing direction; and could even possibly translate them if they knew how to. In this later situation they would be able to see all six numbers at once.

This is the ultimate in dimwitted baboonism. Is THIS geometry?

Yes we can. Or at least, we can when we're talking about macroscopic exaggerated-sized atoms, as I have used for illustrative purposes. If our own 3D atoms were big enough, we could in theory see "through" solid rock simply by looking through the gaps between the atoms (at least, we could see quite some distance into the rock, if not the whole way through, by looking along a line that passes through a large number of said gaps). But atoms being microscopic, and even smaller than visible wavelengths of light, means that these gaps don't really factor into what we see. So I didn't bring up this issue in my previous post - I felt it would distract from the main point of the analogy.

Again you're talking about "atoms" when the question pertains to geometry in 4D. You are in violation of this subsection of the forum rules. In geometry, the objects are made up of planes and lines. Both have directions. Instead of "paining" the faces of a cube, you can assign them chirality and assume that the light is reflected off them, say blue, when looked at clockwise, and red, when counterclockwise. This does not confuse you with adding a dimension to an object where it does not have it, yet allows to view this aspect of the structure and how it relates to your POV.

Again, why wendy does not interfere into this bacchanalia hat has nothing to do with geometry?

I feel sorry for the visitors to this site. They are being misled by what's going on here.

The 4Der does not see the surface of the atom the same way we see it. We see the surface projected onto a circular area in our eye, and we imagine that the 4Der somehow is seeing two circular areas (the two halves of the sphere's surface, projected into two circular areas) simultaneously. But the 4Der doesn't see such a thing at all. He sees the volume (the "hyper-area", if you like) of the sphere, which we can't see.

Pray, how on Earth do you know what your mythical being can or cannot see? There are tools of analytic geometry and they allow us to know without a trace of a doubt what is visible and how, from what POV, in what dimension. You, apparently, have the software that allows you to render the images of n-D objects from whatever N dimensions. Why don't you use them, instead of this endless speculative babble?

[gonegahgah]

I'll have to go find that book QuickFur, thanks.

Step 7. Looking at our perspective of the 2Der from above:



Here's a simple overhead look at us viewing the 2Der's square while they do so as well from their perspective.
Thought I'd add this in to bridge the sequence.

Step 8. Looking at multiple of our perspective of the 2Der from above:



To the 2Der they can only perceive of looking at the square, if they are on level ground, from either it's front or it's back.
However, we can see their square through a whole 360deg of sideways angles while it is standing; including from in front and from behind like our 2Der friend.
I've shown our little 2Der again as a line smeared sideways into our dimension.

When immediately in front or behind we can only see the same as the 2Der but when we move into those other angles we can start to see the two 3-sides that they can't see.
We can also be closer or further away at any of those angles.

The question for this picture then becomes how does the 2Der perceive us standing in relation to themselves?

[quickfur]

[...] If you used analytic geometry instead of lame analogies, you would see very quickly that in 4D, a ray from a POV outside the cube cuts through the middle of the cube, thus allowing to see the "inner side" of its far faces. The front faces are seen from outside.

Hooray! Let's do some analytical geometry! Those are always fun.

Let the cube's corners be defined by the points (±1,±1,±1,0). Its faces, therefore, are the following six squares:
1) the one with the corners (-1,1,1,0), (1,1,1,0), (1,-1,1,0), (-1,-1,1,0). This face is perpendicular to the vector (0,0,1,0).
2) the one with the corners (-1,1,-1,0), (1,1,-1,0), (1,-1,-1,0), (-1,-1,-1,0). This face is perpendicular to the vector (0,0,-1,0).
3) the one with the corners (-1,1,1,0), (-1,1,-1,0), (-1,-1,-1,0), (-1,-1,1,0). This face is perpendicular to the vector (-1,0,0,0).
4) the one with the corners (1,1,1,0), (1,1,-1,0), (1,-1,-1,0), (1,-1,1,0). This face is perpendicular to the vector (1,0,0,0).
5) the one with the corners (-1,1,1,0), (-1,1,-1,0), (1,1,-1,0), (1,1,1,0). This face is perpendicular to the vector (0,1,0,0).
6) the one with the corners (-1,-1,1,0), (-1,-1,-1,0), (1,-1,-1,0), (1,-1,1,0). This face is perpendicular the vector (0,-1,0,0).

Since all the corners of the cube have the last coordinate 0, it follows that the inside of the cube must also have the last coordinate 0. This makes sense, because the cube lies in the hyperplane where the 4th coordinate is zero.

Now let's trace some lines of sight! First, let's put our viewpoint at (0,0,0,5), and orient ourselves so that we're looking at (0,0,0,0) directly. So the vector from our viewpoint to (0,0,0,0) is (0,0,0,-5). Let's assume that our field of vision is 45° (a reasonable assumption, I hope, unless we have tunnel vision. Which we don't.). So that means any line passes through our viewpoint (0,0,0,5) and makes an angle of less than 45° with (0,0,0,-5) represents a line of sight from our viewpoint to what we can see.

So. Let's consider the line:

L1 = (0,0,0,5) + k*(0,0,1,-5), where k is free to vary from 0 to infinity.

When k=0, the line crosses (0,0,0,5), which is our viewpoint. Now let's make sure it actually represents a valid line-of-sight. The dot product of the direction vector, (0,0,1,-5), with (0,0,0,-5) is 25. So using the definition of dot product A dot B = |A|*|B|*cos T, we have:

|A|*|B|*cos T = 25

Since the magnitude of the vector (0,0,1,-5) is |A|=sqrt(26), and the magnitude of the vector (0,0,0,-5) is |B|=5, we plug in these numbers and do a little algebra to get:

cos T = 25 / (sqrt(26) * 5)

So we use our calculator to solve for T by taking the arccos on both sides of the equation, and we get T = approximately 11.3°, which is less than 45°. So this line represents a valid line of sight.

Now let's take a closer look at it. When k<1, the last coordinate is always positive. Since the cube lies in the hyperplane where the last coordinate is always 0, this line doesn't intersect the cube when k<1. Luckily, when k=1, the line passes through (0,0,1,0) -- the last coordinate is zero, so this is where the line crosses our 3D hyperplane. And it so happens that (0,0,1,0) lies on face #1 of the cube. So we have found a line of sight from (0,0,0,5) that crosses one face of the cube. Furthermore, since the line only crosses the 3D hyperplane when k=1, it doesn't intersect the rest of the cube. The only intersection is (0,0,1,0).

Now let's look at L2 = (0,0,0,5) + k*(0,0,-1,-5), again with k free to vary from 0 to infinity.

If we take the dot product of (0,0,-1,-5) with (0,0,0,-5), we will see that it's equal to 25, just like with L1. The magnitude of the vector (0,0,-1,-5) is also sqrt(26), so taking out our calculator and solving for the angle again, we find that it's also 11.3°, which lies within our field of vision.

So let's take a closer look at L2. When k=0, it passes through our viewpoint (0,0,0,5). When k<1, the last coordinate is, again, positive. When k=1, then the last coordinate is zero, and that's where it intersects our 3D hyperplane. Plugging in k to the equation of the line, we find that the intersection is (0,0,-1,0), which happens to lie on face #2. Again, since the line only crosses the 3D hyperplane at k=1, this is the only place where it intersects with the cube.

Now, you'll note that face #1 is directly opposite face #2 (you can check this from the coordinates of the cube). But... wait a minute, how is it possible to see two opposite faces of the cube at the same time?! Whoa, this is weird. Let's check out some other lines to see what's going on. Let's look at the line:

L3 = (0,0,0,5) + k*(0,0,0,-1)

Again, we verify that this line lies within our field of vision, by computing the dot product of (0,0,0,-1) with (0,0,0,-5), plugging into the equation for dot product, and solving for the angle. This time, the angle turns out to be 0°. (Unsurprisingly, since this is the line directly at the center of our field of vision.) So let's see. When k=0, the line passes through (0,0,0,5). OK. When k<1, the last coordinate is positive, so it doesn't intersect the cube. When k=0, the line passes through (0,0,0,0). Aha! This is where it intersects the 3D hyperplane. But wait, (0,0,0,0) lies inside the cube, so that means L3 intersects the cube at (0,0,0,0).

Hold on a second here!! When k<1, the last coordinate is positive, and when k>1, the last coordinate is negative. So k=1 is the only place where the line crosses the 3D hyperplane. That means the middle of the cube is unobscured?! So we can see the middle of the cube from (0,0,0,5) without passing through any other part of the cube. Something strange is going on here. Let's look at more lines to see what else is happening.

To save space (since this post is already growing too long) let's consider the next two lines at the same time:

L4 = (0,0,0,5) + k*(-1,0,0,-5)
L5 = (0,0,0,5) + k*(1,0,0,-5)

I'll leave it up to the reader to confirm that both (-1,0,0,-5) and (1,0,0,5) make an angle with (0,0,0,-5) that's less than 45°. So they both lie in our field of vision. Again, as before, when k=0, both lines pass through the viewpoint (0,0,0,5). When k<1, the last coordinates are positive, so they don't intersect the 3D hyperplane that the cube sits in. When k=1, they both pass through the 3D hyperplane. L4 passes through at (-1,0,0,0) and L5 passes through at (1,0,0,0). Since the last coordinate is non-zero for all other values of k, that means these are the only intersections between each respective line and the 3D hyperplane.

Hmm. This is certainly odd! (-1,0,0,0) lies on face #3 of the cube, and (1,0,0,0) lies on face #4. But if you examine the coordinates of the cube, you'll see that face #3 and face #4 are also opposite each other. Crazy, huh? Apparently from (0,0,0,5) we can see not just one pair of opposite faces, but two!

What about the next two lines?

L6 = (0,0,0,5) + k*(0,1,0,-5)
L7 = (0,0,0,5) + k*(0,-1,0,-5)

I won't repeat the computations here -- by now it's clear how to do it. The angles that these two lines make with (0,0,0,-5) are both <45°, so they are also in our field of vision. When k=0, they cross the viewpoint (0,0,0,5). When k<1, the last coordinates are positive, and when k>1 the last coordinates are negative. When k=0, the last coordinate is zero. So that's where each respective line crosses the 3D hyperplane. A little algebra shows that the intersections are (0,1,0,0) and (0,-1,0,0), respectively. Checking the coordinates of our cube again, we see that (0,1,0,0) lies on face #5, and (0,-1,0,0) lies on face #6. Again, since the only time the last coordinate is zero is when k=1, that means these intersections are not obscured by anything else; there is an unobscured line of sight from (0,0,0,5) to (0,1,0,0), and also from (0,0,0,5) to (0,-1,0,0).

Huh. So now we have a situation here. All six faces of the cube are unobscured!! Not only so, it seems that the middle of the cube is unobscured, too. And there is no secret transparency of the cube's faces that allow us to see all 6 faces at once, either -- the lines of sight we traced aren't even in the 3D hyperplane until they intersect the cube at points on all six faces. The cube is solid and opaque, and yet we can still see all 6 faces and its middle simultaneously. Neither is there any omnipresence; all our computations started from a single viewpoint (0,0,0,5).

I tell ya, this analytical geometry thing is weird. All we did was to follow basic geometric computations, and we end up with such strange conclusions. No wonder people think math is hard.

[gonegahgah]

Step 8b. The changing shape of different viewing angles:



When we hold a square in the air before us perpendicular to our face it looks exactly like a square, in this picture, a red square with green edges.
As we've mentioned, when you turn it end on you can only see the edge so the square, in this picture, ends up looking like a green line.

But also, if you hold it at various rotations in between, the square appears to change from a full square to a line, and narrowing rectangles in-between, before our very eyes.
Of course it retains the same shape and we know it is a square.

Exactly the same thing happens for a 4Der turning a 3D cube around (though I hope to demonstrate that later; not yet).
They will see the green 4-edges (our six faces) surrounding the red inside through the two 4-faces in the extra two directions available to them.
The shape will change as they rotate from the cube's 4-edge on through to it's 4-face on.

On a separate note:
I realised this morning that this series of pictures seems to be leading us full circle to something like what I was considering early on when I joined this forum.
So, I'm now even believing that it will be possible to create a 4D engine afterall that will give as a much better view of what a 4Der sees and helps us to understand better.
It occurred to me that such a new 4D engine should allow us to observe a cube (and other things of course) in the ways I'm describing.
If I'm right then we will even be able to observe the phenomena, that I'm talking about now; where you hold the cube and look at its two 4-faces,...
... and when you rotate it, towards its 4-edge; like we rotate the square to it's edge (or 2-face), it resolves nicely into the cube that we know.
Let's hope so hey? This should hopefully answer the questions, on how it could be plotted, that you were seeking QuickFur.

Unfortunately, it takes awhile to draw each picture. The way QuickFur is doing the pictures is much better than my way.
I'm trying to use a 2D-drawing program whereas QuickFur is more sensibly using a 3D-modelling program.
But, I'll keep plugging away...

[gonegahgah]

I shouldn't be stating an ability to be quite as efficacious as that. Unfortunately we still won't be able to see bulk.
But I do forsee that we will however be able to manoeuver within a 4D environment and also understand what we are doing.
It should be possible to show actual 4D based objects that - through the model used - make sense to the player.
And this should also hopefully help to give us a greater, clearer sense of 4D that is accessible to more people.

[gonegahgah]

Step 9. The 3Der looking at the 2Der's upright square from various different angles:



This should really be the next step rather than the previous image.
Here you have the 3Der standing around the 2Der's square from various angles and distances available to the 3Der.
How is the 2Der meant to squeeze their concept of all our locations into their limited 2D space?

Step 10. Overlapping co-location method:



One of the ways we have and can imagine 4D is to imagine that everything basically exists in the same place as our 3D stuff but is just a distance off somewhere in the 4th direction. In our 3D we could depict a shadow to indicate there is something there and if we move enough into the 4th direction at the right angle then we will bump into the object eventually.
The 2Der might do the same and do this by imagining that 3D is just a series of 2D planes. The could then squeeze all those 2D planes together and have a shadow in their plane where an image in another plane is co-located with their plane when compacted.
Moving at the correct 3-angle would allow them to bump into our objects through a bit of hit an miss movement into the hidden 3rd direction.
They could also walk directly towards the shadow and then try walking 3-left or 3-right, hopefully picking the right one, until they bump into the object.
Also, objects more in line with their front would obscure objects that were behind; though for our 3D world we have more resources available to avoid that problem a bit.

Step 11. 3Der's shadowy presence in the 2D world:



This picture gives a few indications of the problems of co-locating. The orange fellow is our original side on view.
When you have two 3Ders perpendicular to the 2Der's plane of existance behind each other then you can not see that the 2nd person is there.
This also occurs when they are on opposite sides of the 2Der's plane and in line with each other on the perpendicular.
You can see by the shadows depicted that our little 2D person is co-located with one of the 3Ders already in the picture.

So this approach is a bit hit and miss. It also doesn't really help them to conceive how 3D really works.

I'll show them both side by side for better comparison:



But, the good thing this picture shows us is that the 3Der does not have to be inside the square to see it's insides.
The 3Ders also don't look through the edges - which are the 2Der's only perspective - to see the insides.
When we do look through the edges we can only see those edges too; just like the 2Der is limited to seeing.

[quickfur]

[...] Unfortunately, it takes awhile to draw each picture. The way QuickFur is doing the pictures is much better than my way.
I'm trying to use a 2D-drawing program whereas QuickFur is more sensibly using a 3D-modelling program.
But, I'll keep plugging away...

Actually, I don't know how much better using povray is. It took me a whole afternoon and evening just to set up the models used for rendering the images that I posted. And I was already using very simplistic models for the 2Der and 3Der. The only real advantage that I see is that once you have your models set up, it's very easy to render the scene from all kinds of different viewpoints. But the models do take a long time to set up.

[gonegahgah]

Actually, I don't know how much better using povray is. It took me a whole afternoon and evening just to set up the models used for rendering the images that I posted. And I was already using very simplistic models for the 2Der and 3Der. The only real advantage that I see is that once you have your models set up, it's very easy to render the scene from all kinds of different viewpoints. But the models do take a long time to set up.

Oh okay. It is easier to throw a few circles and lines together in CorelDraw but then, as you say, the trick for me is trying to imitate those viewpoints. Yours are certainly much better looking views. I don't think that is just because of the 3D (which does look nice) but also because once you have your actors set up you can move the camera and you get the best shots.

I was dreading trying to draw the multiple 3D actors in the last picture that are appearing to look from different directions. So I cheated a little and mainly only turned the head (or the eyes in fact). Done in povray the same scenes would probably look fantastic.

I don't imagine it would be very difficult to produce arrows and dotted lines in povray?

[gonegahgah]

Step 12. The 2Der can only see the 3Der in front; or their shadow if they are off in third direction:



Still using the co-location planes then our 2Der is only going to be able to depict a shadowy figure in front of themselves in this picture.
This tells them that there is 3Der in front of them but who is off in the 3rd direction somewhere.
If they walk through the shadow they will be able to see the 3Der, behind the shadowy depiction, whom I've given a nice maroon shirt too.
If instead they move themselves somehow through the sideways adjacent 2D panes then they will eventually see the 3Der whom I've give a nice blue shirt too.
The 2Der would have to depict the 3Ders pretty much like a 2Der. This is fairly unavoidable no matter what we do.

Step 13. The 3Der can see many more 4Ders and the shadowy depictions of those off in the 4th direction:



The same thing happens for us if we use the co-location planes method of overlapping other 3D planes over ours to create a 4D universe.
We will see our 4D friends who intersect with our 3D plane and will only see shadowy images for those who are off in the fourth direction (either ana- or kata-wards).
You can see a shadow inside the cube. 4D creatures can no more exist inside our solid cubes than we can...
... but using the co-location method they can be co-located with the cube but be off somewhere away in the 4th direction.

Also (just like for the 2Der) when the higher dimensional beings are behind each other in the 4th direction, we can only see the foremost 4Der closest to us in the 4th direction.
Again, they could either be ana-wards or kata-wards to us; so might be either side of us in the 4th direction.

The shadowy 4Der that is behind the cube can't be seen by us because the cube is in the way.
Again I've got a 4Der co-located with us; as I had a 3Der co-located with the 2Der.
This is fine because they are not in us but off in the 4th direction somewhere in an adjacent 3D plane.

I've got our 4Ders looking up to show that they don't look at us from the same direction that we look at things.
They look at us from 2 inside directions that we can't see.

We would have to depict the 4Ders pretty much like us 3Ders.
This is fairly unavoidable no matter what we do just as the 2Der will probably have to draw a 3Der pretty much like themselves.