Dimensional Baby Steps

[quickfur]

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.

True. Povray models are nice if you know you're going to be reusing them to produce many views. Otherwise it just feels like so much work just for one miserable picture that you could've drawn by hand in 5 minutes.

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

They're not difficult, but they are annoyingly tedious. Basically, there is no 2D in povray; to draw an arrow you have to model it with cylinders, cones, spheres, etc.. So just for drawing a single dotted line, you have to create a whole string of cylinders (or spheres) lined up one behind the other with gaps in between, perhaps with a cone at the end for the arrowhead. That's quite a lot of objects for a single logical dotted line. (Plus, you need to design textures to go with the objects, with unusual material properties so that they come out as flat colors instead of the usual realistic shading that povray produces. The colors have to be chosen to be high-contrast with the general background of your scene. It wouldn't do, for example, to have diagrammatic arrows show up like physical spears in your scene.)

Of course, you can write macros for automatically creating a dotted lines or arrows given two endpoints, but it takes some effort to design the macros so that they work in a variety of situations that you might want to use them for. So again, it's really only worthwhile if you're going to be reusing it in multiple scenes. Otherwise it's a lot of effort for something that can be sketched in 5-10 mins.

[4Dspace]

[...] 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.

.....

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).

You forgot to mention that the 3 near faces are seen from the outside and the 3 far faces are seen from inside. Thus the question that ac2000 asked you above, you answered incorrectly. The question was, if you put some lettering on the faces of the cube, will the 4Der be able to read it. You answered with a long and tedious tirade that because the faces are seen simultaneously from both sides (which is patently wrong), the 4Der will not be able to read a thing. And now, with a bit of analytic geometry, which you overdid and overstated, you got the right answer: yes the 4Der will be able to read the lettering on the cube. He will see what's written on 3 near faces and then, after turning the cube around, he will read what's on the other 3 faces.

Congratulations! You're improving

[quickfur]

[...] He will see what's written on 3 near faces and then, after turning the cube around, he will read what's on the other 3 faces. [...]

According to the calculations I made, all 6 faces are visible simultaneously. As well as the middle of the cube. There is no turning around involved here.

[gonegahgah]

Step 14. Under co-location we see them when in plane; but see them shadowy when in adjacent plane at some angle:



For the 2Der in the left image, they can only see us when we actually intersect with their plane. They won't see our innards of course; just a slice of our outside.
When we are off in the 3rd direction (that gives us 3D) we are depicted only as a shadowy outine in their 2D world.

The same goes for the 3D image. The 4Ders, shown as not shadowed, are actually intersecting our 3D plane. The shadowy figures are 4Ders off in the 4th direction (their 4D).
Of course, in 3D we can show them in more places than the 2Der can; but the 4Ders can be off at as many angles to us in 4D as we are off at an angle to a 2Der in our 3D.
That's interesting.

When a 4Der intersects our plane they can only see the cube as we see it; just as when we instersect a 2Der's plane we can only see the square as the 2Der sees it.

Again in both diagrams, for the 2Der and the 3Der, there is a shadow depicted in the middle of the solid object in our respective worlds.
We obviously can not be inside a square; we can only observe it from outside the square; so it could only be a shadowy depiction of us.
The same goes for a 4Der. They can not actually be inside a solid cube; they can only observe it from outside the cube; just as we are constrained to doing.

But, just as we can observe the inside of the square from any angle in our two 3D directions; they can observe the inside of our cube from any angle in their two 4D directions.
But, again I highlight that the co-location method suffers from not providing us with any information about how far a 4Der is off in the 4th direction.
I guess there are ways to do that such as size and perhaps even different colours or shades for being kata-wards or ana-wards.

Again we can see that, just as for the 4Ders who are able to have multiple co-locating persons; but whom are actually at different distances off in the 4th direction, the 2Der can also have multiple co-locating 3Ders whom, as we have seen, are actually at different distances from the 2Ders plane; and not really in the same location. ie.:



The other interesting thing to note is that in the co-location model it shows that we are all on the same ground level - 2Ders, 3Ders and 4Ders, whether we are forward-backwards, left-right or ana-kata. It is possible that the hill slopes down at an angle in the ana or kata directions, or forward-back or left-right for that matter. The model does help us to realise that ground level is ground level; whether you are in 2D, 3D or 4D.

[4Dspace]

[...] He will see what's written on 3 near faces and then, after turning the cube around, he will read what's on the other 3 faces. [...]

According to the calculations I made, all 6 faces are visible simultaneously. As well as the middle of the cube. There is no turning around involved here.

No one denies that all 6 faces are seen simultaneously. The point is --and always has been-- from what side do you see those faces?

Now, you are doing Euclidean geometry here. Which means that from any given POV you can see only one side of a 2d-plane, no matter in what N-space you are in. This is basic. Now, explain to your readers here how is that possible for a plane to show both of its sides to a given POV in Euclidean space.

Impossible. Either you're standing on both sides of the plane simultaneously, which does imply the omnipresence, or you see its only one side, or you're not in Euclidean space.

This, by the way, also proved you wrong on your another "deduction" on this thread. You said that a 3d object has a 0 4d bulk and => does not exist in 4D. But a cube in 4D merely looks sort of flat from the 4D prespective. However, its 3d volume does not disappear. Your 4Der can not only see a cube in 4D, he can also read what's written on the faces of the cube, albeit no more than a half of what's written at any time, just like we can see only 3 faces of a cube at most.

You've been making this mistake for years, I showed you where and how you were wrong --that was more than a week ago already-- but you still persist in your error. It's hard for you to change gears and let go of wrong reasoning. That's cause you invested in it so much. So you will persist in it. It's so human (in a baboonish short of way)

[quickfur]

[...] He will see what's written on 3 near faces and then, after turning the cube around, he will read what's on the other 3 faces. [...]

According to the calculations I made, all 6 faces are visible simultaneously. As well as the middle of the cube. There is no turning around involved here.

No one denies that all 6 faces are seen simultaneously. The point is --and always has been-- from what side do you see those faces?

You said there are 3 'near' faces.That is wrong. All 6 faces are near, because if you calculate the distance of each face from (0,0,0,5), you will find them all equal. None are "behind" the other from the 4D viewpoint.

Furthermore, every part of the cube's volume is also visible. I have shown that the middle of the cube is unobscured. To see the center of the cube from 4D does not require the line of sight to pass through any of its faces. In fact, if you calculate the distance of the center of the cube to the viewpoint, you will find that it is closer than all 6 faces. Therefore it cannot be obscured by any of the cube's faces. The same is true of any other part of the inside of the cube. Every line that passes through the 4D viewpoint can only intersect the 3D hyperplane that the cube lies in at most once, which means nothing in the 3D hyperplane can obscure anything else in the same hyperplane. So a (hypothetical) native 4D being can see the entire 3-bulk of the cube simultaneously -- every part of it. No omnipresence is involved.

As for writing on the faces of the cube, the writing has to exist somewhere, so it is reasonable to assume that we can represent this writing as line segments (or curves as the case may be) whose coordinates can be specified in the form (x,y,z,0). Given any such coordinates, we can construct the line:

L = (0,0,0,5) + k*(-x,-y,-z,-5), where k is free to vary from 0 to infinity.

Since L passes through the viewpoint (0,0,0,5), it is a line of sight. A quick calculation shows that L intersects the 3D hyperplane exactly at (x,y,z,0), and nowhere else. Therefore, whatever the coordinates of the said writing are, it is unobscured from the 4D viewpoint. It follows that the writing on all 6 faces of the cube are simultaneously visible.

[...] You said that a 3d object has a 0 4d bulk and => does not exist in 4D. But a cube in 4D merely looks sort of flat from the 4D prespective. However, its 3d volume does not disappear.

I never said its 3d volume disappears. I said it has 0 4D bulk. 4D bulk and 3D volume are not the same thing. What I said about 3D objects not existing in 4D was in the context of material existence. You can't make a zero 4-bulk object with 4D atoms which fill non-zero 4-bulk. If you're talking about mathematical existence, that's a different story. Nobody denies that you can mathematically embed lower dimensional objects in a higher dimensional space.

[4Dspace]

I see you're not familiar with basics of higher geometry, yet you position yourself as an expert in 4D seeing. You are confused, which is not a crime in itself. The fact that you generously spew your confusion onto others is.

The point of my previous post you forgot to address was:

Now, explain to your readers here how is that possible for a plane to show both of its sides to a given POV in Euclidean space.

What you apparently do not know is the following basic of analytic geometry, which will help you in the future decide from what side, i.e. "inside"/"outside" you see a plane that belongs to an object you're viewing. And that is --attention!-- you draw an imaginary dividing plane through the very center of the object, such that this plane is perpendicular to your POV. So, please note that the angle of that plane to the object depends on your POV. Given a position of the object in space and your POV, there is only one such plane. It is transparent and infinite in all directions. It cuts the object in 2 halves and it is perpendicular to your POV. Got it?

Now, the ray from your POV, before hitting a face of a cube, will either pass through this plane, or it will not (there are no other cases in Euclidean spaces). If the ray does not hit this plane before hitting a face of the cube, this means that you 're seeing OUTSIDE face. If the ray does pass through this plane before hitting the face, this means you see this face FROM INSIDE.

You're welcome.

[quickfur]

[...] The point of my previous post you forgot to address was:

Now, explain to your readers here how is that possible for a plane to show both of its sides to a given POV in Euclidean space.

What you apparently do not know is the following basic of analytic geometry, which will help you in the future decide from what side, i.e. "inside"/"outside" you see a plane that belongs to an object you're viewing. And that is --attention!-- you draw an imaginary dividing plane through the very center of the object, such that this plane is perpendicular to your POV. So, please note that the angle of that plane to the object depends on your POV. Given a position of the object in space and your POV, there is only one such plane. It is transparent and infinite in all directions. It cuts the object in 2 halves and it is perpendicular to your POV. Got it?

You're the one who is confused. This cutting plane that you describe is not specifically a 2D plane, but an (n-1)-dimensional hyperplane. (Why? Because a 2D plane does not divide space for all dimensions higher than 3. In fact, 3D is the only dimension where a 2D plane divides space; in 2D space, it doesn't divide space, it fills it.) In n-dimensional space for n>3, a 2D plane is not the same as the dividing hyperplane. That is what I have been trying to tell you, and all I get is verbal abuse, mockery, and ad hominem attacks.

When you draw this dividing hyperplane in the given example (the cube at (±1,±1,±1,0) and the viewpoint at (0,0,0,5)), the cube lies in the hyperplane. This hyperplane does not divide the cube at all, because of the plain and simple fact that it is co-planar with the hyperplane. Therefore there is no "inside" or "outside". The cube is a flat hyperface (i.e., a 3-face).

Now, the ray from your POV, before hitting a face of a cube, will either pass through this plane,

Correction. The word you want here is hyperplane.

or it will not (there are no other cases in Euclidean spaces). If the ray does not hit this plane

Correction. Hyperplane.

before hitting a face of the cube, this means that you 're seeing OUTSIDE face. If the ray does pass through this plane before hitting the face, this means you see this face FROM INSIDE. [...]

You missed one case. The ray can hit the object at the same time it hits the dividing hyperplane.

The cube in our example lies in the hyperplane. The ray hits the face of the cube precisely when it hits the dividing hyperplane. Therefore, there is no division of the cube's faces into "inside" or "outside".

You were the one who had a misconception about Euclidean geometry, yet not only do you fail to see your mistake (which in itself is not a crime) and accuse others of being wrong (which isn't a crime either, though it is rather annoying), but, instead of proving them wrong, you verbally abuse them, call them names, tell them to shut up, and resort to ad hominem attacks. Yet you call yourself a logical, analytical person. Does a logical, analytical person dish out abuse to whoever doesn't agree with him? Resort to ad hominem attacks? Judging from your actions, that is apparently what you believe. Was I not right in saying that it's a waste of time trying to explain things to you?

[quickfur]

And just in case anyone doubts that 2D planes do not divide n-dimensional space for n>3, here's a proof for the case where n=4.

Coordinates in 4D have the form (x,y,z,w), so we can consider the 3D hyperplane within 4-space where w=0. Points on this 3D hyperplane have the form (x,y,z,0).

Now place our viewpoint at (5,0,0,0). This viewpoint lies in our 3D hyperplane, since the last coordinate is zero. Draw a line from (5,0,0,0) through the origin (0,0,0,0). Let P be the 2D plane (within the 3D hyperplane where w=0), perpendicular to this line. As 4DSpace has said, this plane is uniquely defined. A simple calculation will show that points that lie on the plane P have coordinates of the form (0,y,z,0). So far so good, there are two free variables, corresponding with the 2 degrees of freedom in this plane.

Now since we're looking from (5,0,0,0), the ray passing through the origin will intersect the plane P at the origin. Given any point A along this ray, if the ray passes through A before it intersects P, then A must be in front of P; if the ray intersects P first and then passes through A, then A must be behind P. So let's pick two points, say B=(1,0,0,0) and C=(-1,0,0,0). It should be obvious that B is in front of P, and C is behind P.

It's clear that P divides the 3D hyperplane where w=0 into two halves, the half where B lies in, and the half where C lies in. How do we know this? Because, if we remain within the 3D hyperplane, it's impossible to connect B to C without crossing P. If you draw a line from B to C, it will intersect P at the origin. If you draw a circle that touches both B and C, then it must also intersect P at two points.

Now the important question: does P divide 4D space? No, it doesn't. Why? Consider the circle whose coordinates are defined thus:

D = (cos T, 0, 0, sin T), where T varies from 0 to 2*pi.

When T=0, cos T = 1 and sin T = 0, so the circle passes through (1,0,0,0), which is B.
When T=pi, cos T = -1 and sin T = 0, so the circle passes through (-1,0,0,0), which is C.

If P does divide 4-space, then this circle must intersect it at some point, because it's passing through two points that lie on opposite sides of P (that's the definition of dividing space).

Notice that when 0<T<pi, sin T > 0, so the circle is outside the 3D hyperplane (its last coordinate is non-zero). Since P lies entirely within the 3D hyperplane (its last coordinate is 0), the circle obviously does not intersect P for these values of T. Furthermore, when pi<T<2*pi, sinT < 0 (again the last coordinate is non-zero), so that part of the circle doesn't intersect P either.

So we have here a circle that passes through two points that, purportedly, lie on two opposite sides of the plane P, yet it does not intersect P at all. Therefore, P does not divide 4D space.

[4Dspace]

On top of everything else, you have reading comprehension deficiencies. Where did I say that the plane divides the space? I said the plane divides the object. It cuts it in half. So, everything else you said after that error applies to your error.

And you still did not answer this:

[...] The point of my previous post you forgot to address was:

Now, explain to your readers here how is that possible for a plane to show both of its sides to a given POV in Euclidean space.

[quickfur]

On top of everything else, you have reading comprehension deficiencies. Where did I say that the plane divides the space?

Because you said this plane is unique. The only way for it to be unique is if it is an (n-1)-dimensional hyperplane. Given a viewpoint and an object in 4D, there are many 2D planes that are perpendicular to the line of sight. There is no unique plane.

I said the plane divides the object. It cuts it in half.

Correction. There is more than one such plane.

Given our cube at (±1, ±1, ±1, 0) and our viewpoint at (0,0,0,5), there are at least three 2D planes that both divide the cube in half and are perpendicular to the line of sight:

P1: the plane whose coordinates are of the form (x,y,0,0)
P2: the plane whose coordinates are of the form (x,0,z,0)
P3: the plane whose coordinates are of the form (0,y,z,0)

The last coordinates are all zero, so no matter where you are on any of these planes, the dot product with the viewpoint-to-object vector, (0,0,0,-5), is always zero. Since the viewpoint-to-object vector is non-zero, the only possibility is that the cosine term of the dot product is zero. That means the angle is 90°.

In fact, every plane that passes through the origin and lies in the 3D hyperplane is a plane that's perpendicular to the line-of-sight. Therefore there is no unique plane that divides the cube. The uniqueness exists only with 3D viewpoints. Your persistent mistake is to assume this uniqueness holds in other dimensions, which it doesn't. Using a plane as the criterion for determining which faces are "in front" or "behind" depends on the choice of planes. This choice is arbitrary, since there are an infinite number of planes to choose from.

Furthermore, since the planes do not divide 4D space, using them to determine "in front" and "behind" is meaningless. "In front" and "behind" is meaningful only when the object divides space, since otherwise, you can stand beside it. Then are you "in front" or "behind"? The answer is neither. You can't use the object to determine "in front" and "behind" since it doesn't divide space.

So, everything else you said after that error applies to your error.

And you still did not answer this:

[...] The point of my previous post you forgot to address was:

Now, explain to your readers here how is that possible for a plane to show both of its sides to a given POV in Euclidean space.

Why should I answer anything, since you don't even bother to read what I wrote?

I have been trolled. I keep taking the time to reply and explain myself, and all for what? To be ignored. Not even read. Do you think I'm doing this because I like wasting time speaking to the air? That I have too much free time on my hands so I just love writing stuff that nobody reads? So let everyone who reads this forum be the judge. You criticize us for using dimensional analogy, and so I have presented solid mathematical proof of my position. You have yet to present a valid disproof of what I presented, or, for that matter, a proof of your own wild claims. You can rail against it all you like, it does not change the facts. A fair-minded reader will know who is the one who got it wrong, and who is the one who is misrepresenting the situation. There is no need for me to defend anything. As I've said many times before, the mathematics speaks for itself.

So go ahead with your smear campaign and ad hominem diatribe against me, and against whoever has the nerve to disagree with you. Do your worst. Let the readers see who is the one who presents the mathematical facts and who is the one who rails and rants and accuses others. I will not bother answering you anymore, as you have proven to be impossible to reason with. Anyone who reads this forum will see that. I have wasted enough time on this already. As they say, do not reason with the unreasonable; you lose by definition. Timely advice, and one I should have taken since the beginning of this unfortunate exchange. My only mistake was to be kind enough to give you the benefit of the doubt, to assume you're a reasonable human being, and to take the time to try to explain things. And my only reward? To be smeared, mocked, and verbally abused. And then to be nitpicked over trivialities and summarily ignored. Well, thank you very much for your gratitude. I hope you're happy.

[gonegahgah]

Apart from the co-location method for demonstrating 4D I have also seen the slice method, the multi-frame method, the projection method, oh... and the math method of course . I guess I should also include the multi-coloured cube method by CH Hinton provided to us courtesy of AC2000. If I have missed any others please add them for me.

The downside of the slice method is that we get all these intriguing but mind beguiling morphing shapes. It's probably a fair comment to say that the slice method, in general, tells us as much about 4D as a blind man's hands tell him about an elephant. Perhaps even less...

The multi-frame method provides us with the insight that the extra dimension exists as adjacent planes to the lower dimension's plane of existance. The 3D world is simply an infinite series of 2D planes side-by-side in the 3rd direction. A 4D world is simply an infinite series of 3D planes side-by-side in the 4th direction. The main downside of the multi-frame method is that you have to provide bounded frames of limited size and that you can only show a small number of the frames. The co-location method is simply a compaction of a few multi-frames into a single frame with a bit of simplification too. The co-location method's downsides have already been explored.

The projection method involves, in general, treating things as prisms. This is far from being totally correct but it is a general fall-back position. Generally, objects are first drawn as we would depict them in 3D and then another identical copy is generally drawn at an offset and all the corresponding points are connected together. The main downside I see of the projection model is that it conveys a, difficult to avoid, biased 3D sense of 4D and the concept of bulk has to be heavily inferred.

The math method - which is extremely sound and good to see - detours the issue of directly making something visual. Everything in the math method, though models can be drawn using one of the other models, is handled in the head. So, again, like the slice method, there is a fair amount of covering our eyes and feeling around with our hands. Describing a rose mathematically is not always the same as smelling it. Mathematical proof is ultimately important in its role as a proving tool and in providing 'accurate' drawings.

I'm not targeting any method in the above but just thought it would be useful to provide a summary at this stage.
I am hoping that I have discovered a new method that will hopefully help to make things clearer. We'll only be able to tell once we see the end result.

Step 14. The 2Der trying to locate a 3D house:



The left picture shows the scenario. The right picture shows the co-location method of presenting the house to the 2Der. They will just see the shadowy effects of the lines vertically in front of them of course. They have to move in the 3rd direction to the right until their plane intersects with the house to actually see it. In this instance, without any other clues provided, they have to hope they pick correctly between choosing left or right and don't know how far it will be to cross planes.

[4Dspace]

On top of everything else, you have reading comprehension deficiencies. Where did I say that the plane divides the space?

Because you said this plane is unique. The only way for it to be unique is if it is an (n-1)-dimensional hyperplane. Given a viewpoint and an object in 4D, there are many 2D planes that are perpendicular to the line of sight. There is no unique plane.

I said the plane divides the object. It cuts it in half.

Correction. There is more than one such plane.

I said, the plane is the unique result of interaction of the POV and the object in space. You get confused with your analysis. When you analyze 4D, you forget the 3d subspace where cube lives. Thus your errors.

Given our cube at (±1, ±1, ±1, 0) and our viewpoint at (0,0,0,5), there are at least three 2D planes that both divide the cube in half and are perpendicular to the line of sight:

P1: the plane whose coordinates are of the form (x,y,0,0)
P2: the plane whose coordinates are of the form (x,0,z,0)
P3: the plane whose coordinates are of the form (0,y,z,0)

Why such lame planes? The plane in question is the diagonal plane: (x,y,z,0). You cannot admit that you made an error, so you will continue your verbal incontinence for the benefit of... whom? Who are those dimwitted 2Ders who cannot say by themselves what's what is and need to be told by you what to think? If there are such people, who cares what they think? People are not stupid and it is plainly obvious that you do not know what you're talking about. You quickfur do not see 4D. The trouble is, you do not want to learn. Like a true troll, your main purpose is self-aggrandizement.

And you still did not answer this:

The point of my previous post you forgot to address was:

Now, explain to your readers here how is that possible for a plane to show both of its sides to a given POV in Euclidean space.

Why should I answer anything, since you don't even bother to read what I wrote?

Again I looked through the last 2 pages and I do not see where you addressed this. Could you please point me to it? I do not see it. That would be interesting to see, since this proposition is impossible in principle, lol.

I have been trolled. I keep taking the time to reply and explain myself, and all for what? To be ignored....

You are the troll. You're the resident troll on this site. You've been spewing your nonsense on people for a long, long time. It's time you either learn so that you can spew truth for a change or go simply away. In order to learn, you gotta admit where you were wrong. Which you seem incapable of doing.

[quickfur]

Apart from the co-location method for demonstrating 4D I have also seen the slice method, the multi-frame method, the projection method, oh... and the math method of course . I guess I should also include the multi-coloured cube method by CH Hinton provided to us courtesy of AC2000. If I have missed any others please add them for me.

There's also the "unfolding" method, in which you take a 4D object like a tesseract, and "unfold" it into an assembly of 8 cubes that lie within our 3D space, in much the same manner as we can take a cube and "unfold" it into an assembly of 6 squares that lie in the 2D plane, for example like this:



However, I think everyone will agree that the unfolding method does not really help us visualize 4D at all. A 600-cell, for example, unfolds into what looks like a big amorphous tangle of tetrahedra (600 of them in total), leaving the reader with practically no clue as to how such a thing could assemble itself into a 4D object, much less what such a thing would look like.

Not to mention that unfolding only works for a limited number of geometrical objects. It's of no help if you're dealing with less symmetric objects, like rivers or mountains, since it is unclear how such things can be "unfolded" in the first place.

The downside of the slice method is that we get all these intriguing but mind beguiling morphing shapes. It's probably a fair comment to say that the slice method, in general, tells us as much about 4D as a blind man's hands tell him about an elephant. Perhaps even less...

<shameless plug> My favorite example showing the limitations of the slice method is this series of slices of a 4D object:

Each of these objects represent a distinct slice of a particular 4D object, and if you stack them on top of each other along the 4th direction in the above sequence, you will obtain the object in question. Assuming, that is, you have any idea at all as to how these 3D objects could possibly be "stacked" on top of each other "in the 4th direction", and what manner of shape would result from such an assembly. In all likelihood, you wouldn't have a clue as to what it might be, unless you knew the answer beforehand, much less what features (number of vertices, number and shapes of faces, cells, etc.) it may have.
</shameless plug>

The multi-frame method provides us with the insight that the extra dimension exists as adjacent planes to the lower dimension's plane of existance. The 3D world is simply an infinite series of 2D planes side-by-side in the 3rd direction. A 4D world is simply an infinite series of 3D planes side-by-side in the 4th direction. The main downside of the multi-frame method is that you have to provide bounded frames of limited size and that you can only show a small number of the frames. The co-location method is simply a compaction of a few multi-frames into a single frame with a bit of simplification too. The co-location method's downsides have already been explored.

The multiframe method has the advantage that it lets you show, for example, the entire contents of the 4Der's 3D retina in an unambiguous way, and in theory, see 4D "as the 4Der would". It can also be used in another way: make a 2D array of 2D images, each of which represents a 2D slice of the 4D world. Collapsing either rows of columns of these slices will produce 3D slices of the 4D world, and, according to theory, putting them all together will result in a 4D array representing a pixelated version of the 4D world. I have actually attempted this before, in fact.

But the problem, as gonegahgah said, is that you can only show a limited number of slices. The 2D array of images idea, for example, requires so many images that even on a modern high-resolution screen, you can only fit about 25 or 30 images across the screen, and a little less vertically, with each image sized approximately 25x20 (in accordance with the number of images across and down, to simulate a roughly equal-sided hypercube slice of 4D space). So that is like trying to depict scenery with a 25x20 image, which is the size of a small Windows icon (or less). The effective resolution is so poor so as to be of little practical use. Plus, you still suffer from the same problem as the slices method: yes you see the individual frames. But how do they fit together? It's non-obvious.

The projection method involves, in general, treating things as prisms. This is far from being totally correct but it is a general fall-back position. Generally, objects are first drawn as we would depict them in 3D and then another identical copy is generally drawn at an offset and all the corresponding points are connected together. The main downside I see of the projection model is that it conveys a, difficult to avoid, biased 3D sense of 4D and the concept of bulk has to be heavily inferred.

This is not quite accurate; the prism effect really only happens for objects that are prism-like, for example the tesseract. It doesn't quite produce the same effect with other shapes. There are also different types of projections -- the one you're describing sounds like the oblique projection method. I personally prefer perspective projections, in which farther objects appear smaller, much like they do in our familiar 3D world. Or, when I'm exploring the structure of something, I use parallel projections -- sorta like engineering drawings with 2-point and 3-point perspectives.

But you're right, that the projection method may give a false sense of understanding, since it produces compelling 3D images that we tend to interpret with our natural 3D instincts, which often leads to a wrong conception of how 4D works. It is possible to get an accurate visualization of 4D with projections -- I do that all the time -- but it requires conscious effort to not interpret the image in a 3D-centric way, but a 4D-centric way. This means that looking at the images will not make you "suddenly see 4D" (like stereograms).

Projections also may suffer from illusions -- the same thing happens in 3D with our own eyes, for example, something which street magicians use to their advantage when they wish to produce an unusual effect. Certain 4D rotations, when viewed from the wrong angle, may appear to transform into a 3D rotation of a distorted version of the original object -- due to the illusions misleading our brain's 3D-centric wiring.

The math method - which is extremely sound and good to see - detours the issue of directly making something visual. Everything in the math method, though models can be drawn using one of the other models, is handled in the head. So, again, like the slice method, there is a fair amount of covering our eyes and feeling around with our hands. Describing a rose mathematically is not always the same as smelling it. Mathematical proof is ultimately important in its role as a proving tool and in providing 'accurate' drawings.

Ultimately, all our efforts with dimensional analogies and visualization methods, etc., are aimed to develop a better intuitive grasp of the mathematical models (and what they represent, if you believe that there's an actual 4D space "out there"). The very reason for using analogies and other such non-mathematical tools is because of the difficulty of developing a visual conception of the mathematical equations and symbols that one is manipulating. It's easy enough to push the symbols around, but at the end of the day, it's nice to be able to have some glimpse of just what it is we have been manipulating all day.

I'm not targeting any method in the above but just thought it would be useful to provide a summary at this stage.
I am hoping that I have discovered a new method that will hopefully help to make things clearer. We'll only be able to tell once we see the end result.
[...]

I like your little teaser. I just read through all your previous steps again, and I like where this is going. Now I'm waiting to find out what's coming next! I'm especially eager to find out what your new method is. Judging from what you've presented so far, it promises to be very interesting indeed.

[gonegahgah]

However, I think everyone will agree that the unfolding method does not really help us visualize 4D at all. A 600-cell, for example, unfolds into what looks like a big amorphous tangle of tetrahedra (600 of them in total), leaving the reader with practically no clue as to how such a thing could assemble itself into a 4D object, much less what such a thing would look like.

LOL. Cool, I didn't think of the unfolding method.

This is not quite accurate; the prism effect really only happens for objects that are prism-like, for example the tesseract. It doesn't quite produce the same effect with other shapes. There are also different types of projections -- the one you're describing sounds like the oblique projection method. I personally prefer perspective projections, in which farther objects appear smaller, much like they do in our familiar 3D world. Or, when I'm exploring the structure of something, I use parallel projections -- sorta like engineering drawings with 2-point and 3-point perspectives.

True, and that does sound like the best variant of that method.

I like your little teaser. I just read through all your previous steps again, and I like where this is going. Now I'm waiting to find out what's coming next! I'm especially eager to find out what your new method is. Judging from what you've presented so far, it promises to be very interesting indeed.

Hopefully it measures up to the pre-hype .

Drum roll please... LOL, just kidding.

Step 15. Looking right high; low left - the rotation method, 2Der style:



This hopefully starts to begin to give a beginning idea of a new method that I am hoping will allow us explore 4D to a greater degree and in person (well 1st person that is).
There is much more to be explained regarding the method but this picture demonstrates a simple example of making a scene that makes 3D come more alive for a 2Der.
It's not the scene but starts to explain how it would be made. I'll explain it further soon...

The method makes certain assumptions about our respective dimensional worlds such as there being an up and forward, and that objects have a usual purpose and shape due to this. I think in most cases that the assumptions can be drawn by logical inference and that the rotational method helps to make this more apparent.

For example, it is fairly safe to assume that a tree can be depicted as having a vertical trunk whether in 2D, 3D or 4D. This makes it easier to recognise across dimensions.
The rotational method should allow us to see the trunk pretty much as we do now even as we move through the 4th direction.
The branches however, I'm forseeing, would be quite spectacular; whilst still being familiar - in the rotational model.
The rotational method also makes it easier to explain a 4Ders retina and how they see; which I'll do soon.

This first example of a rotational method approach may help to start giving ideas of what I'm looking to have...

[4Dspace]

Therefore, whatever the coordinates of the said writing are, it is unobscured from the 4D viewpoint. It follows that the writing on all 6 faces of the cube are simultaneously visible.

Is this your answer to this?

[...] The point of my previous post you forgot to address was:

Now, explain to your readers here how is that possible for a plane to show both of its sides to a given POV in Euclidean space.

So, you changed your mind, congratulations. That's quite a departure from your opinion just a week ago, when you claimed that both sides of the cube 6 faces are visible simultaneously. Now you decided that all 6 outer faces of the cube are visible from 4D at the same time, lol. It did not bother you, the supposed expert, that a cube defines a hyperplane in 4D -- you still claim that you're able to see this hyperplane from both sides, just like you did a 2d plane, lol. That's why I keep calling your vision omnipresent.

But look, you're talking about the hyperplane here, in different context:

When you draw this dividing hyperplane in the given example (the cube at (±1,±1,±1,0) and the viewpoint at (0,0,0,5)), the cube lies in the hyperplane. This hyperplane does not divide the cube at all, because of the plain and simple fact that it is co-planar with the hyperplane. Therefore there is no "inside" or "outside". The cube is a flat hyperface (i.e., a 3-face).

So, you admit yourself here that by seeing what's on all 6 faces of the cube at once, you see the dividing hyperplane from both sides, even though your POV is fixed on one side only. This is a definition of omnipresent vision.

The cube in our example lies in the hyperplane. The ray hits the face of the cube precisely when it hits the dividing hyperplane. Therefore, there is no division of the cube's faces into "inside" or "outside".

That's because you forgot that the cube's adjacent faces have the same orthogonal relationship in the 3d subspace they occupy. The 3 adjacent faces are perpendicular to each other, even in 4D, and they form convex or concave surface to your POV. The convex structure you view from 'outside' and concave, from 'inside'.

So, by claiming that you see the writing on all 6 faces at once, again you claim the impossible trick of seeing both sides of a (hyper)plane at once. In Euclidean spaces this trick requires omnipresent vision.

Furthermore, since the planes do not divide 4D space, using them to determine "in front" and "behind" is meaningless. "In front" and "behind" is meaningful only when the object divides space, since otherwise, you can stand beside it. Then are you "in front" or "behind"? The answer is neither. You can't use the object to determine "in front" and "behind" since it doesn't divide space.

First, above I already showed you where and how you're wrong. Second, this 'front' and 'behind' pertains to the POV and the object. A plane can intersect the line of sight to a given POV. The fact that you can change your POV and "walk around it" is meaningless in this context.

Why should I answer anything, since you don't even bother to read what I wrote?

You already answered plenty. You showed that you do not see 4D and have no understanding of geometry, starting with its most basic tenets. Thus, according to you, you are able to see both sides of a (hyper)plane from a single POV, lol.

I have been trolled. I keep taking the time to reply and explain myself, and all for what? To be ignored. Not even read. Do you think I'm doing this because I like wasting time speaking to the air? That I have too much free time on my hands so I just love writing stuff that nobody reads? So let everyone who reads this forum be the judge. You criticize us for using dimensional analogy, and so I have presented solid mathematical proof of my position. You have yet to present a valid disproof of what I presented, or, for that matter, a proof of your own wild claims. You can rail against it all you like, it does not change the facts. A fair-minded reader will know who is the one who got it wrong, and who is the one who is misrepresenting the situation. There is no need for me to defend anything. As I've said many times before, the mathematics speaks for itself.

You have no shame. You need to apologize to all those people you misled, to all those who tried to argue with you in the past and whom you have-a-nice-dayed into leaving. To all those whom you managed to convince that they were wrong and you were right, while in fact it all was the other way around.

You should start with apologies to ac2000 who asked you a simple question in this very thread.

[Hugh]

Step 15. Looking right high; low left - the rotation method, 2Der style:



This hopefully starts to begin to give a beginning idea of a new method that I am hoping will allow us explore 4D to a greater degree and in person (well 1st person that is).
There is much more to be explained regarding the method but this picture demonstrates a simple example of making a scene that makes 3D come more alive for a 2Der.
It's not the scene but starts to explain how it would be made. I'll explain it further soon...

Whoa, this is looking awesome gonegahgah... really enjoying your evolving pictures!

[quickfur]

[...]

This is not quite accurate; the prism effect really only happens for objects that are prism-like, for example the tesseract. It doesn't quite produce the same effect with other shapes. There are also different types of projections -- the one you're describing sounds like the oblique projection method. I personally prefer perspective projections, in which farther objects appear smaller, much like they do in our familiar 3D world. Or, when I'm exploring the structure of something, I use parallel projections -- sorta like engineering drawings with 2-point and 3-point perspectives.

True, and that does sound like the best variant of that method.

There are other variants as well. One of my least favorite (I admit I'm biased here) is the so-called Schlegel diagram, where you do a perspective projection to a point not outside the object proper, but on its surface. Of course, you can't really do this in real-life (if your eye is right on the surface you can't really see much), but mathematically you can compute the coordinates of the result. It turns out that when you do this, the nearest face of the object will expand such that its projection image covers the image of the rest of the object. The rest of the object then appears, highly-distorted, inside this nearest face. For example, here's a dodecahedron from the usual "outside" point of view:



Here's a dodecahedron represented as a Schlegel diagram:



The nice thing about Schlegel diagrams is that you avoid having overlapping faces, which will happen when you do the "outside" point of view projection, plus it shows you how all the faces are connected.

The not-so-nice thing IMO, is that you don't really get a good idea of the global shape of the object. Based on the second image above, for example, you can hardly recognize the overall shape of the dodecahedron, even though its connectivity structure is crystal-clear. The first image has multiple faces occupying the same 2D areas, but it's much more helpful in seeing the dodecahedron's overall shape. Plus, in the Schlegel diagram the shape of the faces suffer from too much deformation that their actual shape is not so obvious.

[...]Step 15. Looking right high; low left - the rotation method, 2Der style:


[...]

Aha! I see where this is going, based on what you wrote before in other threads. This is a nice alternative approach to the multi-frames method, except that instead of doing multiple parallel frames, you're doing rotated frames around the vertical. Let's see how you develop this idea.

[ac2000]

Now place our viewpoint at (5,0,0,0). This viewpoint lies in our 3D hyperplane, since the last coordinate is zero. Draw a line from (5,0,0,0) through the origin (0,0,0,0). Let P be the 2D plane (within the 3D hyperplane where w=0), perpendicular to this line. As 4DSpace has said, this plane is uniquely defined. A simple calculation will show that points that lie on the plane P have coordinates of the form (0,y,z,0). So far so good, there are two free variables, corresponding with the 2 degrees of freedom in this plane.

I like this example with the plane and the circle, because I think I understand most of it. Since I'm bad at maths in general the reason for this must be that you explained it very well .

This hopefully starts to begin to give a beginning idea of a new method that I am hoping will allow us explore 4D to a greater degree and in person (well 1st person that is).

That would be cool . Your pictures are quite interesting, I'm not sure if I have fully understood the rotational concept yet, though. I think I have to reread the posts again and think a bit more about them.

There are other variants as well. One of my least favorite (I admit I'm biased here) is the so-called Schlegel diagram, where you do a perspective projection to a point not outside the object proper, but on its surface.

I wanted to know more about the Schlegel diagram and so I looked it up.
On the wikipedia Page on Schlegel diagrams it says about the hypercube:
"There are 8 cubic cells visible, one in the center, one below each of the six exterior faces, and the last one is inside-out representing the space outside the cubic boundary."
I don't understand the last part at all: why is the eighth cube supposed to be "inside-out" and why does it represents the space "outside" the cubic boundary?
Assuming this is true, I wonder if it's still true with other kinds of 4d=>2d projections of the hypercube or whether it has something to do with the kind of projection?

Also, the Schlegel diagram, as well as any other kind of visualisazion of the hypercube I have come across so far, do not seem to take into account the presumed 3d-retina of a 4d being.
According to what I have read so far, the hypercubes and/or normal cube (the latter with a thin "helper"-4d Dimension) should somehow represent the parts with volume (or "bulk") more prominently whereas the faces and edges should look more small/marginal.
I'm pretty sure that it isn't possible to represent it mathematically correct in that way or to represent a complete cube/hypercube all at once. But maybe it would be possible to represent at least a section of it in a different way, so one could get a feeling of how "bulk" looks like if we assume a 3d-retina.

[gonegahgah]

Thanks Hugh, QuickFur & AC2000.

There are other variants as well. One of my least favorite (I admit I'm biased here) is the so-called Schlegel diagram, where you do a perspective projection to a point not outside the object proper, but on its surface. Of course, you can't really do this in real-life (if your eye is right on the surface you can't really see much), but mathematically you can compute the coordinates of the result. It turns out that when you do this, the nearest face of the object will expand such that its projection image covers the image of the rest of the object. The rest of the object then appears, highly-distorted, inside this nearest face.

Cool, thanks for that QuickFur. I was curious and had a look at Wikipedia for Spatial 4D. I couldn't see that they mentioned the different modelling techniques. That might be useful for someone to add or suggest one day.

Aha! I see where this is going, based on what you wrote before in other threads. This is a nice alternative approach to the multi-frames method, except that instead of doing multiple parallel frames, you're doing rotated frames around the vertical. Let's see how you develop this idea.

That's it exactly. I'm hoping that some computer games can be developed using this technique to give us a new expanded 4D experience...
...and I'm thinking a bit about how I would like the story, for the first game, to unfold which I'll write about later...

Just as a slight diversion for the moment I had one of those moments of clarity last night as I was going to sleep. Well, two moments actually.
I'll explain the 2nd moment first and I'll post the 1st moment later today because it requires several diagrams.
The 2nd could use diagrams too but I'll just explain it here.

Both moments had to do with how a 4Der pictures a cube. The 2nd moment brought time into things temporarily (or maybe that should be temporally) to help us understand.
This is probably something that has already been written about but it just occurred to me last night.

Another separate point that occurred to me (maybe we should call it moment of clarity 3 ) was that ana and kata need to be separated from viewing with a 4th directional ability.
A 4Der can, for example, view a cube from one of our 3 dimensional directions and still see a cube how they would see it...
The pre-requisite to this is that they need to hold and rotate the cube with them so that they are still not looking at it edge on.
(The result of the 4Der turning it like this - as QuickFur has mentioned - is that we would only be left seeing a square).
So, I'm feeling their needs to be another two terms to indicate whether a 4Der is looking at an object from one direction or it's opposite.
I guess front and back could remain useful. ie. We look at the front of a square and then we turn it around and look at the back.

The same terminology then needs to be expanded when we talk about cubes. We 3Ders can look at the front of a cube, turn it around and look at the back. We can also turn it quarter around and look at it from either side, or turn it up or down to look at the top or bottom. To us these are all equivalent to looking at the cube from its prominent views.

A 4Der, however, instead thinks in terms of looking at a cube's front or turning it around and looking at its back.
For them to look at it any other way, they would conceive as themselves looking at the cube edge on.
The equivalent for us would be to look at a square edge on which is not our usual thing.
The same goes for the 4Der. They will generally only look at a cube front on or turn it around and look at its back. They generally won't look at a cube edge on.

And as I mentioned, when a 4Der turns a cube edge on they see it as we would see a cube; just as we also see a square how a 2Der does when we look at that square edge on.

Now back to the moments...

Moment 2:

To help explain how a 4Der sees things I am replacing the 4th direction with time.

If we do this we can throw ourselves into the picture as well with a little bit of modification to ourselves.
The modification is that we have to be able to see past, present and future all at once.

If you look at a cube over time the cube remains unchanged. It looks exactly the same.
Your retinas also travel through time, like the cube, otherwise you would only see the cube while your retinas existed.
ie. We don't see the cube before we were born and we can't see it after our death.

Now, if instead of travelling through time our retinas were able to connect all those moments of time, and our brain could process all these images as if at one time, then this is how a 4Der would see a cube. If they look at - or we look at all at once through time - a particular square face then they/we would see the same square face connected through time to form a time cube. So basically through this process they are seeing one of the cube faces of a tesseract.

Of course they can do this with any face of the resultant tesseract so that they see six cube faces by turning the tessearact around.
Like we can only see 3 cube faces at at one time they can only see 4 tesseract 'faces' at any one time.

To be a tesseract it has to have all equal sides so at some point of time the tesseract would have to pop into existance and then pop out of existance.
I'm not sure how you would measure centimetres into the timeline but the tesseract would have to be n centimetres long, wide, high and through time.
Our eyes and brain would have to be able to see through time with the left of time and right of time (ana/kata) disappearing off into the perpendicular time horizon.

Now, how about a cube cube; and not a tesseract?

Well, for us to see a cube with a 4Der's eyes the cube would have exist at only one point in time... or does it?
The trick is that this view is the same view a 4Der sees when they are looking at the cube edge on.
If the cube existed for only one moment and our eyes were able to see all moments at once then we would basically be looking at the cube edge on as a 4Der does.

So how does the 4Der - and us with our retinas seeing through time - see a cube front on then?
This is the part that really explains how a 4Der normally looks at our cube front on or via its back.
What we do is take a single square face and put that square face through multiple adjacent time moments for a distance equal to the side of the square.
If our eyes can see all those moments at once then we are effectively looking at a cube front on as a 4Der sees it.

Except that you have to remember that all those time frames are connected to our up-down and left-right views and this forms a cube that we can see in its entirety all at once.
We don't see a series of individual time frames with eyes in each time frame. We see all those time frames at once to form a time cube.

Just as a square is a series of lines but we don't think of them as lines; the 4Der sees a cube from front not as a series of squares but as a cube that they can see all at once.
You have to remember that the first time frame is not the front and the last time frame is not the back; or vice versa.
ie. that equally means the last time frame is not the front and the first time frame is not the back. To the 4Der all these time frames would be equal.
So instead, if our eyes could see across time then the first square face would just be the time-left of the cube and the last square face would just be the time-right of the cube.
Just as for us the first line of a square is just the left of the square and the last line of the square is just the right of the square.

[quickfur]

[...]
So, I'm feeling their needs to be another two terms to indicate whether a 4Der is looking at an object from one direction or it's opposite.
I guess front and back could remain useful. ie. We look at the front of a square and then we turn it around and look at the back.

And here it is important to understand that what we think of as front and back is different from what the 2Der thinks of as front and back. When a 2Der looks at the square, they see one of the square's edges (or two if they are looking at a corner). This edge, to them, is the "front" of the square, and the edge opposite it is the "back".

However, when we 3Ders look at the same square face-on, we do not see any of the edges as "front" or "back"; all of the edges are equivalent from our viewpoint. To us, "front" refers to the area of the square that's facing us, not any of its edges, and "back" refers to the area facing away from us. But the 2Der will have a hard time with this concept. To them, area fills space; so how can an area possibly "face" any direction? Much less to say that a square's area faces two directions. This concept is very hard to understand for the 2Der.

Now translate the situation to 3D. We are looking at a cube placed in front of us, and we see one of its 6 faces (we may see more faces if we're looking at an edge, or a corner, but let's just consider this simple case first). To us, this face is the "front" of the cube, and the face directly opposite it is the "back". The rest of the faces are on the "side".

When a 4Der looks at the same cube, though, from their viewpoint outside of 3D space, they don't see any of the cube's faces as "front" or "back". All 6 faces are equivalent from their viewpoint. To them, "front" does not refer to any of the cube's faces at all; it refers to the volume of the cube that faces them. Likewise, "back" refers to the volume of the cube that faces away from them. To us, this is something mysterious and hard to understand. Volume fills space; how can volume "face" any direction? The answer is that in 4D, volume does not fill space; it acts as "4-area", that is, the boundary of a 4D bulk (i.e., hypervolume). The cube's volume lies within a 3D hyperplane, and every hyperplane in 4D has two sides, a "front" and a "back". This "front" and "back" correspond with the 4D directions ana and kata; they have nothing to do with what we consider as "front" and "back".

The same terminology then needs to be expanded when we talk about cubes. We 3Ders can look at the front of a cube, turn it around and look at the back. We can also turn it quarter around and look at it from either side, or turn it up or down to look at the top or bottom. To us these are all equivalent to looking at the cube from its prominent views.

A 4Der, however, instead thinks in terms of looking at a cube's front or turning it around and looking at its back.

There is an additional dimension (har har) to this. When we 3Ders are looking at a square, we may spin it clockwise or counterclockwise without changing the direction it's facing. If we're looking at the square face-on, we can rotate the square and still have it facing us.

To a 2Der, though, when we do this, the edge of the square that was facing them before will be moved away, and another edge will come into view and face them. Then it will move away too, and a third edge will come into view, etc.. So to a 2Der, when the square is spinning, which edge(s) are "front" and "back" are also changing. To us, however, the "front" of the square is still the same front, and the "back" is still the same back. They just appear to us to be spinning. But the same face continues to face us.

When a 4Der looks at a cube, there are several ways it can be spun around while still facing the 4Der. These rotations correspond with all the possible 3D rotations (including the principal rotations around the X, Y, Z axes, respectively). The 4Der can rotate the cube in any of these directions and still have the same side of the cube's volume face them. From our viewpoint, however, the cube is tumbling around; what used to be the front face is rotating out of view, while another face is coming into view in its place. To us, which faces are "front" and which are "back" is constantly changing as the cube rotates; but to the 4Der, it's still the same "side" (i.e. 4-area) of the cube facing them. It's just the cube's 6 faces that are spinning around the cube's volume.

[...]Moment 2:

To help explain how a 4Der sees things I am replacing the 4th direction with time.

This is quite a common method people use for understanding 4D. It's somewhat equivalent to the multiframes method, except that you can treat time as a continuous quantity, so there's not that breaking-up into multiple discrete frames.

[...] Now, if instead of travelling through time our retinas were able to connect all those moments of time, and our brain could process all these images as if at one time, then this is how a 4Der would see a cube. If they look at - or we look at all at once through time - a particular square face then they/we would see the same square face connected through time to form a time cube. So basically through this process they are seeing one of the cube faces of a tesseract.

I like the twist you added to this method. Most people just stop at identifying the 4th dimension with time, and think of 4D objects as morphing 3D objects. You really have to consider the entire thing -- every "morph" of the object over time -- all at once, in order to get the right understanding of how 4D works. It has to be considered as one, unchanging object that extends over the 4th dimension, not a 3D object that's changing over time.

The limitation to this method, though, is that it is essentially the slicing / multiframe method in disguise. Each point in time is basically a "slice" or "frame" through the middle of the object. As such, it suffers from the same limitations as slicing: familiar objects like the tesseract, if placed in the usual orientation lined up with all 4 axes (x,y,z and time), will give the familiar cube-shaped slices. Rotate the tesseract a bit, though, and things get considerably hairier. Remember that sequence of slices that I showed earlier?



These are actually slices of the tesseract, rotated so that two opposite corners line up with the 4th direction. If you were to watch these slices over time, they would start with a point that grows into a tetrahedron, then the tetrahedron gets truncated into an octahedron, then the octahedron becomes an inverted truncated tetrahedron, which shrinks into an inverted tetrahedron, which then shrinks back into a point.

See how different this is compared to the usual view of the tesseract as a cube that exists over a fixed amount of time? They are the same object in 4D!

(I bet you are having a hard time imagining just how the above sequence of point -> tetrahedron -> truncated tetrahedron -> octahedron -> truncated tetrahedron -> tetrahedron -> point can possibly be the same object as the sequence cube -> cube -> cube -> ... -> cube. Or, if you like, how an object that starts from a point, morphs into a tetrahedron, ... octahedron ... tetrahedron ... point can be the same as the object that appears as a cube, stays as a cube, and then disappears.)

[...]Now, how about a cube cube; and not a tesseract?
[...]
What we do is take a single square face and put that square face through multiple adjacent time moments for a distance equal to the side of the square.
If our eyes can see all those moments at once then we are effectively looking at a cube front on as a 4Der sees it.

You can also see it as something that starts out as a point, grows into a triangle, then truncates into a hexagon, then an inverted triangle, then back to a point. Like this:



Except that instead of separate frames like the above, it's a polygon that's morphing over time between those shapes.

(Wait, what? How? The above shapes form a cube?! Yes they do:

This may be the first time for some of you to realize that a cube can make a hexagonal cross-section. )

[...]Except that you have to remember that all those time frames are connected to our up-down and left-right views and this forms a cube that we can see in its entirety all at once.
We don't see a series of individual time frames with eyes in each time frame. We see all those time frames at once to form a time cube. [...]

Yes. For all its limitations, this method does give us some insight into how the 4Der's vision works. Their retina covers a 3D volume.

[wendy]

One of these people have admitted in another thread that he is stringing quickfur et al along.

The real problem is that terms like 'side' is freely let to slide from side (margin) of a square to side of a cube (face) to side of a piece of paper or polytope (facing half), and that saying any allows one to slide meaning to any other.

It's more to their playing nimm rather than any real enquiry into four dimensions.

[gonegahgah]

When a 4Der looks at the same cube, though, from their viewpoint outside of 3D space, they don't see any of the cube's faces as "front" or "back". All 6 faces are equivalent from their viewpoint. To them, "front" does not refer to any of the cube's faces at all; it refers to the volume of the cube that faces them. Likewise, "back" refers to the volume of the cube that faces away from them. To us, this is something mysterious and hard to understand. Volume fills space; how can volume "face" any direction? The answer is that in 4D, volume does not fill space; it acts as "4-area", that is, the boundary of a 4D bulk (i.e., hypervolume). The cube's volume lies within a 3D hyperplane, and every hyperplane in 4D has two sides, a "front" and a "back". This "front" and "back" correspond with the 4D directions ana and kata; they have nothing to do with what we consider as "front" and "back".

Exactly. I would just elucidate that ana and kata, although unavailable to us, are just another direction to a 4Der. They could look at their specific front or back of a cube from any direction including our left-right, forward-back, up-down or their ana-kata. Ana and kata are just directions that we can't use. However as QuickFurs animation shows - that I sadly can't find - if they look at their specific front or back of a cube from any of our available directions then it will not look like a cube to us.

When a 4Der looks at a cube, there are several ways it can be spun around while still facing the 4Der. These rotations correspond with all the possible 3D rotations (including the principal rotations around the X, Y, Z axes, respectively). The 4Der can rotate the cube in any of these directions and still have the same side of the cube's volume face them. From our viewpoint, however, the cube is tumbling around; what used to be the front face is rotating out of view, while another face is coming into view in its place. To us, which faces are "front" and which are "back" is constantly changing as the cube rotates; but to the 4Der, it's still the same "side" (i.e. 4-area) of the cube facing them. It's just the cube's 6 faces that are spinning around the cube's volume.

Cool.

I like the twist you added to this method. Most people just stop at identifying the 4th dimension with time, and think of 4D objects as morphing 3D objects. You really have to consider the entire thing -- every "morph" of the object over time -- all at once, in order to get the right understanding of how 4D works. It has to be considered as one, unchanging object that extends over the 4th dimension, not a 3D object that's changing over time.

Thanks QuickFur. Yes that's exactly right.

The limitation to this method, though, is that it is essentially the slicing / multiframe method in disguise. Each point in time is basically a "slice" or "frame" through the middle of the object. As such, it suffers from the same limitations as slicing: familiar objects like the tesseract, if placed in the usual orientation lined up with all 4 axes (x,y,z and time), will give the familiar cube-shaped slices. Rotate the tesseract a bit, though, and things get considerably hairier. Remember that sequence of slices that I showed earlier?

Cool, another method. I didn't even think of the time method at the time nor to add it to the list... What you say is exactly true too. I had thought about this in relation to my rotation method as well. I'm hoping that the rotation method exposes more bulk; but even if it doesn't then I'm expecting a common concept of up-down to help keep objects familiar. For example, I don't think my rotation method would work for a space game where there is no down. I was thinking with a computer game that what you mention could actually be a feature. First off you could start with familiar objects presented in regular easy to resolve ways; even when they are four dimensional or somewhere off in the 4th direction. The last section of the game, once the user has their head around four dimensions (which I'm hoping the game will help to deliver), would then start to throw in some unobvious shapes that the player was hopefully better equipped to handle by then.

(I bet you are having a hard time imagining just how the above sequence of point -> tetrahedron -> truncated tetrahedron -> octahedron -> truncated tetrahedron -> tetrahedron -> point can possibly be the same object as the sequence cube -> cube -> cube -> ... -> cube. Or, if you like, how an object that starts from a point, morphs into a tetrahedron, ... octahedron ... tetrahedron ... point can be the same as the object that appears as a cube, stays as a cube, and then disappears.)

Well if its any hint, I wasn't willing to take that bet . Thanks for showing the sequence to elucidate on how it fit.


Now, here's the 1st thing that I thought of as I was going to sleep last night:



It is the closest I think we can get to conveying to us how a 4Der sees a 3D cube. It may also be another rotational method; though it's not specifically what I was thinking of with the one I am exploring. I had been wondering if something similar to it would emerge somehow as I expand on my rotational model. I'll have to wait and see too.

What the pictures shows is how a 4Der sees our cube from their 4th direction if we presented the result of rotating the multiple (seemingly simultaneous) layers of their retinas. They see the red insides primarily - which the picture manages to do - whilst still seeing the green edges of the cube. It should be remembered that the 4Der still sees the green as connected and not cut into slices like this. However they see the red inside which we can't. However you have to remember, as per previous examples provided for the 2Der, that the 4Der could paint the inside of a cube on its two 4-faces and, even if we cut into the cube, we would be none the wiser. We would still see red, even if they painted the insides from their perspective blue.

The main intent of this picture is to present a cube in such a way as to allow us to see its insides almost as a 4Der can if we rotated the layers of their retina.
But it's a bit too much for my brain at the moment. Sorry. If I can I'll clarify later.

However, you should note that the green sides are all the same length as denoted by the arrows on the picture. You should also note that although the squares are overlapped they are not actually in front of or behind each other to the 4Der. Each slice is exactly on the same level and it should again be noted that the slices are purely for our benefit. The 4Der doesn't see the slices but instead sees a continuous inside. So you have to imagine that the cube slices and the 4Ders retinas are purely the result of rotating the layers of the 4Ders eyes rather than superimposing them as we would normally want to do to get a good understanding of how they see.

Also, the following are both the same then in this example:



I haven't explained this exactly as I wan't to as I'm a bit tired. I'll see how it looks in hindsight.

[gonegahgah]

What the pictures shows is how a 4Der sees our cube from their 4th direction if we presented the result of rotating the multiple (seemingly simultaneous) layers of their retinas. They see the red insides primarily - which the picture manages to do - whilst still seeing the green edges of the cube. It should be remembered that the 4Der still sees the green as connected and not cut into slices like this. However they see the red inside which we can't. However you have to remember, as per previous examples provided for the 2Der, that the 4Der could paint the inside of a cube on its two 4-faces and, even if we cut into the cube, we would be none the wiser. We would still see red, even if they painted the insides from their perspective blue.

It is worth re-iterating QuickFur's pictures - those ones depicting the 3rd side of a flat 2Ders plane with both a 2D and 3D observer - and to say, as QuickFur has also said in relation to comparing from 3D->4D in relation to those pictures, that the consecutive slices I've shown in my last post only give us an approximation of what a 4Der sees. We can still only depict 3D atoms and the 4Der would have 4D atoms. Those atoms would then have a fourth set of sides that can be painted by the 4Der whereas we can only paint three sets of dimensional sides.
That fourth set of sides - in a free form 4D space - could however be rotated into our 3D plane for viewing but we would only see a tiny bit of it at a time.

Our little 2Der will demonstrate for us again:

Step 16. Rotating higher dimensions into ours:



In the left image the 2Der is looking at his square. Then in the right image we rotate the square into the 3rd dimension and the 2Der can now see a little of the third side.

So the 2Der can see the 3D parts of our atoms but they can only see <1deg of our sideways dimension of them at once.
So even when the square is rotated into our dimension the 2Der still only sees a single line; though they can now see the blue paint we used.
You notice that in the right image that we now see the square as the 2Der sees it.

The same goes for us. We could observe the 4th side of our 4Der's atom but only at <1 deg of the extra direction at a time.

[quickfur]

[...]
It is worth re-iterating QuickFur's pictures - those ones depicting the 3rd side of a flat 2Ders plane with both a 2D and 3D observer - and to say, as QuickFur has also said in relation to comparing from 3D->4D in relation to those pictures, that the consecutive slices I've shown in my last post only give us an approximation of what a 4Der sees. We can still only depict 3D atoms and the 4Der would have 4D atoms. Those atoms would then have a fourth set of sides that can be painted by the 4Der whereas we can only paint three sets of dimensional sides.
That fourth set of sides - in a free form 4D space - could however be rotated into our 3D plane for viewing but we would only see a tiny bit of it at a time.
[...]

And as wendy pointed out, we need to be careful when using the word "side", because it can mean different things (and does mean different things when in other dimensions). A "side" of the cube can mean one of its faces, or the near half of the cube which is visible to us. So we should be clear which meaning is meant. Furthermore, the meaning changes across dimensions: a "side" in 2D can mean an edge, or the set of edges facing a particular 2D viewpoint. In 4D, a "side" may refer to a cell (a 3D construct) or the part of the hypersurface (which forms a 3D manifold) that is facing a particular 4D viewpoint.

So the "three sets of dimensional sides" is really not on the same level as the "fourth set of sides", because they refer to things of different dimensions. The former refers to 2D surfaces that we can see, paint on, etc., but the latter refers to no mere 2D construct, but to hyper-areas which are 3D in their extent; in fact, to hyper-areas whose margins are the "three sets of dimensional sides". So we need to be careful of our terminology, lest we cause unnecessary confusion.

[quickfur]

Sorry for the late response. I've been having a hard time keeping up with everything that's going on here.

Now place our viewpoint at (5,0,0,0). This viewpoint lies in our 3D hyperplane, since the last coordinate is zero. Draw a line from (5,0,0,0) through the origin (0,0,0,0). Let P be the 2D plane (within the 3D hyperplane where w=0), perpendicular to this line. As 4DSpace has said, this plane is uniquely defined. A simple calculation will show that points that lie on the plane P have coordinates of the form (0,y,z,0). So far so good, there are two free variables, corresponding with the 2 degrees of freedom in this plane.

I like this example with the plane and the circle, because I think I understand most of it. Since I'm bad at maths in general the reason for this must be that you explained it very well .

Well, I tried. But apparently not well enough for 4DSpace to get it.

[...] I wanted to know more about the Schlegel diagram and so I looked it up.
On the wikipedia Page on Schlegel diagrams it says about the hypercube:
"There are 8 cubic cells visible, one in the center, one below each of the six exterior faces, and the last one is inside-out representing the space outside the cubic boundary."
I don't understand the last part at all: why is the eighth cube supposed to be "inside-out" and why does it represents the space "outside" the cubic boundary?

Well first, let's be clear that the Schlegel diagram of the tesseract (hypercube) is a 3D diagram. The 2D image you see on the wikipedia page is just a 2D view of that 3D diagram (since obviously the computer screen can't display a 3D diagram).

As for the inside-out thing, this is an artifact of the viewpoint being on the eighth cube. When this happens, some parts of the interior of that cube projects to points at infinity (this is a result of the way the projection is calculated). The Schlegel diagram has the property that every hypersurface of the tesseract projects uniquely to a section of 3D space; that is, the images of the tesseract's 8 cubes form a tiling of 3D space. Seven of the cubes tile the inside of a cubical hyper-area (i.e., volume at the center of the Schlegel diagram); the 3D space outside of that cubical boundary is the image of the interior of the 8th cube.

Assuming this is true, I wonder if it's still true with other kinds of 4d=>2d projections of the hypercube or whether it has something to do with the kind of projection?

This is only true of the Schlegel diagram; it arises because the viewpoint sits on the surface of the object being projected.

As I said, I'm not particularly fond of Schlegel diagrams, precisely for this reason: the cells are shown highly-distorted, and, in the case of the 8th cube, even inverted in a way that is hard to understand for non-mathematical types. I much rather prefer perspective projection from a distance, i.e., from a point outside the object being viewed, preferably far enough away that it an analogous viewpoint to how we normally look at things in our physical 3D world (e.g., when you look at a cube on your desk, it's at least 2-3 feet away from your eyes; holding it against your face generally isn't helpful in seeing its overall shape).

Also, the Schlegel diagram, as well as any other kind of visualisazion of the hypercube I have come across so far, do not seem to take into account the presumed 3d-retina of a 4d being.
According to what I have read so far, the hypercubes and/or normal cube (the latter with a thin "helper"-4d Dimension) should somehow represent the parts with volume (or "bulk") more prominently whereas the faces and edges should look more small/marginal.
I'm pretty sure that it isn't possible to represent it mathematically correct in that way or to represent a complete cube/hypercube all at once. But maybe it would be possible to represent at least a section of it in a different way, so one could get a feeling of how "bulk" looks like if we assume a 3d-retina.

Oh no, representing it mathematically is definitely possible -- very easy, in fact. The difficulty lies in how to display such a thing on the computer screen, because the computer screen can only show a 2D image. And it's no fault of the computer screen, either -- our own eyes can only see 2D (3D is inferred, as I've mentioned on many occasions), so even if we were to build a 3D model representing the 3D image that forms in the 4Der's retina, we could only see its boundary: the inside would be opaque to us.

When we 3Ders look at a 3D object, say a cube or an icosahedron, our eyes capture a 2D image that has a square or hexagonal boundary, with some edges and corner lying internal to this boundary. The region of interest really lies in what's inside the boundary; but a 2Der looking at a 2D model of this image could only see the square or hexagonal boundary. They might be tempted to identify the shape of that boundary with the 3D shape, but that is inaccurate, because they're missing out on the most important part of the image: what's inside the boundary.

So there are several ways to mitigate this problem. One approach, the one I like to use, is to make the 3D image semi-transparent, so that we can see through its parts. This approach is the one used on the wikipedia page for the Schlegel diagram of the tesseract -- only the edges are shown, not the faces or the volumes. Keep in mind, though, that it's actually not possible to directly show the volume, because again, our eyes wouldn't be able to see the entirety of that volume. The best we can do is to show the faces that form the boundary of that volume, and allow our brain to infer the shape of that volume based on how the faces enclose it.

Another approach is to form the 3D image as a 3D array of pixels stored in the computer, and have the computer show us that array one 2D slice at a time. This is the approach used by Patrick Stein (who used to frequent this forum years ago, but sadly has been inactive since) in his n-dimensional raytracer, which can ray-trace objects of any dimension.

And you're quite right that volumes would feature prominently in the 4Der's vision, rather than edges and faces, so when we look at these projection images, we need to consciously keep that in mind, and think in terms of volumes, rather than merely the edges and faces, which are there only to help our brains deduce where the volumes are.

[ac2000]

Sorry for the late response. I've been having a hard time keeping up with everything that's going on here.

Yes, it's quite a lot of information and it's difficult to keep up with it. I wonder how you all can come up with so many ideas and write so much in such a short time. Nevertheless I really enjoy reading it .

[...] I wanted to know more about the Schlegel diagram [...]

As for the inside-out thing, this is an artifact of the viewpoint being on the eighth cube. When this happens, some parts of the interior of that cube projects to points at infinity (this is a result of the way the projection is calculated).

The Schlegel diagram has the property that every hypersurface of the tesseract projects uniquely to a section of 3D space; that is, the images of the tesseract's 8 cubes form a tiling of 3D space. Seven of the cubes tile the inside of a cubical hyper-area (i.e., volume at the center of the Schlegel diagram); the 3D space outside of that cubical boundary is the image of the interior of the 8th cube.

OK, thanks for explaining! I think I got that part about the inside-out thing at least theoretically. I still have to think some time about the tiling stuff ...

This is only true of the Schlegel diagram; it arises because the viewpoint sits on the surface of the object being projected.
As I said, I'm not particularly fond of Schlegel diagrams, precisely for this reason: the cells are shown highly-distorted, and, in the case of the 8th cube, even inverted in a way that is hard to understand for non-mathematical types. I much rather prefer perspective projection from a distance, i.e., from a point outside the object being viewed, preferably far enough away that it an analogous viewpoint to how we normally look at things in our physical 3D world (e.g., when you look at a cube on your desk, it's at least 2-3 feet away from your eyes; holding it against your face generally isn't helpful in seeing its overall shape).

I see. There is this other kind of projection called "stereographic projection" which is sometimes used for 4D representation, too. The 4D objects are first kind of bloated into spheres and then projected to 2D. I don't know in what aspects it's better or not compared to the other types of visualizations. It seems to be an interesting approach, though.
This kind of "stereographic projection" of four 4D objects is made use of in the dimensions films. I suppose you and/or gonegahgah already know them? If not, and you might be interested, here are the links:
This is a general overview about the films: http://www.dimensions-math.org/Dim_tour_E.htm
Here the films can be downloaded: http://www.dimensions-math.org/Download_Lyon.htm
(They can also be watched online, but the quality is somewhat bad compared with the download versions. I'd recommend the "American English" version, because the speaker/presenter sounds much better than in the "English" version.)

So there are several ways to mitigate this problem. One approach, the one I like to use, is to make the 3D image semi-transparent, so that we can see through its parts. This approach is the one used on the wikipedia page for the Schlegel diagram of the tesseract -- only the edges are shown, not the faces or the volumes.
[...]
Another approach is to form the 3D image as a 3D array of pixels stored in the computer, and have the computer show us that array one 2D slice at a time. This is the approach used by Patrick Stein (who used to frequent this forum years ago, but sadly has been inactive since) in his n-dimensional raytracer, which can ray-trace objects of any dimension.

Thanks for the link! I have checked it out but for one thing, there seems to be only source code and no executable (but I have no compiler) and also the images are not so pretty.
I was rather thinking of dumping some code into POV-Ray, when at some distant time in the future I have some more than a vague idea of a simple visualization of, say, a 3D cube in 4D.
I'm familiar with POV-ray (though not so much with the functions/macros stuff) and the pictures are generally nicer. It seems to work well, at least for some kind of projections, when one feeds it with code from a program called "Mathematica" (I have never used this) because the pictures on Paul Nylanders homepage, e.g. of the 120 cell, look very well, and they are made with Mathematica and POV-Ray, it says there. (The 120 cell is there, when scrolling down a bit: http://nylander.wordpress.com/tag/mathematica/
Do you use Mathematica together with POV-Ray, too, or how do you figure out all those formulas and/or zillions of coordinates for the many beautiful Polychorons on your homepage?

And you're quite right that volumes would feature prominently in the 4Der's vision, rather than edges and faces, so when we look at these projection images, we need to consciously keep that in mind, and think in terms of volumes, rather than merely the edges and faces, which are there only to help our brains deduce where the volumes are.

I have a very vague idea of visualizing such a thing in a way that all the voxels of a 3d retina are filling the whole field of view, so that one gets the impression that one is somehow inside a bulk of many voxels (even though the depiction might not be mathematically 100% correct) and the edges and faces all looking very small and distant and marginal. Just to convey the impression of the bulkiness of such a 4D view.

[quickfur]

Sorry for the late response. I've been having a hard time keeping up with everything that's going on here.

Yes, it's quite a lot of information and it's difficult to keep up with it. I wonder how you all can come up with so many ideas and write so much in such a short time. Nevertheless I really enjoy reading it .

Well, the writing is the easy part. It's mostly trying to put in words what one sees in the mind.

But the ideas didn't just come overnight... at least for me, it took a long time to piece the puzzle together. For me, it all started when I suddenly realized one day that 4Ders have 3D retinas, and that just as we see the surfaces of 3D objects as areas, so they would see the surfaces of 4D objects as volumes ("hyperarea"). Although that was a "moment of clarity", as gonegahgah calls it, it took a long time to work out the consequences (and to check with the math to make sure it matches up!). I can't say I've fully explored all of the consequences yet; hence why these discussions are so interesting to me.

[...] I see. There is this other kind of projection called "stereographic projection" which is sometimes used for 4D representation, too. The 4D objects are first kind of bloated into spheres and then projected to 2D. I don't know in what aspects it's better or not compared to the other types of visualizations. It seems to be an interesting approach, though.
This kind of "stereographic projection" of four 4D objects is made use of in the dimensions films.

The main advantage of stereographic projections is that the angles between edges are preserved, unlike perspective/parallel projections, where right angles no longer show up as right angles when seen from an angle. So this gives you some idea about the actual connectivity of these cells in their native 4D space.

I still can't say I'm too fond of it, though, because it stills represents a distortion that doesn't happen with our own seeing of 3D, so it is not so easy to interpret the images.

[...] Do you use Mathematica together with POV-Ray, too, or how do you figure out all those formulas and/or zillions of coordinates for the many beautiful Polychorons on your homepage?

I'm a programmer by profession. I write programs to make the computer do all the work for me.

But yes, I do use povray. I have a tool for computing coordinates, which generates a 4D definition file that contains all the information about the vertices, edges, ridges (2D faces), and cells, which are then fed to another program that does the computations for projecting the object into 3D based on a given viewpoint. This projector program also lets me assign textures to various elements of the polytopes, to highlight/hide certain parts of the object for clarity's sake, etc.. Once the projection is done, it writes the 3D model of the result into a povray scene file that can be raytraced from any camera angle (specified externally by a .pov template file).

[...] I have a very vague idea of visualizing such a thing in a way that all the voxels of a 3d retina are filling the whole field of view, so that one gets the impression that one is somehow inside a bulk of many voxels (even though the depiction might not be mathematically 100% correct) and the edges and faces all looking very small and distant and marginal. Just to convey the impression of the bulkiness of such a 4D view.

Hmm. That is actually an excellent idea!! It's not hard to do with my setup: I just have to place the povray camera inside the 3D model produced by the 4D->3D projection program. Perhaps move it around a bit and make an animation out of it. The 3D model will thus surround you all around, as though it were a virtual 3D environment that you can explore. You'll be literally "moving inside the volumes" produced by the 4D->3D projection, and, by exploring these volumes from inside, hopefully develop an accurate idea of what sits in the 3D retina of the 4Der's eye.

I shall try this when I get some free time!

[Hugh]

I just have to place the povray camera inside the 3D model produced by the 4D->3D projection program. Perhaps move it around a bit and make an animation out of it. The 3D model will thus surround you all around, as though it were a virtual 3D environment that you can explore. You'll be literally "moving inside the volumes" produced by the 4D->3D projection, and, by exploring these volumes from inside, hopefully develop an accurate idea of what sits in the 3D retina of the 4Der's eye.

I shall try this when I get some free time!

I'm very interested in this quickfur, and would love to see the results!

[ac2000]

But yes, I do use povray. I have a tool for computing coordinates, which generates a 4D definition file that contains all the information about the vertices, edges, ridges (2D faces), and cells, which are then fed to another program that does the computations for projecting the object into 3D based on a given viewpoint. This projector program also lets me assign textures to various elements of the polytopes, to highlight/hide certain parts of the object for clarity's sake, etc.. Once the projection is done, it writes the 3D model of the result into a povray scene file that can be raytraced from any camera angle (specified externally by a .pov template file).

Aha, OK, so even three programs are needed for these images. I actually thought it would be easier, because POV-Ray is usually so versatile when dealing with all kinds of formulas and functions (at least from what I've seen of the more experienced POV-ers.)

[...] I have a very vague idea of visualizing such a thing in a way that all the voxels of a 3d retina are filling the whole field of view, so that one gets the impression that one is somehow inside a bulk of many voxels (even though the depiction might not be mathematically 100% correct) and the edges and faces all looking very small and distant and marginal. Just to convey the impression of the bulkiness of such a 4D view.

Hmm. That is actually an excellent idea!! It's not hard to do with my setup: I just have to place the povray camera inside the 3D model produced by the 4D->3D projection program. Perhaps move it around a bit and make an animation out of it. The 3D model will thus surround you all around, as though it were a virtual 3D environment that you can explore. You'll be literally "moving inside the volumes" produced by the 4D->3D projection, and, by exploring these volumes from inside, hopefully develop an accurate idea of what sits in the 3D retina of the 4Der's eye.

I shall try this when I get some free time!

I'm glad you like the idea. Actually I made a mistake above, when describing the idea "in a way that all the voxels of a 3d retina are filling the whole field of view". I rather meant "part of the voxels/voxel array of a 3d retina are filling the whole field of view". I don't know exactly which part that could possibly be (maybe half of the voxel array for representing something like a 180 degree 2d viewing angle). I guess to put the whole voxel array into one image would look too messy.
But I was thinking of placing the camera inside, yes. Maybe experimenting with some fish_eye or ultra_wide_angle camera or something like that. And maybe some colour cues (i.e. different colour for volume with difference light shading for distance within the volume, and different colour for the edges and still different colour (or maybe edge colour but darker) for vertices.