Building 4D objects and worlds

[gonegahgah]

It should be possible for someone to write a program to render 4D worlds shouldn't it; and to determine the shape of 4D devices and their purposes?
I would like to talk about the shape of objects in a 4D world in this thread as well but I just wanted to ask these questions first.

[Mrrl]

gonegahgah,
there is no problem to write 4D viewer/renderer. We may select one of many options, that will give more or less adequate representations on 4D. You may draw 3D section of 4D and give opportunity to shift view direction to parallel sections (turn head in 4D), or project 4D to 3D (by parallel projection) and render the resulting space, or make double central projection, or build central projection (3D) and draw result by semitransparent colors...
Main problem will be in model constructions. I can imagine 4D editor with features "add/remove tetrahedron to the surface" and "move vertex parallel to screen", but I'm not sure that somebody will be able to draw something usable by it. Or we may design high-level markup language and implement features for construction of 4D objects. In that case you have to imagine your model in all details, calculate coordinates of points and code the model in text. Third way is high-level parametric 4D editor where you can edit coordinates of anchor points of different features in graphic view. Possible, but not very easy to write such thing as well as use it.

[gonegahgah]

hi mrrl
What I would mainly want is for a viewer app to allow us to view a 3d cross section of a 4D built world but that would allow us to move and rotate our 3D cross section in the 4D space as though we had 4D control over our 3D space. Just like for a 2D worlder being able to move their plane around our 3d world by rotating it left/right around themselves &/or moving it sideways in ours.
I wonder also what a 4d world builder would look like. As well as the x, y, z views would you also need xy, yz, zx views to depict the 4th dimension relative to each of these?

[gonegahgah]

It's much easier to think like a 2Der in a 3D world and that helps us to think in a 4D world. The 2Ders rendering program would simply have a front edge and an above edge (whereas we have front face, top face, and the extra side face. The 2Der would be fully dependent on shade to depict different 2D shapes in their world whereas we can simulate faces of 3D objects with just lines; and only need shading where curve is involved.
However, we/they can imagine lines going from the edges of those 2D shapes to the edges of other 2D shapes in another dimension.
The same goes for us. We don't have to connect the insides of our solids only the surface area.

Having straight lines and corners makes it even easier as we only have to connect lines from the corners then to other corners of 3D objects in the forth dimension to make our 4D shapes. In this case we can decide the co-ordinates of other corners in the forth dimension and simply connect to those.
The problem in depicting it is that you will have these co-ordinate points occuring through simultaneous space if you overlay them all at once.
I would suggest having different size dots at corners to depict how far into the fourth dimension a co-ordinate is.

You can see the challenge when it comes to a simple tesseract (or the extension from dot to line to square to cube to...). If you overlay them all you will still only see a cube in reality on the screen (unless you throw in a additional level of psuedo-perspective). But by having bigger dots behind the dots in the 'nearer' 3rd dimension then you can see that the cube is extended into the 4th direction as well; although we still have to understand that there are a line length number of cubes between the front cube and the back cube of our tesseract.

[gonegahgah]

The fun really begins when your 4D object moves through a series of 3D shapes. To us it would simply look as if the object were morphing before our eyes.

It is further fun when the 4D corner points don't align up in the same 3D spaces. That is that a 4D corner arrives on one line extended into the 4th dimension before it arrives on the other lines extended into the 4th dimension.

Again back to the 2D/3D world to explain better. If we had a cube but cut off just one corner and lead with one of the full faces through the 2D worlder's plane at 45 deg to them so they can see 2 edges, then they will first see no change as it passes through but at some point they will see the 2 edges change into 3 edges with the 3rd middle edge starting off as nothing and growing bigger at an even rate until the whole thing suddenly disappears.

[gonegahgah]

Further confusion arises when you start turning an object in 4 dimension. Things then become harder to recognise; as if it weren't difficult enough already.

Going back to the 2D world that we are passing a cube through. If you keep a square face parallel to the 2Ders world as you pass it through then they will always see two edges of equal length or otherwise of varying shade to show they are really of equal length because one may look longer because it is turned more to the 2Der. So they can still figure out at a known constant rate of movement that there is a cube.

If however you tilt the cube towards their 2D world, so that one edge enters first, then they will no longer see equal length edges as one edge will be a part diagonal of the cube and the other edge will be the width of the cube. It will no longer appear to be the edges of a square; which it isn't in their 2D world. As it moves through they will see the 2D edges get longer, remain constant while moving, and or get shorter; while not appearing to match what they would expect from an object in their world rotating. So for shapes with some regularity, the trick is to try to align your view of it so that it is more meaningful in you dimensions. Oddly shaped things become even harder to discern.

This is the same for our steering ourselves through a 4D world. We have to try to align ourselves so that we can find some regular shapes and move our 3D world through these to try to discern what they are.

[quickfur]

Cross-sections are fun, but ultimately limited in usefulness especially when it comes to visualization:

http://eusebeia.dyndns.org/4d/vis/03-xsec

[gonegahgah]

Hi quickfur,

Lovely diagrams, thanks. The other thing for the 2Der is that they don't even get that much information that we are seeing for them. That cross-section of the cylinder passing through their plane; they only see a shaded line for. So, whereas we can see their cross-section in all its fullness they can only see the edge of the cross section closest to them. So you are definitely right that they have limited information to work with.

On the other hand, if they were trying to depict a 3D cylinder in their world, all they could do would be to draw part of a shaded circle edge and draw a line from this to part of a shaded circle edge above. Obviously, this is a fairly poor representation of a 3D cylinder. Perhaps one of the artists here could show what I mean?

The same thing occurs when we try to draw 4D objects as a single object. There is just so much information lost that we get a false impression of what the 4D object actually looks like. In some respects I tend to think that it can lead to greater miscomprehension of the nature of 4D objects.

One thing we can do is to add a memory feature. A memory feature is a way of showing us remembering what we saw as we pass our scanner through an object. They are simply time lapse scans of the object placed side by side; or above each other for the 2Der. Fortunately, we can put the 2Der's scans side by side to get a fairly good picture of their memorised scans of an object even from their perspective.

If we apply this to 2D objects it gives us a better chance of discovering what they are. Let's take an aerosol can. Now, as you point out, orientation is everything, and one of the first tasks is to try to find an orientation that will be meaningful to us. This is where being able to rotate the various axes and do experimental scans becomes vitally important so that we can find memorised scans that become more meaningful. For our aerosol can example I would suggest some good 'angles' to determine the object would be side on or front on. If you saw these lines one above the other it would be harder to tell what you are seeing but in our 3D world we can put them side by side and see a fairly good representation of an aerosol can from its shading. Perhaps one of the artists here could show what I mean by this too?

Fortunately for us we have 3D but even as for the 2Der - who stacks their scans above each other - we must stack our scans side by side, which does make it a little harder to determine what the 4D object is. But, we can apply the memory scans also and do experimental scans of an object in various rotations until we are able to determine what an object does, hopefully.

I think it would be good to first try and explore what a 2Der would make of some of our objects. The first one I'd like to look at is an outdoor tap with 2 prongs on top. Perhaps someone could kindly provide a picture?

[wendy]

The two most common representations of 3d onto 2d, are maps and pictures. In pictures, things at the top of a picture will fall to the bottom, as is what we see in 3d. In maps, things at the top don't fall to the bottom.

When you start making things in four dimensions, this is the starting point. You visualise in four dimensions, the way ye read maps, and look ate

[quickfur]

[...]On the other hand, if they were trying to depict a 3D cylinder in their world, all they could do would be to draw part of a shaded circle edge and draw a line from this to part of a shaded circle edge above. Obviously, this is a fairly poor representation of a 3D cylinder. Perhaps one of the artists here could show what I mean?

This touches on the central issue of 4D visualisation. The point is that even though 2D creatures can only see in 1D, their mental model of the world is ultimately 2D. At the very least, they should be able to comprehend the shape of the boundary of things in 2D. For example, even though they only see a maximum of 3 edges at a time, they should have no problem visualizing a hexagon in their mind. And that is the key word: mental model.

You are right that no drawing in 2D (which would necessarily be 1D since otherwise you can't draw it) can adequately capture a 3D object. However, if you could somehow convey the 2D-ness of a 3D projection into their mind, so that in their mental model they see the projection as we see it, then they will be able to, with training, infer 3D depth in the projection and construct a 3D model in their mind.

So I'm not saying that just by staring a pretty pictures you'll be able to magically understand 4D. After all, we are projecting from 4D to 3D, then rendering the 3D on a 2D screen. Losing 2 dimensions in the process doesn't convey enough information to be able to reconstruct the 4D object accurately. However, the idea is that our mind is very facile at constructing 3D models from 2D images. So if we could somehow convey the shape of the 3D projection using 2D images so that our mind can reconstruct a mental model of the 3D projection, then we shall be able to see that object as a 4D being sees it. Then we just need to learn to infer 4D depth and we will be able to reconstruct, with practice, a model of the actual 4D object.

So by using this method, projections will actually help you to visualize 4D, since the reduction to 3D gives you the mental bridge to make that transition. With cross-sections, there is no way you can make any sense of a series of slices unless you already have some facility with assembling slices in 4D. So there is no bridge here; you have to already have made the leap into 4D before this information is of any use. Moreover, the information it provides is piecemeal: things like facets, edges, vertices, etc., are all indirect: you have to already be able to manipulate objects in 4D in your mind before you can easily infer such things from a series of slices. The projection method, in comparison, is a far superior method for making that initial transition into 4D visualization.

[gonegahgah]

Hi quickfur. I agree with both of us. It is very much so that even what I propose fails to help people to understand the 'sideways' nature of a 4th dimension in many respects. I think both approaches have their pitfalls that it is difficult to address. It's a bit of talk across on my posts but even gravity remains with one down direction despite the extra dimension. Admittedly it is a lot stronger down as we have a whole added magnitude. That is a line has a certain mass, then you go to square which has the next magnitude of mass more, then you go do a cube which has the next magnitude of mass more, and then you go to a hypercube and you have the next magnitude of mass. This corresponds to the 'volume' of course. And it's very easy to see that the 'projection' approach drastically fails to convey this mass. It also fails to convery the downward-ness as the objects are projected without respect to downward-ness. A cube, even if rotated, is in respect to the ground; but a 4D cube projection isn't because part of the ground is off in the 4th dimension and fails to be represented. Whether you have a 2D circle, a 3D sphere, or a 4D hypersphere, the centre of gravity is always towards the one single centre in one direction only. The memory slice certainly shows the mass a little better but fails to show the connectedness of the innards to the extra dimension; whereas the connectedness of the innards of an ordinary 3D object is fairly apparent. The connectedness of the innards is a little better shown in your projections than memory slices. Now, the 4D worlder would see their hypercube as having 8 'sides' but they wouldn't see them as cube 'sides' in the sense that we recognise a cube; just as we don't see a cube as having 'square' sides in the sense of what a 2Der views a square to look like. Neither approach - projection or memory slice - can give a good representation of the 4D world. But I think the real question is which one will allow us a greater chance to determine and/or discern what is happening in a made up 4D world. I would enjoy discussing this if we may.

[quickfur]

Gravity is not stronger in 4D; in fact, it is weaker, because it falls off in an inverse cube relation as opposed to the inverse square relation in 3D. Of course, this is partially compensated by the fact that a full-bodied 4D object has mass proportional to r^4, so the bulk (hypervolume) of larger objects increase very quickly.

I fail to see how projections "convey" this mass. Our own eyes in 3D see only 2D projections of objects; from the perspective of a 2D creature, this only conveys the area of the object, not its volume (which mass is proportional to). Even we ourselves can only deduce the volume of what we see based on previous experience. For example, when you look at a cube and see its 3 tetragonal faces, based on previous experience with cubes you infer that there are another 3 faces on the far side, and based on that assumption you infer the approximate volume of the cube. However, if those 3 tetragonal faces are the only thing you see, you have no way of knowing whether it is even a cube... for all you know, it could just be 3 squares glued together, and there is nothing on the other side, there is no volume at all. So in this sense, even our own eyes deceive us; to be absolutely certain that we're looking at a cube and not just 3 squares glued together, we would have to handle the object with our hands to feel out its shape, or look at it from another angle to confirm that, yes, there is something else behind those 3 faces. This is certainly a limitation of projections, but based on our everyday experience, we can fairly confidently assume that when we see that configuration of 3 tetragons, we're actually looking at a 6-faced cube. So the idea behind visualizing 4D via projections is to learn what kind of assumptions are reasonable from a 4D being's perspective; then when we see that rhombic dodecahedral projection of the 4-cube, we can fairly confidently assume that we're actually looking at a 4D cube, and not just 4 cubes glued together with nothing else behind them, and based on this, we can infer the 4D hypervolume that is enclosed.

And about visualizing the ground - well, this depends on your view angle. Even in 3D, if a cube lies on the desk and you look from the edge of the desk, the surface of the desk projects to a single line, so you don't see the "ground" that the cube is rotating on either. But the usual, more comfortable view angle is by looking from a point above the desk, so that the surface of the desk projects to a tetragon over which the projection of the cube is overlaid. From a 2D being's point of view, it looks like the cube is "embedded" in the tetragonal image of the desk's surface, but of course, we know that the cube lies above the desk in the 3rd direction. Similarly, a more comfortable 4D viewpoint would be from a point above the 4D desk, where the surface of the desk would project to a hexahedral volume in the background, with the image of the 4D cube in the foreground, apparently "embedded" in the hexahedral volume. But by dimensional analogy it is easy to see that the 4D cube is actually sitting above that hexahedral surface.

As for "sides"... a healthy dose of dimensional analogy will easily show that the volumes in a 4D projection are merely surfaces in 4D. Just as the image of a cube in our 3D eye's retina occupies a hexagonal volume -- and if a 2D creature were to look at it, it may mistakenly think that the cube is merely a hexagon, but we know better because we realize that those tetragonal areas in the projection are actually perfect squares in 3D, and they only appear as "squished" tetragons because they are being seen from an angle -- so in the same way the image of a 4D cube in a 4D being's eye occupies a rhombic dodecahedral volume. When we look at the 4 parallelopipeds in that volume, we tend to interpret it in a 3D-centric way, as though the 4D cube were merely a rhombic dodecahedron. But the 4D being knows that these parallelopipeds are actually cubes, and they appear as "squished" parallelopipeds because they are being viewed from an angle. So obviously, for us to correctly interpret 4D projections, we will have to learn how the 4D being interprets them; once we have mastered that, then these projections help us to have the same visual experience as a native 4D being living in 4D space in a way that no slice-based method can. You may regard this as an inherent bias due to the fact that our own eyes see in projections; if our sight were to be based on slices instead, we would definitely find 4D slices the easiest way to understand 4D space. The usefulness of projections lies in the fact that our brains are wired to interpret 3D projections -- it is so instinctive and spontaneous for us, that we can exploit it to great advantage by generalizing it to 4D projections. The similarity, at least in principle, of 4D projections to our native instinctive visualization by 3D projections is what makes the projection method arguably the "most intuitive" method of 4D visualization.

As for how 4D beings interpret 4D projections, I did write up a little discussion here, which does address some of the points you brought up.

[gonegahgah]

Hi quickfur. I think we agree on most points. But there are a few things I would like to distinguish.

I just want to clarify something. I used the term 'volume' (with quotes for that reason though it bares explanation) to generally refer to each levels comparative term (ie. area, volume, hyper-volume). Also, just quickly on the gravity - though its not the main point I'm thinking of - at the 'surface' the 4D gravity would have as units L⁴; whereas at the surface the 3D gravity would have as units L³. (L is used to refer to length; 4D 'surface' is hypersurface or volume). A number raised to 4 gives a much greater value then the same number raised to 3. So gravity is a magnitude greater on a 4D surface. It is only once you move away from the surface that the gravity drops off at the much greater rate for the 4Der. So on the surface a 4Der would weight much more than a 3Der, which is where we tend to think of something's weight.

But back to the main thing that I think that is important and it is exactly the page that you have provided. We may think that we are giving some impression of what a 4Der is seeing through drawing hyper-perspective but I hope the following shows how fraught that is.



The first image is our psuedo image of a cube. It gives us a sense of volume and a sense of its 3Dness. No problem for us. But when we represent a 4D object in 3D with your projection method it is akin to drawing a cube like the right hand diagram. The front face of the cube is shown as the four frontmost diagonals, the rear face as the four rearmost diagonals, and the 4 side faces are shown as each respective group of side facing diagonals. Suddenly, from this image, we lose the whole sense of the true volume of a cube.

The same basically goes for depictions of tesseracts. All we are showing basically, and as you refer to of course, is the surface (or hypersurface) of the tesseract. Just as for my right hand diagram - where the volume is hidden in the cube diagonals; or connecting congruences (shown as lines) between the 'faces' - the hyper-volume for tesseract depictions is actually hidden in the lines between the hypersurface cubes that are depicted. It might even be better to depict a tesseract with fattening lines to give a hint that the hyper-volume actually resides in the connections between the cubes. Maybe like this:

[quickfur]

[...]
I just want to clarify something. I used the term 'volume' (with quotes for that reason though it bares explanation) to generally refer to each levels comparative term (ie. area, volume, hyper-volume).

It's obvious that terms like "volume", "face", etc., will eventually lead to ambiguities when we cross dimensions. That's where things like Wendy's polygloss comes in - i believe she has "bulk" for the n-dimensional volume in n-space. In any case, terms like "volume" or "area" should be defined when there's a possibility of ambiguity, to avoid confusion.

Also, just quickly on the gravity - though its not the main point I'm thinking of - at the 'surface' the 4D gravity would have as units L⁴; whereas at the surface the 3D gravity would have as units L³. (L is used to refer to length; 4D 'surface' is hypersurface or volume). A number raised to 4 gives a much greater value then the same number raised to 3. So gravity is a magnitude greater on a 4D surface. It is only once you move away from the surface that the gravity drops off at the much greater rate for the 4Der. So on the surface a 4Der would weight much more than a 3Der, which is where we tend to think of something's weight.

You're conflating mass with gravity. Gravity as a force is constant if you assume surface gravity on a spherical planet. Whether it has greater or lesser value has nothing to do with the weight of objects; it depends only on its distance from the planet's center. Now, mass on the other hand, will increase faster with increasing radius -- I say radius because we are presuming that increasing size means increasing the measurement of the object in all 4 dimensions. If you're only increasing measurement in 1 dimension, then obviously mass only increases linearly.

So because of the additional dimension, things with slightly larger radius will increase in mass much faster than in 3D, so in that sense they will feel their weight more. That doesn't mean gravity is stronger; it just means that you need to add a lot more mass in order to increase the size of something.

But back to the main thing that I think that is important and it is exactly the page that you have provided. We may think that we are giving some impression of what a 4Der is seeing through drawing hyper-perspective but I hope the following shows how fraught that is.



The first image is our psuedo image of a cube. It gives us a sense of volume and a sense of its 3Dness. No problem for us. But when we represent a 4D object in 3D with your projection method it is akin to drawing a cube like the right hand diagram. The front face of the cube is shown as the four frontmost diagonals, the rear face as the four rearmost diagonals, and the 4 side faces are shown as each respective group of side facing diagonals. Suddenly, from this image, we lose the whole sense of the true volume of a cube.

OK, you need to explain yourself more clearly. I don't understand where the diagonals come in. Where in this image do we see any diagonals?



All the lines in this image are actual edges on the tesseract, and lie on its surface. I don't understand where your diagonals come from.

The same basically goes for depictions of tesseracts. All we are showing basically, and as you refer to of course, is the surface (or hypersurface) of the tesseract. Just as for my right hand diagram - where the volume is hidden in the cube diagonals; or connecting congruences (shown as lines) between the 'faces' - the hyper-volume for tesseract depictions is actually hidden in the lines between the hypersurface cubes that are depicted.

OK, I still don't understand where the diagonals come from.

But I think I get what you're saying. The 3D-ness of the volume enclosed by a cube (or equivalently, the 4D-ness of the hypervolume enclosed by the tesseract) is not immediately obvious from the projection. You could interpret the above tesseract projection in a purely 3D sense: it's just a bigger cube enclosing a smaller cube with sticks joining the two together. Similarly, if a 2D being were faced with this projection of the cube:



It could interpret this as a completely flat square with a smaller square stuck inside, with 4 sticks joining the two. So in that sense, the projection doesn't really convey the 3D volume enclosed by the cube (and similarly, the 4D projection doesn't convey the 4D volume enclosed by the cube).

So we must ask ourselves, why is it that when we look at the above image, we see the 3D depth of the cube, but when a 2D creature looks at it, it only sees a flat square? The reason is simply because our mind is reconstructing a 3D model based on what it sees in the 2D projection image. This reconstruction is done based on certain principles, which is essentially a set of assumptions built from both instinct and previous experience, that allows our mind to (most of the time) reverse the projection back into the 3D object.

So in this sense, projections are no silver bullet to magically make you see 4D; you have to actively interpret the projection image in a 4D way. Basically, the idea is to try to imitate a 4D being's mind in reconstructing the 4D object from its 3D projection, so that, hopefully, we can also do the same and thereby grasp the 4D-ness of the image. You can't just stare at the image and think that you will suddenly see 4D; you have to learn and actively apply the reconstruction principles that a 4D being would use to recover the 4D depth information from the 3D projection.

It might even be better to depict a tesseract with fattening lines to give a hint that the hyper-volume actually resides in the connections between the cubes. Maybe like this:

I'm not sure how fattening the lines help you see in 4D? Wouldn't you just see that instead of lines for edges, you have lines that are fat in the middle? I'm not sure I understand how that helps you visualize a 4D hypervolume.

[quickfur]

P.S. With respect to the difference between gravity and mass: the force of gravity is constant; but the total amount of force experienced by an object depends on how massive the object is. This is because every atom in the object experiences the same constant force, so if there are more atoms in the object, the total amount of force experienced is greater. This doesn't mean gravity is stronger; it just means the object is bigger (i.e. there are more atoms in it).

[gonegahgah]

Hi quickfur. I'll get back to the gravity later. For now I just want to focus on the representation of things.

Just to clarify, the rightmost image of:


Rather than depicting in any form the tesseract, it is instead representing the depiction of a cube with the same level of detail that presenting a tesserat in 3D achieves.
Back to the leftmost normal diagram, each face of a cube is connected to each of the other faces by the volume in between. This is the '3rd dimension volume' between each of them. The squares serve only as faces.

The same goes for the tesseract. The 8 'cube' 'faces' serve only as faces and not the connection between those faces. The 8 faces of a tesseract are all cubes but the faces are not the space in-between just as the square faces of a cube are just the faces and not the space in-between. And just as the area of the 6 square faces does not represent the volume of the cube; the volume of the 8 cubes does not represent the 'bulk' of the tesseract.

The diagram on the right is simply a representation of a cube removing the space in between; just as we are forced to leave out the hyperspace of the tesseract when depicting it in 3D. The only way to remove the space inbetween for a cube is to suck out the insides. Using this metaphor, this then (if we leave the corners where they are) turns the cube into an 8 pointed star where each point is a line to the centre. Each 'face' of the cube is then represented only by a set of 4 adjacent lines joining at the centre.

In a similar fashion we can't depict the 'insides' or 'bulk' of a tesseract in 3D. The right hand diagram shows what a cube would look like if we couldn't depict its insides either.

This is then where I say that not only are the faces of the cube represented by sets of adjacent lines radiating out from the centre but also these lines represent also the volume or 'bulk' of the cube; because there is nothing else in the diagram to represent them. And this also occurs when we depict a tesseract in 3D. The drawn 'cubes' represent - not well - the hyperfaces of the tesseract. But they don't represent it's 'insides' or 'bulk'. Instead each of the edges connecting each cube to each other cube actually represents the 'insides' or this 'bulk'.

Unfortunately, in 3D I can't depict a cube with square faces but with no space given to the insides (except as I have done). Unfortunately in 3D we can depict a tesseract with cube 'faces' but with no space given to the insides. The right hand diagram is meant to give us an impression of how different a cube looks to us if we try to draw it in the same way we are attempting to draw tesseracts. What we see a tesseract's cube 'face' like is nothing like what a 4Der sees those same cube 'faces' looking like. Just as the 2Der sees the sides of a square completely differently to how we see the sides of a square.

That is also why I was trying to depict that the connecting 'lines' as actually representing more than they appear to represent by making them not look like normal lines. Instead of just representing lines connecting the cube 'faces' together; they also represent the 'insides' of the tesseract; just like the lines in my anorexic cube do for its insides.

So basically, the rightmost picture is more towards what a cube would look like if we tried to draw it the same way that we are drawing tesseracts in 3D.

[quickfur]

[...]
The diagram on the right is simply a representation of a cube removing the space in between; just as we are forced to leave out the hyperspace of the tesseract when depicting it in 3D. The only way to remove the space inbetween for a cube is to suck out the insides. Using this metaphor, this then (if we leave the corners where they are) turns the cube into an 8 pointed star where each point is a line to the centre. Each 'face' of the cube is then represented only by a set of 4 adjacent lines joining at the centre.

OK, this is an interesting transformation of the cube, but I don't see what it has to do with the matter at hand. What you've done is to reduce each face to zero and bend the edges of the cube inwards so that they touch the center. But these lines don't represent the volume of the cube at all. They are just the cube's edges (not even faces) that got bent inwards so that they touch each other at the center.

[...]This is then where I say that not only are the faces of the cube represented by sets of adjacent lines radiating out from the centre but also these lines represent also the volume or 'bulk' of the cube; because there is nothing else in the diagram to represent them.

OK, I'm not following this logic at all. How do these lines represent the volume of the cube? As far as I can tell, they are just the 12 edges of the cube bent inwards into L-shape so that they merge with each other. I don't see how this has anything to do with volumes.

And this also occurs when we depict a tesseract in 3D. The drawn 'cubes' represent - not well - the hyperfaces of the tesseract. But they don't represent it's 'insides' or 'bulk'.

OK, I follow up to this point.

Instead each of the lines connecting each cube to each other cube actually represents the 'insides' or this 'bulk'.

But this I don't understand. What do you mean by "lines connecting each cube to each other cube"? There are no such lines in the tesseract; its cube facets touch each other at their square faces. The cubes are not separated from each other by any lines at all.

Unfortunately, in 3D I can't depict a cube with square faces but with no space given to the insides (except as I have done). Unfortunately in 3D we can depict a tesseract with cube 'faces' but with no space given to the insides.

Perhaps if you shrink the 8 cubes into nothing, and then collapse its faces and edges inwards until they collapse into diagonals, so you end up with a 16-pointed star? That would be the analogous process you applied to the cube. Although I have to admit I don't quite understand how this relates to the matter at hand.

The right hand diagram is meant to give us an impression of how different a cube looks to us if we try to draw it in the same way we are attempting to draw tesseracts. What we see a tesseract's cube 'face' like is nothing like what a 4Der sees those same cube 'faces' looking like. Just as the 2Der sees the sides of a square completely differently to how we see the sides of a square.

OK, you do have a point here. When a 2Der looks at a square, it can't see the inside of the square. In essence, it can only see the 4 edges that surround the square. Whereas when we 3Ders look at a square, we can see the entire area inside the square bounded by the 4 edges. So in that sense there's a difference between what we see and what a 2Der would see, looking at the same square. Nevertheless, the 2Der understands the difference between a hollow square (only 4 edges with no content inside) and a filled square (4 edges filled up inside with area). When looking at the projection of a cube, for example, if the 2Der understands that the tetragonal images of the cube's faces are filled, not hollow, then I would say that it has a pretty good understanding of what we 3Ders see when we look at a cube, wouldn't you say?

Similarly, when we 3Ders look at a cube, we only see its 6 faces (at most 3 at a time, to be precise). We don't see the inside of the cube at all! For example, if the cube was made of wood, we wouldn't be able to see the structure of the wood grain in the center of the cube; we would only see the part of the grain structure that is exposed on the 6 faces. But when a 4Der looks at the same cube, she can see not only the square faces of the cube; she can see every point on its inside, and she can see the entire 3D wood grain structure simultaneously. In this sense, we 3Ders are handicapped when it comes to 3D visualization, because we can only see surfaces, not volumes! However, that doesn't stop us from having a very good idea of the difference between a solid cube and a hollow cube. Now, when a 4Der looks at a tesseract, she sees the cubical facets in their entirety, including every point inside each cube. Obviously, we can't do that; but nevertheless, if we understand that the cubes we see in the 3D projection are solid (they are only made transparent for our benefit, so that we can see through them to perceive the structure of the whole projection), then I would say we have a pretty good grasp of what the 4Der sees in her retina.

Now of course, being able to perceive the faces of the cube as filled tetragons is not the same as perceiving the volume enclosed by the cube itself, as you pointed out. This is where depth inference comes in. When the 2Der looks at the projected cube, it may understand that the tetragonal faces are filled, not hollow, but this doesn't mean it understands what the enclosed 3D volume is. In order for this next step to take place in the 2Der's understanding, it would have to understand firstly that these tetragons are, contrary to its expectations, squares, and that they appear as squished tetragons only because they are being viewed at an angle. It would have to draw an analogy between looking at an edge side-on and seeing its entire length, vs. looking at an edge at an angle, where the length of the edge appears shorter than it really is. Once it understands this, then it would have to apply the next step of dimensional analogy: suppose it were to look at a square in its native world, corner-first. It would see two edges meeting at a point, and the two edges would appear shorter than they are because they meet at an angle. By this foreshortening, it would understand that this point is "jutting outwards" towards its eye, and that therefore there is 2D area behind these edges; they are not just segments of a straight line. The dimensional analogy then is that just as these edges appear squished because of the 2D view angle, so the tetragons of the projected cube are squished tetragons because of the 3D view angle, and that this squishing implies 3D depth.

In the same way, when we look at the projected tesseract, we are firstly to understand that those cubes and frustum shapes are all solid in the 4Der's eye. That must be the mental model in our mind when we look at the projection. Then secondly, we must understand that the foreshortening we see of those frustum shapes are caused by the 4D view angle, looking at regular cubes from an angle and causing them to appear squished or distorted in some way. By analogy with what we see of the 3D cube -- its square faces appearing as squished tetragons implies 3D depth -- we are to understand that there is 4D depth involved.

That is also why I was trying to depict that the connecting 'lines' as actually representing more than they appear to represent by making them not look like normal lines. Instead of just representing lines connecting the cube 'faces' together; they also represent the 'insides' of the tesseract; just like the lines in my anorexic cube do for its insides.

OK, this is where I still don't understand how those lines in your "anorexic" cube represent the inside of the cube. They just look like diagonal lines cutting through the cube; how do they represent the entire volume of the cube?

So basically, the rightmost picture is more towards what a cube would look like if we tried to draw it the same way that we are drawing tesseracts in 3D.

And I would have to disagree. If you want to be pedantic and do the same reduction from 4D(object)->3D (projection)->2D (representation of projection on screen) in an analogous way for cubes, then the correct representation of the cube is not your diagonal lines, but a bunch of line segments on a 1D screen. To be precise, 12 line segments on the 1D screen, some of which are superimposed on each other. If anything, that would be the real analogue of our 3D representations of the tesseract.

Obviously, this is exactly what a 2Der would see when it looks at our projected cube. It wouldn't see any areas at all; just a bunch of lines. In order to make any use of projections to understand 3D, the 2Der would have look at the projected 2D image from different angles. If it were to walk around this projection, it would eventually realize that the outer boundary is a square. And if the outer edges of this projection were made transparent so that it could look inside, it would eventually realize that there is another square inside this outer square, and that there are 4 edges connecting the outer square to the inner square, dividing the space between the two squares into trapezoidal areas. So eventually it would form the correct mental model of the projection: it's a large square whose area is divided into a smaller square in the center surrounded by 4 trapezoids. It took a bit of effort to discern all of this -- whereas we 3Ders can simply glance at the projected image from our 3D viewpoint and immediately see this structure without any additional effort. But in the end, what we see and what the 2Der deduces is the same: the image is a square whose area is divided into a smaller square and 4 trapezoids. So sooner or later, both we and the 2Der arrive at the same understanding.

The next step, then, is for the 2Der to understand how we infer 3D depth from this image. Of course, we are so used to doing this that it's unconscious; but it's instructive to understand how we infer depth. As is obvious, as soon as we see the trapezoidal areas, we understand that they are actually squares being viewed at an angle, and the fact that they are at an angle means that they are protruding into 3D, and so by assembling this information together, our minds produce a mental model of the cube in its full 3D-ness --- even though our eye really only sees in 2D. A 2Der, by going through the same steps (albeit manually), would in theory arrive at the same 3D model of the cube -- all with the help of a projected image.

In the same way, what we see when we look at a tesseract projection into 3D is, in a sense, not the same as what a 4Der sees, because a 4Der can see the entire 3D content of the projected image in one glance from its 4D viewpoint. But even though we 3Ders can't do that, we can explore the projected 3D image from various different angles (and this is why on my site I usually do multiple renders of the same object---we need to see it from different perspectives to fully reconstruct the 3D structure of the projection in our mind), and eventually we will arrive at a mental model of the 3D structure of the projection image -- which is exactly what the 4Der sees in her eye. Then we go the next step to understand how the 3D volumes in this projection image imply 4D depth, and in that way form a 4D mental model of the projected object. The entire exercise is trivial for the 4Der, since she can see the entire 3D structure of the projection in one look, and her mind is already so used to inferring 4D depth from it that she doesn't even think twice before her mind reconstructs the 4D model of the object. It takes a bit more effort for us to do the same -- but ultimately we will arrive at the same 4D mental model. That is what I call true 4D visualization.

So really, whichever way you choose to represent the various elements of the tesseract in projection is secondary; what is important is that the projection images should help you to reconstruct, in your mind's eye, what the 4Der sees in her eye. It is that reconstructed 3D model in your mind, not what you see on the screen, that will help you visualize 4D.

[gonegahgah]

Hi quickfur. I can only make a quick comment at the moment. I'll try to get back later. I just wanted to mention that, which you may say anyway; not sure, whereas we see a cube as taking up volume in our space; a 4Der sees a cube as a flat object. I'll refer to gravity again about this soon but just thought I'd drop that in to test the water. Sorry to dash off without any specifics.

[quickfur]

Hi quickfur. I can only make a quick comment at the moment. I'll try to get back later. I just wanted to mention that, which you may say anyway; not sure, whereas we see a cube as taking up volume in our space; a 4Der sees a cube as a flat object. I'll refer to gravity again about this soon but just thought I'd drop that in to test the water. Sorry to dash off without any specifics.

Yes, by dimensional analogy it's quite easy to see that what we perceive to be a solid, full-bodied 3D object is regarded as a mere "piece of paper", as it were. Just as a Flatlander or 2Der perceives a filled square to be rock solid, to us it's merely a thin sheet. This doesn't diminish the 2D-ness of the square, and neither does it diminish the 3D-ness of the cube in 4D, but it does indicate that the availability of another dimension takes "bulk" to a whole 'nother level.

Additionally, we perceive 2D surfaces as dividing space: a wall, for example, is essentially a 2D rectangle (with a little bit of 3D depth) that divides one room from another. In 4D, though, it doesn't divide anything at all. It occupies about as much space as a pole does in 3D: you can simply walk around it. So a 2D wall cannot divide rooms; you need 3D walls (as in, the wall must be extended in 3 dimensions) in order to be able to divide space into rooms.

Such things are important to keep in mind as we learn to infer 4D depth from projections. All those 3D volumes we see in projection do not occupy any 4D bulk at all; they merely enclose the real 4D volume which lies inside. (Analogously, a 2Der should not think that, for example, the tetragonal areas in the projection of the cube occupy 3D volume: they do use up 2D area, which in the 2D world constitutes real estate, but which in 3D only serves as fences. The real 3D volume is enclosed inside.)

[gonegahgah]

Hi quikfur. I have to apologise. This is more what I think I should have been presenting:



The thing that was puzzling me was that I got rid of the mass but I was also getting rid of the faces. I shouldn't have done that.

Does this look more acceptable for what I am saying. This shows a 2.5D cube with faces that are distorted because the cube is shown without any insides.
Admittedly it does look fairly similar to a cube.

But, hopefully it now highlights that a 3D drawn tesseract will have a different look than it will appear to a 4Der. Whereas the cube without any insides distorts the flat faces into inverse pyramids; our 3D tesserat distorts the flat cube 'faces' into 3D cubes and doesn't show any of the insides of the tesseract. The 'bulk' is lost in... Am tired, I'll think about where it gets lost tomorrow...

[quickfur]

[...] our 3D tesserat distorts the flat cube 'faces' into 3D cubes [...]

Um... the 'faces" of the tesseract are 3D cubes! Just as the faces of the cube are 2D squares.

The tesseract's cubical facets are flat only in the 4D sense; even in their native space they are fully 3D. From the 4Der's point of view, we are flat; our entire universe is flat. Flat not in the sense of having extension only in 2 directions, but flat in the sense of having no extension in the 4th direction. Analogously, the cube's square faces are flat only in the 3D sense. In the 2D world, these square faces are full-bodied 2D "volumes" (i.e., areas). From a 2Der's point of view, they are not flat in any sense of the word.

[Mrrl]

Hi all,
There is very simple 4D viewer based on 3D sections: http://astr73.narod.ru/4DView/4DviewV2.zip . This time I wrote it using OpenToolkit library, but it works in Window Forms control, so probably you will have to use Windows again to run it.
Run viewer, select File/Load item and load pzl.txt file. It contains very simple 4D scene. Navigation is by mouse with Ctrl and Shift keys:

Let X be "left/right", Y - "up/down", Z - "ana/kata" and W - "forward/backward" directions.
Then:
Left Button: rotation of object in X/W and Y/W directions (3D rotatiion)
Shift+Left Button: rotation in X/Z and Y/Z directions (rotation of section space preserving direction of view)
Ctrl+Left Button Up/Down: Zoom In/Zoom Out
Ctrl+Left Button Left/Rigth: rotation of object in Z/W direction (rotation of section space along the sceen plane)
Right Button: rotation of camera in X/W and Y/W direction (rotate head in 3D)
Shift+Right Button: same as Shift+Left Button
Ctrl+Right Button Up/Down: Slide Forward/BackWard (don't use it! You'll be lost in no time. And directions are swapped now )
Ctrl+Right Button Left/Right: rotation of head in Z/W direction. It is the most easy way to see a sequence of parallel 3D sections of the object.

The quest is: to describe the scene

[gonegahgah]

Hi quickfur. What we are drawing when we try to draw a tesseract is actually a 3.5D tesseract.

Does the picture I drew look like a cube-like shape with inverse empty square based no bottom pyramids forming the 6 sides?
Can you see that the 2.5D cube that I drew is an attempt to draw a cube without having any insides?
I have turned the faces into inverse pyramids to achieve this.

My inference is that just as a 2Der's attempt to draw a cube looks nothing like what a cube looks like to us; we also have no chance to draw a tesseract that looks anything like what a tesseract looks like to a 4Der; and that is despite having that extra dimension of movement available to us. A 2Der can not draw a square face; and so they do not have the ability to represent a cube to look anything like what we see.

What they will draw for a cube will even look different to what I have drawn. Maybe I should refer to these as 2.5D cube (2D perspective) and 2.5D cube (3D perspective). At least as a 3Der I can add a little bit more detail to the 2.5D cube that the 2Der can never hope to do. The same applies to the tesseract. We can draw a 3.5D tesseract (3D perspective). It won't look anything like a 4D tesseract. The 4Der could even draw a 3.5D tesseract (4D perspective) which will look closer to the real thing but our 3.5D tesseract (3D perspective) will fail to come close to looking even like it.

We do go back to the 2D perspective to better understand our attempts to understand the transition to 4D. Your web article includes these sorts of examinations. Just as we examine these facets to better understand the next dimension; I think we also need to accept that the chasm that exists for the 2Der to faithfully represent 3D objects also translates to an unbreachable chasm to our ability to faithfully represent 4D in any sort of true fashion.

[gonegahgah]

Hi mrrl. Wow, thank you for that! When I get some time I'll have a play with it.
I might even suggest some 'shapes' I might like to explore then too?
Thanks.

[quickfur]

I think we're not on the same page here. Projection is a mathematical operation that reduces the dimension of an object by 1. A cube projected to a 2D screen face-first is a "square within a square", exactly like that diagram I had. It doesn't matter whether we are the ones doing the projection or a 2Der, or even a 4Der, the result is always the same.

Now of course, it is not possible for a 2Der to actually draw such a diagram on their paper (which is 1D). They would have to represent this projected image by doing another projection from 2D to 1D. Or they could build a full 2D model of it, in which case, if we were to look at it, we would recognize that it is the same image that we see with our own eyes when we look at the cube.

You're right in a sense, that given a single image of the 2D projection of a cube, a 2Der would not be able to accurately reconstruct the 2D image, and therefore wiill not fully understand what we see when we look at a cube. That's not the point here. I'm not suggesting that those 2D images of the tesseract projections are a complete representation of what a 4Der would see: that's impossible, because the computer screen is only 2D, but what the 4Der sees is a 3D image!

What I am saying is that we first do the 4D to 3D projection mathematically, and have the computer represent that 3D image in full, and then give us a way to explore that 3D image however we like -- look at it from different angles, cut out some faces, slice it, whatever, but the goal of that is to reconstruct, in our mind, the full 3D image projected from 4D. When we form a clear mental picture of that 3D model -- which our brain is perfectly capable of doing because we understand 3D -- then we can see exactly what the 4Der sees when she looks at the 4D object.

I do not regard those 2D images as anything but a single view into the true projection of the 4D object, which is a 3D model, and which cannot possibly be represented on a 2D screen in full. Their intention is to help us reconstruct the 3D model in our mind, and in this way convey what the 4Der sees into our brain. This is perfectly possible, because we don't see 3D at all; our eyes only see 2D images. Yet our mind is perfectly capable of constructing 3D models of what we see, based on these 2D images.

So what I'm saying is that we make use of this ability of our mind to go from 2D to 3D, by using projections of the 3D projection image of a 4D object, so that our mind reconstructs the 3D projection image. That 3D image is exactly what the 4Der sees, because a 4Der doesn't see in 4D at all; she only sees in 3D. The 4D information is reconstructed by her mind, inferred from the 3D images she sees. But since we can reconstruct that 3D model in our mind, that means we can see what she sees, too, albeit not with our eyes, but by reconstructing that 3D image in our mind. Then we can learn how to infer 4D from that 3D image, and thus arrive at the true visualization of 4D.

In other words, those 2D images of the tesseract projection don't "represent" the tesseract in the sense that they are not the real 3D images of the tesseract that a 4Der would see. But they are, you may say, a mental "crutch" for us poor 3D beings to be able to discern the 3D structure of the tesseract projected into 3D. These images are not a "direct" representation of the tesseract -- they cannot be, because it's impossible to capture the full 3D image of a tesseract as seen in a 4Der's eye with only a 2D image. Rather, they are meant to help us reconstruct the 3D image that the 4Der sees. And how do we know that these images can do just that? It's because our own eyes only see in 2D, yet we fully understand 3D because our mind reconstructs 3D models of what we see based on the 2D images. So with the help of these 2D images, our mind can reconstruct the 3D image which represents the "true" projection of the tesseract. At first, we only see it as a 3D object, but by learning how a 4Der interprets 4D depth from 3D images, we can also do the same: to infer 4D depth from the 3D model that we have developed in our mind.

[Mrrl]

What I am saying is that we first do the 4D to 3D projection mathematically, and have the computer represent that 3D image in full, and then give us a way to explore that 3D image however we like -- look at it from different angles, cut out some faces, slice it, whatever, but the goal of that is to reconstruct, in our mind, the full 3D image projected from 4D. When we form a clear mental picture of that 3D model -- which our brain is perfectly capable of doing because we understand 3D -- then we can see exactly what the 4Der sees when she looks at the 4D object.

It's not so easy. I afraid that most humans will have a problem to reconstruct "3D model". And the problem is that this model is not a set of 3D objects, but the area in 3D covered with different colors. In our life we usually work with solid objects, that have "sharp" surfaces,and we imagine the object as its surface. So the best thing that we can imagine is some kind of "flat" image of 4D scene, without shades, gradient coloring and other beauties of thier world. And even this is extremally difficult for non-trained brain. Of course, some people, like architects and engineers are closer to this ability - they need better 3D understanding for their work. But others look at images of 4D scenes and say "I see just strangely mutating 3D object. Where is 4D?"

[quickfur]

Well, I didn't say it was easy. That's why we need to train the mind to interpret these images as 4D projections, so that the apparent morphing is interpreted as 4D rotation. if you look at it in 3D-centric way, then of course it makes no sense, it's only a strange morphing object. You have to consciously interpret it as 4D rotation for it to make sense.

And you're right that we are missing the shades, the gradient coloring, etc.. These can be remedied by showing different subparts of the 3D projection image -- for example, if you have clouds of colored gas in 3D, they also form a complex 3D structure of different colors. But we can understand their 3D layout by examining 3D models of each type of colored gas, etc.. Same thing with the structure of the human brain: if you only look at it from outside, all you see is the surface. But we can map it with MRI and other kinds of scanning to build 3D models of the structures inside, and the computer can show us parts of this model so that we develop an understanding of the 3D structure.

The renderer that I have right now doesn't have the ability to do these kinds of things yet. But it does show the boundaries of simple objects like polytopes, so you can think of it like the old school dungeon crawl games where they draw the walls with lines and no texture. It looks ugly, but you can still see the 3D-ness of the dungeon. I mean, compare the difference between the wire-and-stick representation of the dungeon to a series of cross-sections. Which do you find helps you see the 3D depth better? I don't know about you, but I'd definitely prefer the wire-and-stick projection. I can imagine playing quake with no textures and only lines for walls, and I would still be able to move around, dodge bullets, etc., with relative ease. But I cannot imagine doing the same with only cross-sections of the map, especially when there are slanting or curved surfaces around. I could still figure out my way around eventually, but it will be way too slow for fast movement like dodging bullets. And the reason is that the wire-and-stick diagram, even though it is deficient and ugly, is still closer to the way we see 3D than looking at slices.

[Mrrl]

Well, I didn't say it was easy. That's why we need to train the mind to interpret these images as 4D projections, so that the apparent morphing is interpreted as 4D rotation. if you look at it in 3D-centric way, then of course it makes no sense, it's only a strange morphing object. You have to consciously interpret it as 4D rotation for it to make sense.

So we need to imagine and understand 4D objects - and only then we'll be able to recognize them on projections and/or cross sections It's OK for me. But I heard often that human can't have 4D imagination because we have no 4D experience. And 4D viewer and renderers looked like a good way to get this experience... So it's really a dead end, and we have to study the linear algebra first?

[gonegahgah]

Hi quickfur. I think we are almost on the same page. The major distinction I'm making is that it is hard to imagine the missing 'bulk' from the 3.5D diagram and that our vision of a cube is as different to a 4Ders vision of a cube as a 2Ders vision of a square is to our vision of one. I'm also thinking that you have a lot more confidence in the 2Der to extrapolate their thinking to 3D than I have when they can never know their world from any other angle.

Taking the square again. We can imagine the inside of the square from a different 'angle' than the 2Der can. This allows us to take a square face and to connect the surface of the square face to some bulk and onto another square face to form a cube. We can intrinsically understand this 'inside' because we can imagine the inside start to the inside finish.

Like us being able to see the 'inside' of a square; the 4Der is able to see the 'inside' of a lonely cube. It's like the cube is somehow missing two more faces than the 6 we know through which the 4Der can peek into the inside of the cube. Just as we can see every 'atom' of a square on one side; the 4Der can see every 'atom' of the cube all at once through one of these peek hole 'sides'. Once we extend a square into a cube; we can only see inside of one face of that square. The other face of that square is now hidden inside the cube so we can't see it anymore. The same goes for the 4Der. They can see into the cube via its hyperface of which there are two. When you extend the cube into the 4th dimension to make a hypercube; they can then only see into one of those hyperfaces for that 'face' cube. The other cube hyperface is now hidden inside the hypercube.

Even the last paragraph shows some terminology confusion. A cube has two hyperfaces in 4D; just as a square has two faces in 3D. But the hyperfaces aren't squares; they are cubes. So a 4Der sees a cube as having 'front' hyperface that is a cube; and a 'behind' hyperface that is a cube. Just like for us that a square has a front square face and a behind square face. How, does that sound so far?

[quickfur]

OK, maybe we are on the same page after all.

You are perfectly right, of course, that to a 4Der, a cube has two hyperfaces, both of which are cubes. If we want to be pedantic, we could imagine that the 4Der's cube is actually a very thin tesseract, so that it has non-zero 4D thickness. A true, mathematical 3D cube has no 4D thickness at all; strictly speaking, a 4Der can't see such a cube because no 4D atom (or whatever it is that comprises the 4D world) can fit in a 0-thickness cube, so it wouldn't be able to reflect light, etc.. But for our purposes, we could just imagine that our 3D world is actually 4D, except that the 4th dimension is ultrathin, so that we can't perceive it. So in that sense, our 3D universe is a hyperplane that we're confined to, and all 3D objects have two hypersurfaces, one "above" the hyperplane and one "below" it. When a 4Der looks at objects in our world, she sees the hypersurfaces, not merely the 3D boundary the bound the object. So in that sense, her view is different from ours, just as our view of a square is different from a 2Der (who only sees it as the area bounded by 4 edges, but can't see the area in its entirety).

But I still think it's possible to construct an accurate 3D model in our mind of what a 4Der sees. After all, even though the 4Der's viewpoint is fundamentally different, what she sees in the end is still just a cube. We cannot physically see every atom inside a cube, but that doesn't mean we're unable to grasp the 3D-ness of the inside of a cube in our mind. I mean, our eyes never see 3D at all, yet we are able to build 3D buildings, make machine parts with intricate 3D shapes, etc.. So we must have some grasp of true 3D, right? So in that sense we can understand the 3D image that forms in the 4Der's eye when she looks at a 4D object. At the very least, if we look at a cube made of glass, we can at least "see every atom" inside the cube, right? At least, we understand the nature of 3D position inside that glass cube, so that we know what it means if there's, say, a piece of dirt stuck inside somewhere -- we can tell accurately where it is in 3D coordinates. So even though the viewpoint is different, the information we get is not fundamentally different from the information a 4Der gets.

Of course, just because we can understand a 3D image doesn't mean we can automatically infer 4D depth from it. Like mrrl said, with no prior training if we were to look at these projections, all we see is a strangely morphing 3D object; we don't see any 4D at all. So in order to truly visualize 4D, we have to train our mind to go another step past the 3D model: we have to learn how to infer 4D depth. Unlike our instinctive, unconscious reconstructing of 3D models from the 2D images in our eyes, we have to consciously exert effort to infer 4D depth, because our brain won't do that for us automatically.

So then the question is, given that we can construct an accurate 3D model in our mind of the 3D image that forms in a 4Der's eye, is it possible to infer 4D depth from it? Some may say it's impossible, but I say it's possible. The main reason is that what happens in 4D when an object is projected onto the 4Der's eye, is completely analogous to what happens in 3D when an object is projected onto our eye. It is a bit more complex because of the extra dimension, but the underlying principle is exactly the same. Thus, we observe the same kind of effects in 4D projections that we observe in 3D projections: foreshortening, horizons, perspective distortion, etc.. So the kind of transformations that the 4D object undergoes when projected onto the 4Der's eye are not foreign to us at all -- we see the same kind of effects, albeit in a simpler form, in 3D projections. So this means that it's possible for us to learn to infer 4D by "looking at" the 3D projections of 4D objects.

I'm not saying that we will ever see 4D exactly the same way as a 4Der -- not unless somebody invents a way to send full 3D images into our brain's neural pathways directly without going through our physical eyes -- but in principle we can infer 4D depth the same way they can. I don't think we'll ever attain to the same level of facility in 4D space as a native 4Der, but in principle, we should be able to perceive all the shapes they can perceive, given adequate training.

The bottomline is that projection from 4D into 3D does not cripple the 4D information in any way more than it already does for a 4Der. What limits us is that we can't see that 3D image directly -- but we can get pretty close, at least to a basic level of being able to perceive elementary shapes like polytopes and simple curved hypersurfaces. In spite of its limitations, the projection method is the closest we can get to understanding the visual experience of a native 4Der; I can't say the same for the other methods of 4D visualization like slicing or unfolding. And IMHO projection does get pretty darned close.