Dimensional Baby Steps

[Mugsy Malone]

I just chanced upon this site. Please forgive my theoretical injection in multidimensional logic. I have a B. A. in English Lit., which I received at the age of 51. I am now 63, a Viet Nam Vet, and have absolutely no one with whom I can discuss these topics. Forgive me for venturing outside the box.

The first of the points I seem to find the most trouble in relating to others is: the 1st dimension of length (or width or height depending on your perspective, but I will call it length); it has direction, but occupies no space. It's not there! [a=l*w; w=0; ergo, a=0] If one dimension reserves no area we can infer it has no volume. This is speculation on my part; call it a romance of many dimensions.

O. K. now that you've overcome the biggest belly-laugh you've ever had; stay with me, there's more.

So I said "Big deal" a couple years back. But for whatever reason I have looked back and freaked myself out even more. I added a perpendicular line to length - we know it as width - and they both were infinitely long and intersected at 90 degree angles. Well, now it marked an area, but it still wasn't there! "Ridiculous!" you say; so did I. But upon reflection, I realized that I could in no way affect any two-dimensional area without adding the dimension of height. The traces of lead from a pencil, ink from a pen, or any element that affects the height of the boundaries used to designate the geometric figure, otherwise impossible to represent without our third dimension, in either a concave or convex manner, introduces the third dimension. It's kind of like the television. All that stuff you're watching ain't there! You are not permitted to enter a scene which exists in any other dimension but your own. Once our two-dimensional object adds a dimension and legally attains citizenship into our three-dimensional reality, we can kick it around. I know what you're thinking; "What about this very computer you are using? You are continuously inserting yourself into this two-dimensional medium, aren't you?" Well, click on
select all, point to cut; then click and this ain't there anymore.

It's obvious that I don't get out much.

Keep in mind! I am a hack with too much time on my hands; so I need some verifiable science to either support or contest my baby-steps through dimensional travel for idiots.

If interested there is more.

Thank You
Mugsy Malone

[gonegahgah]

If you take a peek at other discussions here you may see that a 4D person would see our 3D world as unable to contain space; just as you see the 2Ders world as unable to hold any space. To them, our universe is zero length in the 4th direction giving: Length x Width x Height x 4thWay(0): which results in 0m⁴. The same goes for us where we see a 2Der as having Length x Height x Width(0) = 0m³.

Just as an aside, there is no one direction in our universe that is the x direction, nor any one direction that is the y direction, nor any one direction that is the z direction.
The spatial dimensions don't somehow exist separately from each other and exist only as a collective whole working together anywhere at any orientation without preference. In my opinion, the referral to dimensions provides us with a useful mathematical tool only and they don't constitute individual entities each with their own existence.

But anyhow I digress. So as far as a 2Der would be concerned, if such a creature were to exist, their Universe would be measured in m² which gives them a perfectly valid number and not zero. It's all a matter of how many dimensions you live in that provide you with the measure of your space.

[ac2000]

Hi Mugsy Malone,

I also have this problem with understanding the concept of two dimensions, or for that matter, with those often quoted imaginary "two dimensional beings" that are supposed to shed some light on the characteristics of the 4th dimension by analogy. I think it was discussed some time back in this forum and occasionally I've read about it, that they might be imagined with a slight extension into the third dimension (e.g. the crayon on paper as you mentioned). Otherwise I cannot really imagine them (nor other two dimensional shapes), probably because I am no mathematician, either.
In some texts this slight thickness into the 3rd/4th is taken into account, such as in the foreword by H.P. Manning in "The Fourth Dimension Simply Explained".
He puts it thus:

"Thus we may suppose that what we call two-dimensional matter is really three-dimensional, and that the two-dimensional beings are really three-dimensional, either with a slight thickness in the third dimension, or at least with a thickness which the beings themselves are unable to recognize.
But we may also suppose them all to be really two-dimensional, and we can try to carry out the details of such an existence. It may be that a particle of matter is only a bundle of forces, attractive and repellent, and there is no difficulty in thinking of such forces lying entirely in one plane."

However I do have some "difficulty in thinking" when it comes to the last sentence of the above mentioned quote .


Cheers,

ac2000

[quickfur]

The difficulty in thinking about 2D (or 4D, or any other dimension, for that matter) is in imagining that objects of that dimension must "exist" in some sense that relates to existence in the real world. The problem is that our experience in the real world is entirely 3D, and so our concept of existence is also colored by our 3D experience.

But the "2D objects" that we speak of are mathematical idealizations, where the objects have no 3D thickness, but have only two measurements, length and width. Such things don't exist in our 3D world (i.e. the "real world")... because our world is 3D, and thus everything in it has 3 dimensions. Or at least, everything we can see and touch. That doesn't mean 2D objects can't exist, in the mathematical sense, though. Polygons, for example, are 2D constructs, having only width and length. We may draw them on paper and imagine that the drawings themselves are the 2D objects, but that is a fallacious assumption. What's drawn on the paper is merely a representation of the 2D mathematical idealization; it consists of ink soaked into the paper, both of which are made of 3D atoms with non-zero 3D thickness. So it's not so much the drawings themselves that are 2D objects; one needs to realize that we're talking about the objects they represent.

As for occupying space: obviously, 2D objects occupy no space in the 3D sense -- well, because they aren't 3D objects, for one thing. If our world were 2D instead of 3D, though, we would regard area as space-filling, and would find it strange that some hypothetical "3D being" would think that it doesn't occupy any space. So this objection is just a manifestation of our 3D bias. But let's think about this a bit more. What does it mean for an object to "occupy space"? For example, if an air-inflated ball collides with a block of wood, the two will bounce off each other, and we would say, that's because they both occupy a non-zero amount of space, so if one object is in a particular volume of space, then another can't be there at the same time.

However, if you think about it, the air-inflated ball doesn't really "occupy space" per se -- at least, if you don't regard air as occupying space. (If you want a more rigorously correct example, you could, for example, imagine a hollow steel ball in the vacuum of outer space, colliding with a solid block of wood. But I won't belabor that technicality here.) If the ball were big enough to contain the wood, for example, then the wood could easily occupy the space inside the ball, but the two will still bounce off each other you try to put the wood into the ball by smashing them together. Why? Because the surfaces of these two objects block each from crossing the other's boundary.

Now, imagine if we have not a ball, but a length of rubber tubing that we looped around and glued the ends together, so that it forms a circular length of tubing (in other words, a torus). We place this rubber tubing on the desk, and the same block of wood from before in another place on the desk. Say the circular tubing has a large enough diameter, that the wood can fit entirely inside it. Now, without lifting either object off the desk, we try to move the wood inside the tubing. We won't succeed, of course, because even though there's ample space within the circular tube for the wood to fit into, their boundaries don't permit them to cross over each other. We have to lift one of them off the surface of the desk before we can put the wood in the circle, or the circle around the wood, respectively.

The point here is, this lifting up is possible because we have access to the 3rd dimension. But suppose there's a glass pane laid over the desk, with these two objects pinned underneath it, and this glass pane cannot be removed (say it's screwed on to four short legs nailed to the desk). Now no matter what you do, you can't put the block of wood inside the rubber tubing, because now they are confined to the surface of the desk, and cannot move into the 3rd dimension. Notice that when confined in this way, one of their 3 measurements cease to have any relevance: it no longer matters that the block of wood is, say, 5cm tall, or the small radius of the tubing is, say, 2cm. It only matters that the block of wood occupies a square area on the desk, and the tubing, a circular area, and these two areas can't overlap. There's still space inside the circle, but it's now inaccessible, so for our intents and purposes it might as well be a solid disk of the same major radius. In fact, we can reduce the height of these two objects, and their interactions with each other on the surface of the desk will remain the same. Say we shave the wood block so that its height is reduced to 2mm, and we replace the tubing with a loop of wire about 2mm thick, and confine these two things under the glass pane (which is lowered closer to the surface of the desk accordingly). There is space inside the loop to fit the wood block (now more appropriately called a wooden square, perhaps), but still, due to their confinement, the one can't be lifted up and put within the other. In other words, height has become irrelevant due to the confinement to the surface of the desk.

Now, we don't really need the the glass pane to confine these two objects; we could just as easily agree that the rules of the game are such that neither object is allowed to leave the surface of the desk, and the same situation would hold. And we have established that the height of these objects are irrelevant, as far as this experiment is concerned. Taken to the logical conclusion, we could say that even if we reduced the height to zero, the situation will continue to hold. And thus, when we arrive at the mathematical idealization of the situation, we have a "2D world" of sorts, where 2D objects can't pass through each other because they are "confined to the surface of the desk", as it were, and they need not have any height at all. Neither do they need to "occupy space", for we have shown that it's sufficient for their boundaries to prevent each other from overlapping.

The same thing holds in the opposite direction, though this is perhaps a lot harder to imagine (but it is just as true mathematically). Remember our air-inflated ball that has enough room inside to hold the wooden block, if only the wooden block could pass through its surface? Given that our square and circle, when confined to the surface of the desk, can't pass through each other, even though the circle is big enough to contain the square, but when we permit lifting one object off the surface of the desk, then we can easily place the square inside the circle, so one may also imagine that the reason the wooden block can't be placed inside the air-inflated ball is because they are confined, not to the surface of a desk, but to 3D space itself. If we could somehow "peel the ball off 3D space", then we could ostensibly put it back around the wooden block, thus achieving our goal of putting the wooden block inside it. This is then the analogous 4D case: we could imagine that our 3D world is actually the surface of a 4D desk of some sort, and we are "confined" by something (the equivalent of a "glass pane" if you will) that prevents us from accessing the 4th direction. If we could access such a direction, then it would be trivial to "lift up" our ball off the surface of the 3D world, drag it over the wooden block, and drop it back so that it now encloses the ball. There would no longer be a problem of the boundaries of the ball and the block colliding, since, when lifted up into 4D, the boundary of the ball exists outside of the 3D space that the block occupies, and so there will be no collision.

Sadly, we don't have access to this extra direction, but it doesn't invalidate the mathematical possibility of such a space, which is what we call 4D space. So this, in a (rather long) nutshell, is how dimensional analogy works.

[4Dspace]

we could imagine that our 3D world is actually the surface of a 4D desk of some sort, and we are "confined" by something ... that prevents us from accessing the 4th direction.

Bravo quickfur! You've just described the model of the 4D world in which we live. As I posted couple of days ago, orbitals is the glue that confines each and every nucleus that make up our bodies to the... "desk"? --I call it "display", because it's made up of EM radiation that delivers info about "matter", i.e. other nuclei attached to it elsewhere. That's the general model of 4D space I came to this forum with, to confirm whether my "vision" of 4D in general is right.

Strictly speaking, this is not a new idea (well, it was new to me when I bumped into it in my head last year). But a quick search of physics pointed at several theories, new and old, where the same basic set up comes through. In Strings theory, this 3D "desk", as you call it, called the brane; and that's where Lorenz's luminiferous ether also lives; in ADD model, which is very popular now (not to be confused with attention deficit disorder ) it is also called the "brane" (I personally hate those ugly words); one renown Georgian (from Caucasus) physicist called it "the shell"; and the main MoND guy described it in English as, ...I forgot his exact words, something like, a 3D ...blob? to which the matter is attached floating in 4D.

It is very easy to visualize and understand on the model of Flatland, where the Flatlanders are gliding above the glass display of their world, completely unaware that they are actually move in the 3rd D. All they see is the 2D display, which shows to them what else is attached to it via its version of EM radiation. And so they think that they are moving through their 2D world freely, that their 2D space is "empty" for them to move. But in fact it is a rigid 2d display to which the nuclei of their bodies are attached via their version of orbitals. It's quite simple actually. I was very excited when I saw it first

[ac2000]

If you take a peek at other discussions here you may see that a 4D person would see our 3D world as unable to contain space; just as you see the 2Ders world as unable to hold any space. To them, our universe is zero length in the 4th direction giving: Length x Width x Height x 4thWay(0): which results in 0m⁴.

I like that idea. It's somehow quite funny in a cartoonish way, when I imagine some imaginary 4D creatures, who have some sort of maths at school and they learn to draw 3D objects (just as we've learned to draw triangles and squares at school when we were kids). And at the same time they take for granted that those 3D objects, e.g. spheres and cubes, are of a purely mathematical abstract nature and do not exist in reality because they can't contain space in their 4D world. And we are sitting here down below (or rather somewhere else) on our planet earth which looks strikingly like the spheres the 4D creatures draw in their maths classes and are completely ignored and overlooked ).

But the "2D objects" that we speak of are mathematical idealizations, where the objects have no 3D thickness, but have only two measurements, length and width.

Hm, that sounds all very logical. What I still don't quite understand is this kind of 2D analogy, like in Abbott's Flatland novel. If I recall it correctly, then he describes that the 2D creatures, if they perceive a square or a pentagon, they see it from the side, so they only see a line which fades at the ends (according to how far the edges of the shapes go into distance). But if these lines are of zero thickness, then I wonder how they could be perceived at all by the Flatlanders?

Remember our air-inflated ball that has enough room inside to hold the wooden block, if only the wooden block could pass through its surface? Given that our square and circle, when confined to the surface of the desk, can't pass through each other, even though the circle is big enough to contain the square, but when we permit lifting one object off the surface of the desk, then we can easily place the square inside the circle, so one may also imagine that the reason the wooden block can't be placed inside the air-inflated ball is because they are confined, not to the surface of a desk, but to 3D space itself.

Thanks for these nice examples with the ball and the wooden block. It's fun to imagine them.
So if I got this right, it does mean, that a higher dimension always provides access to the gaps or "holes" of lower dimensions, which are inaccessible in the lower dimension itself. And also to the "inside" of all other objects of a lower dimension.


Best wishes,

ac2000

[quickfur]

If you take a peek at other discussions here you may see that a 4D person would see our 3D world as unable to contain space; just as you see the 2Ders world as unable to hold any space. To them, our universe is zero length in the 4th direction giving: Length x Width x Height x 4thWay(0): which results in 0m⁴.

I like that idea. It's somehow quite funny in a cartoonish way, when I imagine some imaginary 4D creatures, who have some sort of maths at school and they learn to draw 3D objects (just as we've learned to draw triangles and squares at school when we were kids). And at the same time they take for granted that those 3D objects, e.g. spheres and cubes, are of a purely mathematical abstract nature and do not exist in reality because they can't contain space in their 4D world. And we are sitting here down below (or rather somewhere else) on our planet earth which looks strikingly like the spheres the 4D creatures draw in their maths classes and are completely ignored and overlooked ).

In fact, I have some notes with story ideas about a 4D boy who has a strange dream one night about a very flat world, where he encounters strange flat people who appear to be quite unconscious of how claustrophobic it is, and who live in cities in which roads divide the city into blocks, rivers need bridges to be crossed, roads intersect each other at right angles and need convoluted conventions like traffic signals to prevent accidents, and other such strange phenomena. Upon awakening, he tells his parents about the dream, and his father laughs it off as ludicrous childhood fantasy, and his mother thinks it's caused by early childhood trauma when he almost suffocated under a pillow years ago. Dissatisfied with their responses, the boy tells the dream to his math teacher at school, who is of the inquisitive type that would consider the strangest ideas as plausible until proven otherwise. His math teacher tells him that what he saw in his dream was consistent with the mathematics of planar geometry (and by planar he meant 3D hyperplane, of course), and reasons that the dream was probably caused by the recent lessons on the subject.

[...] What I still don't quite understand is this kind of 2D analogy, like in Abbott's Flatland novel. If I recall it correctly, then he describes that the 2D creatures, if they perceive a square or a pentagon, they see it from the side, so they only see a line which fades at the ends (according to how far the edges of the shapes go into distance). But if these lines are of zero thickness, then I wonder how they could be perceived at all by the Flatlanders?

To understand this, we need to consider how sight works. In our 3D world, sight is caused by light rays bouncing off 3D objects, with some of these rays entering our eye and being focused by the lens in our eye onto a 2D array of light-sensitive cells (the retina). The pattern of light that falls upon these cells are then transmitted to our brain and interpreted. Now, the key point here is that, in order for us to see anything at all, there must be an unobstructed path from the object being seen to our eye. Not only does that mean that other objects can obstruct the view; it also means that a 2D array of light-sensitive cells is the best we can get. If we had a 3D array of light-sensitive cells, for example, it wouldn't help us see anymore than we already can, because the light that strikes the cells inside this 3D array must pass through the cells on the outside, so that either no light would reach those inner cells, or the light that does reach them has already passed through the outer cells anyway, so no new visual information is conveyed to the eye.

In light of this (if you'll excuse the pun), consider now a native 2D being as in Abbott's novel. Being confined to the plane, it can only move in the planar 2D world. Likewise, objects in the 2D world are also confined on that plane, and so any light that bouncing off those objects that can reach the eye of the 2D being must also be travelling strictly within that plane. So imagine some rays of light bouncing off, say, a hexagonal object in front of our 2D being. Some of those light rays reach the eyes of the 2D being, and get focused onto an array of light-sensitive cells. The question is, how many dimensions does this array have? Suppose there is a 2D array of light-sensitive cells, just like our own eyes in 3D. But since the 2D being is confined to the plane, the cells it is composed of must also lie within that plane; and since visible light is also confined to that plane, it stands to reason that the only way for light to reach that array of light-sensitive cells must be from the edge of the array. So the light will always strike the boundary of that array first, and so the inner cells are useless, because the light they see has already been seen by the cells on the boundary of the array. Furthermore, the lens cannot focus light so that it strikes the boundary of the array from every direction; at the most, it can only focus light from a particular direction. So we conclude that the 2D being's retina must be merely a 1D array of light-sensitive cells.

Thus, the only thing the poor creature can see is a merely a line -- a 1-pixel thick line, if you wish, though in the mathematical idealization it's really a zero-thickness line, just as Abbott describes. As to how a zero-thickness object can even be visible -- well, that is not a problem as long as light can bounce off it, which it can if the light itself is confined to 2D, so that it cannot pass through the object being seen without striking its boundary, upon which it will either be absorbed or reflected. The light then conveys the presence of that object to the 2D being's eye. So the 2D being only ever sees the edges of objects in its world, and cannot see the polygons as we 3Ders see them, face-on.

In the same vein, we may think that our 3D eyes can see a lot: we have a full 2D view of things, unlike the poor 2D beings! However, we cannot see inside the 3D objects in our world -- much like the 2D beings are unable to see the face of the polygonal objects in their world, and so, our vision is limited to the surface of 3D objects. A 4D being, however, has a much different view. What we consider as solid, bulky objects, to the 4D being is merely a "flat" polyhedron (or some other flat shape) that fits entirely on the surface of a 4D desk, a mere figure that can be drawn on a piece of 4D paper, unable to enclose any hypervolume. They see our 3D objects not as 2D surfaces like we see them; they see the interior of these objects as though looking at the face of a polygon. Every part of the interior of the 3D objects is laid bare before their eyes; if they were to look upon us, they would not see the beautiful visage that we imagine ourselves having; rather, they would see a human-shaped boundary of skin, with muscle tissues, organs, bones, blood vessels, etc., all laid bare before their eyes. They could reach into our innards and remove an organ, and place it on a 3D desk next to us. Or, to use a less morbid example, if we had a bullet wound they could reach inside our body and remove the bullet without needing any kind of surgery -- it would be as simple as reaching into a polygon and removing a piece of dirt embedded within it. To such 4D beings, the fact that we can see anything at all is a miracle; for the objects in our world as so flat in their consideration that it might as well have 0 hyperthickness, and so they would be quite astounded that we appear to have the ability to see things of zero hypervolume!

[...]
So if I got this right, it does mean, that a higher dimension always provides access to the gaps or "holes" of lower dimensions, which are inaccessible in the lower dimension itself. And also to the "inside" of all other objects of a lower dimension. [...]

Correct, as I described above. And so, though our 4D beings would appear to us to have miraculous powers, such as being able to remove a bullet from our body without any need of surgery, or, for that matter, remove things from a 3D safe without needing to unlock or open it, they are unable to do the same in their own world. They would need surgery to remove a bullet in a 4D being's body, and they would be unable to access the contents of a 4D safe without unlocking and opening it. But a 5D being would be able to do those things easily. And so forth, up the dimensions.

This is what dimensional analogy is all about.

[ac2000]

In fact, I have some notes with story ideas about a 4D boy who has a strange dream one night about a very flat world, where he encounters strange flat people who appear to be quite unconscious of how claustrophobic it is, and who live in cities in which roads divide the city into blocks, rivers need bridges to be crossed, roads intersect each other at right angles and need convoluted conventions like traffic signals to prevent accidents, and other such strange phenomena. Upon awakening, he tells his parents about the dream, and his father laughs it off as ludicrous childhood fantasy, and his mother thinks it's caused by early childhood trauma when he almost suffocated under a pillow years ago.

Nice idea . Though I suppose it might be quite difficult to bring across the idea that bridges and right angled roads are very strange for a 4d boy. Because when familiar words like "bridge" are used, a reader of our world couldn't help to associate it with ordinary things, well things like a "bridge" .
I like the part with the "suffocated under a pillow" trauma best . That bridges the gap between the 4D creatures and the 3D reader because the 4D creatures are sort of "humanized" and besides it's funny.

So we conclude that the 2D being's retina must be merely a 1D array of light-sensitive cells.

That's quite clear and also logical.

Thus, the only thing the poor creature can see is a merely a line -- a 1-pixel thick line, if you wish,

Hmm, thinking again about this I'm wondering now whether this would be perceived as a line. Because a line somehow implies there is still an empty area, vertically to the line where there is "no line". If there's no such area, because the line (of, say, 1-pixel thickness) fully takes up all the space of the (also 1-pixel thick) field of vision of the 2d creature, then I guess it might also be perceived as a kind of wall or plane, because it takes up all the creatures visual field in that hypothetical 1-pixel thick direction. (Though to the left and right there might be some empty space, depending on the distance of the edge of the perceived object).
But thinking about this further, neither "line" nor "plane" seem to be adequate words to decribe what the 2D being might be seeing, because it doesn't look like a line to him (because he can't see any empty spacy vertically to the line), and he's not capable to see ordinary planes at all.

though in the mathematical idealization it's really a zero-thickness line, just as Abbott describes. As to how a zero-thickness object can even be visible -- well, that is not a problem as long as light can bounce off it, which it can if the light itself is confined to 2D, so that it cannot pass through the object being seen without striking its boundary, upon which it will either be absorbed or reflected.

Being curious I've just spent some time searching the web for some matter, which could exist in such a 2D space and finally came across an "anyon".
"In physics, an anyon is a type of particle that occurs only in two-dimensional systems" says Wikipedia.
That's pretty cool, I didn't know something like that exists. Although, as it seems to be the case with many particles in physics, their actual existence is not fully proven so far.
As far as the light is concerned, are you sure, that light can be reflected off of 2D matter/particles? When the light is traveling as a wave, I wonder if the wave might also slip over or under the 2D matter, and never hit it, or when the light is somehow artificially confined to 2D, whether it can travel at all (as a ray or wave). I have too little knowledge of physics, and all these waves/photons stuff seems to me to be quite complicated.

They see our 3D objects not as 2D surfaces like we see them; they see the interior of these objects as though looking at the face of a polygon. Every part of the interior of the 3D objects is laid bare before their eyes;

So, if a 4D creature looked upon a 3D sphere containing a 3D cube from a certain angle in 4D space, would it simply look like a circle with a square inside? Or rather like a sphere which is cut open at the top and inside it a cube which is also cut open at the top? Or would it look still completely different?

Another thing I'm interested in is, if I have a 3D cube and write a number on opposite sides of this cube, would it be possible for a 4D being to read both numbers from a single visual angle? Or couldn't it read any of the numbers at all because they are too flat and it can only perceive the sides of the cubes as edges?
Hmm, assuming that a 2D being would paint some dots or lines (numbers would probably not work because of the missing dimension) on opposite borders of a square we could see them at the same time, I think, at least if they had a minimum thickness of, well an ink spot or something.
So it should also be possible for the 4D being to see the numbers on the cube, shouldn't it?

-------

[quickfur]

In fact, I have some notes with story ideas about a 4D boy who has a strange dream one night about a very flat world, where he encounters strange flat people who appear to be quite unconscious of how claustrophobic it is, and who live in cities in which roads divide the city into blocks, rivers need bridges to be crossed, roads intersect each other at right angles and need convoluted conventions like traffic signals to prevent accidents, and other such strange phenomena. Upon awakening, he tells his parents about the dream, and his father laughs it off as ludicrous childhood fantasy, and his mother thinks it's caused by early childhood trauma when he almost suffocated under a pillow years ago.

Nice idea . Though I suppose it might be quite difficult to bring across the idea that bridges and right angled roads are very strange for a 4d boy. Because when familiar words like "bridge" are used, a reader of our world couldn't help to associate it with ordinary things, well things like a "bridge" .

Yeah, well, the 4D boy wouldn't call it a "bridge"; he'd describe it as a strange way of building an elevated section of road to cross over a river, which he finds utterly strange that such a thing would be necessary.

I like the part with the "suffocated under a pillow" trauma best . That bridges the gap between the 4D creatures and the 3D reader because the 4D creatures are sort of "humanized" and besides it's funny.

Actually, I wanted to write it in such a way that the 4Dness of the boy is not made explicit until later in the story. It would read like a regular story, but then little things here and there would seem misfitting, and then when the boy's father starts objecting that it's perfectly logical that space must have exactly 4 dimensions (for various reasons which I won't spoil here), the perceptive reader might start to clue in to what's going on.

[...]

Thus, the only thing the poor creature can see is a merely a line -- a 1-pixel thick line, if you wish,

Hmm, thinking again about this I'm wondering now whether this would be perceived as a line. Because a line somehow implies there is still an empty area, vertically to the line where there is "no line". If there's no such area, because the line (of, say, 1-pixel thickness) fully takes up all the space of the (also 1-pixel thick) field of vision of the 2d creature, then I guess it might also be perceived as a kind of wall or plane, because it takes up all the creatures visual field in that hypothetical 1-pixel thick direction. (Though to the left and right there might be some empty space, depending on the distance of the edge of the perceived object).

Correct! So what to us a mere line, the edge of some object, is to the 2Der a solid wall. Similarly, what to us 3D beings is a solid wall (i.e., a 2D surface), is to a 4D being a mere "ridge", it is to them as a 1D edge is to us, just a thin edge of something.

But thinking about this further, neither "line" nor "plane" seem to be adequate words to decribe what the 2D being might be seeing, because it doesn't look like a line to him (because he can't see any empty spacy vertically to the line), and he's not capable to see ordinary planes at all.

Correct. To us he sees a line, but to him, he's seeing a panoramic view of the 2D world. He would have a hard time believing that the grand scene before his eyes is but a merely line to us!

Similarly, a 4D being's eyes have 3D arrays of light-sensitive cells. They see 3D directly, face-on, just as we can see 2D face-on. To them, our sight is a ridiculously constricted slice of "real" sight; a mere 2D plane capable of only seeing what is to them a mere thin ridge. We think our sight is seeing a grand view of the 3D world, but this is quite laughable to the 4D being.

[...] Being curious I've just spent some time searching the web for some matter, which could exist in such a 2D space and finally came across an "anyon".
"In physics, an anyon is a type of particle that occurs only in two-dimensional systems" says Wikipedia.
That's pretty cool, I didn't know something like that exists. Although, as it seems to be the case with many particles in physics, their actual existence is not fully proven so far.

That's interesting! I never knew of the existence of such things!

As far as the light is concerned, are you sure, that light can be reflected off of 2D matter/particles? When the light is traveling as a wave, I wonder if the wave might also slip over or under the 2D matter, and never hit it, or when the light is somehow artificially confined to 2D, whether it can travel at all (as a ray or wave). I have too little knowledge of physics, and all these waves/photons stuff seems to me to be quite complicated.

When we speak of 2D or 4D worlds, although we might draw analogies with the "real" 3D world, we should always keep in mind that ultimately they are just devices to help us understand the geometry of the different dimensions. I doubt there is an actual 2D world in which living beings dwell. Perhaps there is, somewhere "out there", but it wouldn't be accessible to our universe (since if it were, it wouldn't be a 2D world anymore; it'd be a slice of the 3D world).

So when we speak of light constricted to 2D, we are, in essence, taking some artistic liberty (or mathematical liberty?) in analogizing familiar things in our world to the 2D world, in order to develop some intuitive understanding for it. Thus we can conceive of "2D light" that bounces off the boundary of 2D objects, and that is confined to the 2D world, unable to escape it. So all interactions happen within 2D itself, and any living beings in such a world would be utterly unconscious of the possibility of existence in higher dimensions.

Similarly, in using dimensional analogy to understand 4D geometry, we're taking the liberty to imagine living 4D beings who interact with their 4D environment in analogous way to our own interactions with the 3D world. So we postulate "4D light" that propagates in 4 dimensions rather than 3 -- something quite foreign to physics as we know it (since if light in our world were to actually propagate in an additional dimension as well, it would fade according to an inverse cube law, rather than an inverse square law as we observe it). Of course, the ultimate object of these little fantasies, as they were, is to gain an intuitive understanding of 4D geometry (which has quite a number of very unique features that are interesting to both professional mathematicians and hobbyists alike). So while we could, in the name of artistic liberty, postulate arbitrary things such as 4D names all start with "tetra", it wouldn't yield very much insight into the geometry of 4D itself. Much more insightful would be to consider, for example, how 4D cities might be organized, given the extra degree of freedom available for laying out the buildings. Considering questions of this nature often leads to very interesting conclusions about the nature of 4D itself, such as the fact that city blocks are unnecessary in 4D, because roads don't divide land into two. Taking this a step further, one might consider how the road system would be organized, and perhaps notice that intersections are unnecessary, because the 3D-ness of the land surface allows us to build "on-ramps" and "off-ramps", like the highways of our own world, yet without any need for elevation. Cross-walks are also superfluous, because one can simply walk around the road without crossing through it.

So, we don't really expect that things in other dimensions must necessarily correspond with physics in our own world. Especially since in 4D, it has been proven that planetary orbits are unstable, as are atomic orbitals, so if a 4D universe with living beings actually exist, it must be of a far more alien organization than we may imagine. What kind of organization this is, is of course a very interesting subject in its own right, but it doesn't stop us from considering in the meantime things like how seasons and weather patterns would be laid out in a 4D planet, if planetary orbits were stable -- because such investigations often yield a lot of insight into the geometry of the 4D hypersphere, even though they are untenable in an actual 4D universe.

They see our 3D objects not as 2D surfaces like we see them; they see the interior of these objects as though looking at the face of a polygon. Every part of the interior of the 3D objects is laid bare before their eyes;

So, if a 4D creature looked upon a 3D sphere containing a 3D cube from a certain angle in 4D space, would it simply look like a circle with a square inside? Or rather like a sphere which is cut open at the top and inside it a cube which is also cut open at the top? Or would it look still completely different?

This is not easy to describe, because our language is, necessarily, shaped by our own visual experiences, so that visual terms would necessarily biased towards our 2D-based sight. However, we may gain some insight into a 4D being's sight if we understand that their eyes must have a 3D array of light-sensitive cells. A 3D array of cells shouldn't be too hard for us to imagine; now imagine that these cells are lit or not, according to whether they fall near enough to the surface of a sphere and the cube enclosed inside it. So this array of cells would, in effect, have some kind of "pixelated image" of the sphere and the cube (sorta like those old 3D voxel-based games, in which everything is made of little blocks, with a jagged appearance).

Now the key here is to understand that every "voxel" in that 3D array is visible to the 4D being simultaneously. That is to say, they do not merely see the surface of a sphere or the surface of the cube; they see the entire 3D volume of both objects, every point on the surface, and every point the interior, all at the same time. This may take a while to sink in. It's just as gigantic a leap of imagination as it would be for a 2D being to wrap his mind around the concept that we 3D beings are able to see the 2D plane directly. The 2D being may slowly grasp the concept that our retina has a 2D array of light-sensitive cells, and so we see not just the 1D panorama he is familiar with, but what amounts to many, many such 1D lines, all stacked together. But eventually, he may realize that this isn't really an accurate description, because we don't see things as little 1D strips stacked on top of each other, but rather, we see the entire 2D image as a whole.

And so it is with the 4D being's sight. They do not see the surfaces of 3D objects as we do, and neither can their vision be accurately described as many 2D images stacked on top of each other. Rather, they see 3D as a whole -- every single point in the 3D volume of their retina is simultaneously visible to them. Thus, when they gaze at the cube within a sphere, they instantly perceive that the sphere is but a hollow shell, and enclosed within it is a cube. And not a cube as we see it; they see the cube in its entirety, every point of its surface and every point in its interior. If the cube were made of wood, for example, they would see the entire structure of the wood grain laid bare before their eyes.

This may sound like some kind of magical "x-ray vision", but that is still an inaccurate understanding of this -- because it implies that the 4D being's line-of-sight somehow passes through the surface of the cube in order to see into its interior. However, this is not the case at all. The line-of-sight doesn't pass through the surface of the cube at all; it goes directly from the 4D eye to the interior point of the cube. The cube is actually just as opaque to them as it is opaque to us; the difference is that while our vantage point is restricted to always seeing the cube from its outward-facing surfaces, the 4D being is able to look at the cube from a point outside the 3D space that the cube resides in, and thus see its volume "face-on" (just as we can see a polygon face-on, whereas a 2D being can only ever see the edges that form the polygon's boundary).

Another thing I'm interested in is, if I have a 3D cube and write a number on opposite sides of this cube, would it be possible for a 4D being to read both numbers from a single visual angle? Or couldn't it read any of the numbers at all because they are too flat and it can only perceive the sides of the cubes as edges?

I would imagine that the numbers would escape their notice unless they were consciously looking for it. Just as a jagged edge on a polygon would escape our notice unless we realized that somebody had etched a pattern of little notches on them, like a secret code of sorts. That is to say, a 2D being had been painting dots on the edge of the polygon in its analogue of handwriting. Once we look for it, though, it's easy enough to read it, provided we understand what the pattern of dots signify. Similarly, our writing to a 4D being would be barely noticeable, though I'd imagine it's easily readable once they know to look for it.

To catch the 4D being's attention, we'd need to build shapes made of solid blocks -- it takes 3D volume to fill up enough of their field of vision to catch their attention. (Just as a 2D being would need to build space-filling shapes, perhaps carve something out of wood or mold it from clay or cement or something substantial, in order to make letterforms that cover enough 2D area for us 3D beings to notice it.)

Hmm, assuming that a 2D being would paint some dots or lines (numbers would probably not work because of the missing dimension) on opposite borders of a square we could see them at the same time, I think, at least if they had a minimum thickness of, well an ink spot or something.
So it should also be possible for the 4D being to see the numbers on the cube, shouldn't it?

Yes, it should be easy enough to see -- once they know to look there.

Of course, writing is another very interesting topic of speculation when you cross dimensions... the thing is, in 4D letterforms are identical to their mirror images via a simple rotation (something that cannot be done in 3D -- no matter how you rotate a capital P, it will never become a lowercase b). This strange effect of an extra dimension is, in fact, what caused the mathematician Möbius to realize that we could flip 3D objects into their mirror images, if only we had access to a 4th dimension. This is one of the first things in a series of steps that eventually led mathematicians from Euclid's 3D geometry to our 4D geometry, and beyond. Anyhow, this means that the 4D being may see our writing in mirror-image instead of the "right way round", or, in fact, anything in between. So they would have to learn which is the correct way to read our writing, and either mentally rotate it the right way round, or shift their viewpoint so that they see it in the correct orientation.

[4Dspace]

This is not easy to describe, because our language is, necessarily, shaped by our own visual experiences, so that visual terms would necessarily biased towards our 2D-based sight. However, we may gain some insight into a 4D being's sight if we understand that their eyes must have a 3D array of light-sensitive cells. A 3D array of cells shouldn't be too hard for us to imagine; now imagine that these cells are lit or not, according to whether they fall near enough to the surface of a sphere and the cube enclosed inside it. So this array of cells would, in effect, have some kind of "pixelated image" of the sphere and the cube (sorta like those old 3D voxel-based games, in which everything is made of little blocks, with a jagged appearance).

But how does the 4D light gets into the 4Der retina? Doesn't it still consist of perfectly straight rays? And if so, what 4Der sees, is just a projection onto 3d hyperplane. I thought we discussed it already

Now the key here is to understand that every "voxel" in that 3D array is visible to the 4D being simultaneously. That is to say, they do not merely see the surface of a sphere or the surface of the cube; they see the entire 3D volume of both objects, every point on the surface, and every point the interior, all at the same time.

This is wrong. Why are you stubbornly continue to spread this disinfo after the discussion that established that at any given moment, a 4Der sees only one side of the object, just like the rest of the NDers in all possible N-Universes?

The correct answer is: a 4Der will see only one side of each plane of the cube at a moment, but as he will change his POV, he will get to see the opposite sides as well. But not at the same time. Just like we have to turn the cube around to see its far face.

This may take a while to sink in.

Indeed you have such nice, thoughful posts, quickfur. Except for that part of the omnipresent 4Der vision... I don't get it... You did admit your error. You forgot?

[quickfur]

This is not easy to describe, because our language is, necessarily, shaped by our own visual experiences, so that visual terms would necessarily biased towards our 2D-based sight. However, we may gain some insight into a 4D being's sight if we understand that their eyes must have a 3D array of light-sensitive cells. A 3D array of cells shouldn't be too hard for us to imagine; now imagine that these cells are lit or not, according to whether they fall near enough to the surface of a sphere and the cube enclosed inside it. So this array of cells would, in effect, have some kind of "pixelated image" of the sphere and the cube (sorta like those old 3D voxel-based games, in which everything is made of little blocks, with a jagged appearance).

But how does the 4D light gets into the 4Der retina? Doesn't it still consist of perfectly straight rays? And if so, what 4Der sees, is just a projection onto 3d hyperplane. I thought we discussed it already

That is exactly what I just said!

Now the key here is to understand that every "voxel" in that 3D array is visible to the 4D being simultaneously. That is to say, they do not merely see the surface of a sphere or the surface of the cube; they see the entire 3D volume of both objects, every point on the surface, and every point the interior, all at the same time.

This is wrong. Why are you stubbornly continue to spread this disinfo after the discussion that established that at any given moment, a 4Der sees only one side of the object, just like the rest of the NDers in all possible N-Universes?

You're confusing yourself again. A hyperplane has two sides, and the 4Der sees one of them, just as a polygon has two sides, and we see one of them. Every point in the polygon is still visible to us, just as every point of a 3D object is still visible to the 4Der. What we consider as the "sides" of a cube has absolutely nothing to do with the "side" of the cube that the 4Der sees from her vantage point. The 6 square faces of the cube are mere ridges, boundaries of a volume, whose entirety is visible. The volume itself has two sides, both of which are bounded by the same 6 square faces, and the 4Der sees one of these sides at a time. That is still a full 3D array of voxels, and every voxel is visible simultaneously.

The correct answer is: a 4Der will see only one side of each plane of the cube at a moment, but as he will change his POV, he will get to see the opposite sides as well. But not at the same time. Just like we have to turn the cube around to see its far face.

I already asked you to tell me, given a cube with coordinates (±1,±1,±1,0) and a 4D viewpoint at (0,0,0,5), which point in the cube is not visible to the 4Der, and you have not answered me. Please do, since you insist that what I said was wrong. Just follow your own reasoning: the 4D light travels in straight lines from the object to the eye. I claim that every point in the cube has an unobstructed straight line path to the eye at (0,0,0,5). Therefore, every voxel in the 3D volume of the 4Der's retina is simultaneously visible. You seem to think this is wrong. So please show me, which point in the cube is not visible?

[...] Indeed you have such nice, thoughful posts, quickfur. Except for that part of the omnipresent 4Der vision... I don't get it... You did admit your error. You forgot?

It is not omnipresent 4Der vision. Nowhere did I claim such a thing. All I said was that the entire 3D hyperplane is visible to the 4Der simultaneously. But what lies behind the hyperplane is obviously obscured. And please note that "behind" here means behind in the 4th direction. It does not refer to any direction within the 3D hyperplane. To use the cube example above, the 4Der can't see the point (0,0,0,-1), because it is obscured by the point (0,0,0,0) which lies within the cube. But (0,0,0,-1) is a point entirely outside of the 3D hyperplane where the cube sits. Again I say, please tell me which point in the cube is not visible, since you keep insisting that I'm wrong.

I think you're confusing the 4th direction with a direction within the 3D hyperplane. In the above example, the 3D hyperplane is spanned by the vectors (1,0,0,0), (0,1,0,0), and (0,0,1,0). It is a plain mathematical fact that there exists an unobstructed straight line from every point of the span of these vectors to the point (0,0,0,5). The faces of this cube lie perpendicular to these vectors, and all 6 faces have an obstructed path to the 4D vantage point. None of them are "in front" or "behind" from the 4D point of view; they are all "on the side" -- this is a plain simple mathematical fact, because the line-of-sight is parallel to the vector (0,0,0,-1), and all 6 faces lie perpendicular to this vector simultaneously. (Don't believe me? Try it yourself. Take the dot product of any of the spanning vectors with the line-of-sight vector. They are all zero, meaning that the cosine of the angle between them is zero, that is, the angle is exactly 90°.) Therefore, the "front" of this hyperplane is entirely visible to the 4D eye, every voxel of it. What is not visible is what lies to the (0,0,0,-1) direction of this hyperplane, that is, points like (0,0,0,-1), (1,1,1,-2), and so forth. These lie behind the hyperplane, that is, in the (0,0,0,-1) direction from it. Again, you have to understand that all 3 directions within the hyperplane are perpendicular to the line-of-sight, and therefore their span is all in plain sight. This is not omnipresence, this is plain geometry.

[4Dspace]

Dearest quickfur, you're such a sweet person, and I love your thoughtful posts ..but you're also a stubborn type I am an analyst by profession. The gist of your confusion is the same that plagues physics, and that is, points and geometry do not mix well. namely:

Now the key here is to understand that every "voxel" in that 3D array is visible to the 4D being simultaneously. That is to say, they do not merely see the surface of a sphere or the surface of the cube; they see the entire 3D volume of both objects, every point on the surface, and every point the interior, all at the same time.

This is wrong. Why are you stubbornly continue to spread this disinfo after the discussion that established that at any given moment, a 4Der sees only one side of the object, just like the rest of the NDers in all possible N-Universes?

You're confusing yourself again. A hyperplane has two sides, and the 4Der sees one of them, just as a polygon has two sides, and we see one of them. Every point in the polygon is still visible to us, just as every point of a 3D object is still visible to the 4Der.

No, it's you who is confused. Unless your cube is transparent, what is seen is ONE SIDE OF a hyperplane. The points confuse you. When you see a cube, you don't see points, even if it is a transparent cube made of glass faces. Same for your 4Der. He --sorry, you want it to be a she-- does not see points of a plane or hyperplane that make up objects. What is seen are the constituents of the object. If a cube is made of faces, then that is what is seen, one side of a face from a given POV at a time. If the cube is made of an array of small cubes, then one side of each of those cubes' faces is seen from a fixed POV. Depending on the angle, some are seen from inside and others from outside. As the POV changes, the insides of such a cube that is composed of small cubes will undulate, showing inside/outside of each face.

What we consider as the "sides" of a cube has absolutely nothing to do with the "side" of the cube that the 4Der sees from her vantage point. The 6 square faces of the cube are mere ridges, boundaries of a volume, whose entirety is visible. The volume itself has two sides, both of which are bounded by the same 6 square faces, and the 4Der sees one of these sides at a time. That is still a full 3D array of voxels, and every voxel is visible simultaneously.

The cube represents a piece of a hyperplane in 4D. It has 2 sides (just like everything else in the world, except points, which keep confusing you). Nobody --anywhere!-- sees points. Points are invisible by definition. What is seen are planes, or hyperplanes, as the case may be, and they have sides.

I already asked you to tell me, given a cube with coordinates (±1,±1,±1,0) and a 4D viewpoint at (0,0,0,5), which point in the cube is not visible to the 4Der, and you have not answered me. Please do, since you insist that what I said was wrong. Just follow your own reasoning: the 4D light travels in straight lines from the object to the eye. I claim that every point in the cube has an unobstructed straight line path to the eye at (0,0,0,5). Therefore, every voxel in the 3D volume of the 4Der's retina is simultaneously visible. You seem to think this is wrong. So please show me, which point in the cube is not visible?

See above: we don't see points. We see the boundaries of objects, in whatever-D, and those have directions. Again, if your cube is made of small cubes, then you will see their faces undulating as you walk around such a cube.

All I said was that the entire 3D hyperplane is visible to the 4Der simultaneously.

Only one side of the 3D hyperplane is seen at any given time.

I think you're confusing the 4th direction with a direction within the 3D hyperplane. In the above example, the 3D hyperplane is spanned by the vectors (1,0,0,0), (0,1,0,0), and (0,0,1,0). It is a plain mathematical fact that there exists an unobstructed straight line from every point of the span of these vectors to the point (0,0,0,5). The faces of this cube lie perpendicular to these vectors, and all 6 faces have an obstructed path to the 4D vantage point. None of them are "in front" or "behind" from the 4D point of view; they are all "on the side" -- this is a plain simple mathematical fact,

And again, the confusion lies entirely with you, since you're confusing the angle of POV of the observer with the directions inherent in a given D (in addition to seeing the invisible by definition points).

... because the line-of-sight is parallel to the vector (0,0,0,-1), and all 6 faces lie perpendicular to this vector simultaneously.

And, the question is : WHICH SIDE OF EACH 6 FACES DO YOU SEE?

Answer: whatever they are, you see ONLY ONE SIDE AT A TIME.

Again, you have to understand that all 3 directions within the hyperplane are perpendicular to the line-of-sight, and therefore their span is all in plain sight. This is not omnipresence, this is plain geometry.

Again you confuse the directions within a hyperplane with the POV of the observer and a given object.

But it does get tiring indeed...

[quickfur]

Dearest quickfur, you're such a sweet person, and I love your thoughtful posts ..but you're also a stubborn type I am an analyst by profession. The gist of your confusion is the same that plagues physics, and that is, points and geometry do not mix well.

That is your own prejudice, which leads to your own strange conclusions. You, of course, have every right to hold to your opinion, but it does not represent facts well-understood by pretty much everyone here who has learned to visualize 4D. That is perfectly fine by me, actually. I really have no interest in these kinds of unprofitable debates -- it is abundantly clear that what I was trying to say has never really gotten through to you, and not for lack of trying either. I give up. I have limited time to work on stuff, and I do have much more rewarding things to be thinking about right now. So by all means continue to draw strange conclusions from your prejudices, just don't pretend that they represent what is already well-established in 4D geometry, because it's very clear that they are not. I'm not going to spend any more time arguing with you. We both get riled up, and all that for nothing, since nothing profitable ever comes of it.

Have a nice day.

[4Dspace]

Well then, should we put it to a vote?

You said, everyone here sees it this way, but gongahgah on a neighboring thread described very well how objects are seen from one side at a time. Wendy claims to see "up to 8D" but her manner of expressing herself is so cryptic that... who could tell? As for the rest, I have not read old threads, only the current ones, and there are not that many people here.

You, apparently, are unable to admit that you were wrong. I guess you're the type that is always --always!-- right. It is so human. People make mistakes and then, even when they get to see it, are unable to admit it and stubbornly continue on their folly. Physics are full of such... things should we call them? They insists that space gotta be empty and keep calculating wave functions and curvatures, as well as energy and fluctuations of "emptiness". How smart is that? So, you're in a good company

[quickfur]

You continue to refuse to answer a very simple and very reasonable challenge, the one about which points in the cube (±1,±1,±1,0) are not visible from the 4D viewpoint (0,0,0,5). And now I'm the one who's always right? Sure, you can't answer a simple question, so let's take potshots at each other, why don't we.

And you continue to evade the question by insisting that lines can't possibly consist of points, planes can't be reduced to points, etc.. First of all, that already puts you in a position that very few (if any) mathematicians hold. Which means the geometry that you're talking about is not the one that most people understand as geometry. You should at least admit that what you conceive of as 4D geometry (or any other geometry) is quite foreign from the generally-accepted one, instead of jumping into the middle of somebody else's discussion and insisting they're wrong, because according to your personal definition of geometry, what they said can't be true. So conveniently ignore the possibility that they just might have different axioms from what you hold dear, why don't we.

Secondly, the very mention of breaking down higher-dimensional objects into points seems to stir such aversion in you that you would engage in tirades against "the physicists" in a forum dedicated to 4D Euclidean geometry -- not physics. Nevermind the fact that, last I heard, our own physical eyes see by virtue of a 2D array of light-sensitive cells, and therefore the supposed lines and planes that we see are actually constituted of points of light as perceived by these cells. Every point of the 2D image projected by the lens in our eye is visible to us simultaneously. Is it so hard to believe that in 4D, one can have a 3D array of light-sensitive cells, all of which are simultaneously visible? But of course not -- lines can't be broken up into points, that's just reductionist physicist nonsense. OK, but you should at least admit that your views aren't generally accepted, and therefore, for the sake of the most basic human courtesy, you shouldn't be correcting others who aren't following your personal version of geometry.

So you see, it's impossible to communicate with you. You refuse to acknowledge that others' understanding of geometry is different from your own, and you insist on championing your personal opinions everywhere, as though every discussion about geometry is a battleground for your apparent vendetta against "the physicists" (whoever they are). When challenged to simple geometric questions, you consistently refuse to answer, and yet continue to insist that I must be wrong, that I have such a superiority complex that I absolutely have to be right, etc.. I have given you the opportunity, repeatedly, to show where I went wrong, yet the best you can come up with is "you can't decompose planes into points" because that's repeating the mistake of "the physicists".

So I am forced to conclude that either (1) you have no idea what you're talking about; or (2) your definition of geometry is so strange that there is no point in discussing it as if it were the same geometry as others understand; or (3) you're just trolling here. I discounted (1) and (3) because everyone deserves the benefit of the doubt. So that leaves me with (2): there is really no point in continuing this discussion.

So have a nice day. I've wasted enough time writing this post. Don't bother replying, because I've already said everything I could.

[4Dspace]

And you continue to evade the question by insisting that lines can't possibly consist of points, planes can't be reduced to points, etc.. First of all, that already puts you in a position that very few (if any) mathematicians hold. Which means the geometry that you're talking about is not the one that most people understand as geometry. You should at least admit that what you conceive of as 4D geometry (or any other geometry) is quite foreign from the generally-accepted one, instead of jumping into the middle of somebody else's discussion and insisting they're wrong, because according to your personal definition of geometry, what they said can't be true. So conveniently ignore the possibility that they just might have different axioms from what you hold dear, why don't we.

What you're saying is rather strange. Stranger still is that you ascribe the strangeness to me. You position yourself as a mathematician, but you seem unaware of the definition of a point. A point is dimensionless... entity that marks a precise location or position on a plane or a line. In computer graphics, similarly as in physics, a point marks a location of an object in space. This is my understanding, which, to my knowledge, does not differ from what is generally accepted. You, to the contrary, state that planes and lines consist of points, that a cube has points -- it is you who has personal definitions, not me.

I take it, this is your lame attempt to misrepresent the situation as a last ditch to save face.

Secondly, the very mention of breaking down higher-dimensional objects into points seems to stir such aversion in you that you would engage in tirades against "the physicists" in a forum dedicated to 4D Euclidean geometry -- not physics. Nevermind the fact that, last I heard, our own physical eyes see by virtue of a 2D array of light-sensitive cells, and therefore the supposed lines and planes that we see are actually constituted of points of light as perceived by these cells. Every point of the 2D image projected by the lens in our eye is visible to us simultaneously.

Your descriptions lack precision:

First, the rays of light, reflected off the planes of the objects we see are caught by the lens and projected onto retina. There is no points. There are reflections off the planes delivered by beams of light.

Second, what is visible to us simultaneously is not "every point of the 2D image" but every point belonging to the plane that the image represents, seen from a specific direction.

By claiming that you see all points of a 3d object in 4D simultaneously, both inside and out, you publicly gave away the fact that you actually DO NOT SEE. You were asked a simple question up this thread, what lettering, if any will be seen in 4D on the faces of a cube. Your answer was wrong. I corrected it for the benefit of the person who asked the question. That's all. Now, would you please lighten up

[gonegahgah]

No, it's you who is confused. Unless your cube is transparent, what is seen is ONE SIDE OF a hyperplane. The points confuse you. When you see a cube, you don't see points, even if it is a transparent cube made of glass faces. Same for your 4Der. He --sorry, you want it to be a she-- does not see points of a plane or hyperplane that make up objects. What is seen are the constituents of the object. If a cube is made of faces, then that is what is seen, one side of a face from a given POV at a time. If the cube is made of an array of small cubes, then one side of each of those cubes' faces is seen from a fixed POV. Depending on the angle, some are seen from inside and others from outside. As the POV changes, the insides of such a cube that is composed of small cubes will undulate, showing inside/outside of each face.

I thought I would target this paragraph as it seems to be near the centre of your present understanding 4DSpace.
Hopefully it is okay to use our view of a 2Ders world to explore this.

Taking a square then the 2Der obviously just sees a line with one colour at the top part and another different shade, due to lighting, for the remainder.
They can see two edges at once from their POV whereas we can see all four edges from our POV.

They can only see the edges from their outside (ie. outside the square).
If we looked at the edges from their POV we would see the same as them.

However, neither of us can see the inside edges as they are inseparable from the 2D 'volume'.
(I use volume because this is how much something can hold and all a 2Der's holders only have 2 dimensions).
We can only see those outside edges albeit from just less than 360deg of sideways viewing but they are still outside even to us.

At no point can we see the inside of the edges. That's worth remembering.
Again, if we drop back to the 2Ders viewing angle, even though we are in 3D, we still only see two lines like they do.
No matter how we rotate the square so that it remains edge on to us, we will never be able to see the inside of the lines at the back.

Unfortunately this translates to a 4Der who is viewing our flat cube.
If they look at it from our viewing angle (which to them is edge on) they will see only three faces and nothing more and will not see the back of the cube.
No matter how they twist the cube while keeping it edge on themselves; they will still see only the three faces.

It's when they look at it from their extra 'side on' that they can look at the extra 'face' of the cube that is not visible to us.
And that 'face' consumes a cubed amount of space. If they turn it around they will see the opposite 'face'.

That 'face' will be described by the 6 faces of our cube as its edges. They can then turn the cube around and see all the edges (our 6 faces) at once from different rotations and angles but the extra face they see will not change except the angle they are viewing it from. The same happens for us when we view the square. They will not see the inside of 3 of the faces and the outside of the other 3 faces because they can no more see an inside face of a cube then we can see the inside of a line of a square.

Sure, just as turning a square before our eyes turns it into a diamond shape so will a 4Der turning a cube turn it into different shapes but they still will not see the inside faces of any of the cube edges. The extra face they see is like a whole foreign land to us though it is worth describing and that is what we should try to do... But first, just see how these arguments here sound to you.

[4Dspace]

Thank you gonegahgah! As usual, your vision is crisp and descriptions are vivid. Except one thing still, I would have to chew on it before replying.

My apologies to quickfur for not answering hiss challenge sooner. I was so dismayed at our mutual misunderstanding that I had to check and doublecheck it, and I was short on time. But the answer is the same as I saw it first.

So, we have a cube described in 4D as (±1,±1,±1,0) and our POV is (0,0,0,5). The outside faces of the cube are colored red and inside, green. Question: what sides of the faces of the cube we see?

Answer: From the given POV, the 3 faces that lie on the (+1,+1,+1,0) hyperplane is seen as red (="outside") and the 3 faces of the (-1,-1,-1,0) hyperplane is seen as green (="inside").

I will re-read your post again, gonegahgah. There was something there that struck me as..."not quite".

Later.

[quickfur]

[...] So, we have a cube described in 4D as (±1,±1,±1,0) and our POV is (0,0,0,5). The outside faces of the cube are colored red and inside, green. Question: what sides of the faces of the cube we see?

Answer: From the given POV, the 3 faces that lie on the (+1,+1,+1,0) hyperplane is seen as red (="outside") and the 3 faces of the (-1,-1,-1,0) hyperplane is seen as green (="inside").
[...]

It would have helped greatly if you had bothered to say this earlier. Now it's a bit clearer where our miscommunication lies. You're still thinking in terms of 4D->2D projections. I'm thinking in terms of 4D->3D projections, as I've stated before (and that you even acknowledged, apparently). You apparently believe that all vision must be 2D, but I've tried to tell you many times that that is a necessity only in 3D. It is not necessarily true for native 4D beings, which is what the discussion was about prior to your rude interruption.

I'm not sure I understand what you mean by "the (+1,+1,+1,0) hyperplane"; I assume you mean the hyperplane perpendicular to (+1,+1,+1,0)? If so, then let me ask: why (+1,+1,+1,0) and (-1,-1,-1,0)? Why not (+1,+1,-1,0) and (-1,-1,+1,0)? Or (+1,-1,-1,0) and (-1,+1,+1,0)? Nothing about the viewpoint (0,0,0,5) dictates one over the others, since its first 3 coordinates are 0. So whence this arbitrary choice?

[4Dspace]

gonegahgah, thank you for your clear explanation. My take on our cube in 4D agrees with yours. The first part of your post is exactly how I see it. The second part, I have some trouble with:

However, neither of us can see the inside edges as they are inseparable from the 2D 'volume'.

Well, from ~45° POV we do see, sort of, "inside" the edge of the square, even though, of course, this is only if we assume for the benefit of our analogy that the edges are not 1-D lines but, say "tubes", or very long and thin parallelepipeds that make up "walls" bounding the square. Then from a side angle we see the 2 "walls"(=edges) from "inside" as we see the other 2 "walls" from "outside".

We can only see those outside edges albeit from just less than 360deg of sideways viewing but they are still outside even to us.

Yes, if edges were walls, our viewing would be limited to 180°, which we would describe as 'front' and 'back'.

At no point can we see the inside of the edges. That's worth remembering.

This is the part that stuck me as "not quite". See above.

Again, if we drop back to the 2Ders viewing angle, even though we are in 3D, we still only see two lines like they do.
No matter how we rotate the square so that it remains edge on to us, we will never be able to see the inside of the lines at the back.

agree

Unfortunately this translates to a 4Der who is viewing our flat cube.
If they look at it from our viewing angle (which to them is edge on) they will see only three faces and nothing more and will not see the back of the cube.
No matter how they twist the cube while keeping it edge on themselves; they will still see only the three faces.

Agree, except, why "unfortunately"?

It's when they look at it from their extra 'side on' that they can look at the extra 'face' of the cube that is not visible to us.
And that 'face' consumes a cubed amount of space. If they turn it around they will see the opposite 'face'.

Agree again. The "faces" 4Ders see are 2 hyperplanes that define the boundary of the cube. They constitute its "front" and "back". Only the "front" is seen from a given POV (which determines 'front/back' due to the fact that POV has a well-defined direction).

That 'face' will be described by the 6 faces of our cube as its edges.

6? I thought they were 3 -? Well, from the example defined by quickfur above, all 6 faces are seen from the given POV. It's like.. the 2+2 edges of a square that are seen from "inside"+"outside" from 3D at ~45°POV are analogous to 3+3 faces of a cube seen in 4D. 3 are seen from "inside" and 3 are seen from "outside". no?

They can then turn the cube around and see all the edges (our 6 faces) at once from different rotations and angles but the extra face they see will not change except the angle they are viewing it from. The same happens for us when we view the square. They will not see the inside of 3 of the faces and the outside of the other 3 faces because they can no more see an inside face of a cube then we can see the inside of a line of a square.

I suppose by the "extra face" above you mean the hyperplane that bounds the cube from one side. Basically, a cube's "sides" are the 2 hyperplanes, each consisting of 3 faces of a cube. It's like a square has 2 "sides", one is seen from below, and one is seen from above. no?

Sure, just as turning a square before our eyes turns it into a diamond shape so will a 4Der turning a cube turn it into different shapes but they still will not see the inside faces of any of the cube edges.

Here we get a bit confused, due to the lack of precision of terms, rightly pointed out by quickfur in one of the neighboring threads. Which are faces and which are edges in what D gets smudged in this analogy.

The extra face they see is like a whole foreign land to us though it is worth describing and that is what we should try to do... But first, just see how these arguments here sound to you.

Yes, totally agree with you. I find these discussions very helpful in honing my vision of 4D. I do want to learn. Let's call the "extra faces" in 4D... say, hyperfaces. And similarly, we could use "hyperedges" -? I think it would be best to use the proper term. Perhaps quickfur could suggest it to us

[4Dspace]

The other thing about 2D -> 3D -> 4D I am thinking about is the vast difference between 2D and 3D, which is not that large between 3D and 4D. And that is, 2d object has only 1 orthogonal vector going into the 3rd D, while a 3d object is bound by 3 planes, and => has 3 such vectors (and 4D has 4). So, you see the difference in the number of these vectors: 1<->3, and 3<->4? There is no such difference between 3 and 4 (and 4 and 5, etc), as between 1 and 3 (in 2D and 3D respectively).

And this is the part that often gets lost in our analogies going from one D to another. It is compounded by the fact that a POV --in any D-- separates the n-space into 2 halves, 'front' and 'back'. So in case of 2D viewed from the 3rd D, this separation into 2 halves, due to the POV, also coincides with the 2 directions of the orthogonal vector rising from the 2d plane. But in 3D, there are 3 such vectors (each has 2 directions), but still, the same 2 halves, 'front' and 'back', determined by the POV.

This simple fact, I feel, has significant implications. I am still sort of chewing on this, I wonder what you think.

[4Dspace]

[...] So, we have a cube described in 4D as (±1,±1,±1,0) and our POV is (0,0,0,5). The outside faces of the cube are colored red and inside, green. Question: what sides of the faces of the cube we see?

Answer: From the given POV, the 3 faces that lie on the (+1,+1,+1,0) hyperplane is seen as red (="outside") and the 3 faces of the (-1,-1,-1,0) hyperplane is seen as green (="inside").
[...]

It would have helped greatly if you had bothered to say this earlier. Now it's a bit clearer where our miscommunication lies. You're still thinking in terms of 4D->2D projections. I'm thinking in terms of 4D->3D projections, as I've stated before (and that you even acknowledged, apparently). You apparently believe that all vision must be 2D, but I've tried to tell you many times that that is a necessity only in 3D. It is not necessarily true for native 4D beings, which is what the discussion was about prior to your rude interruption.

I'm not sure I understand what you mean by "the (+1,+1,+1,0) hyperplane"; I assume you mean the hyperplane perpendicular to (+1,+1,+1,0)? If so, then let me ask: why (+1,+1,+1,0) and (-1,-1,-1,0)? Why not (+1,+1,-1,0) and (-1,-1,+1,0)? Or (+1,-1,-1,0) and (-1,+1,+1,0)? Nothing about the viewpoint (0,0,0,5) dictates one over the others, since its first 3 coordinates are 0. So whence this arbitrary choice?

Oops, I just saw this post, quickfur. I guess we were replying at the same time (yeah, the page was open since I saw gonegahgah's post and I did not reload it when I returned to it). I have to run now and will get to it later.

[quickfur]

[...]

Sure, just as turning a square before our eyes turns it into a diamond shape so will a 4Der turning a cube turn it into different shapes but they still will not see the inside faces of any of the cube edges.

Here we get a bit confused, due to the lack of precision of terms, rightly pointed out by quickfur in one of the neighboring threads. Which are faces and which are edges in what D gets smudged in this analogy.

And this is where wendy comes in. She realized the problems caused by imprecise terminology a long time ago, and so invented the "polygloss" which tries to remedy this by introducing systematic, consistent terms. Unfortunately, that also means we have to forego our usual 3D-centric terminology, so some effort is needed to learn the terms. But since we aren't dealing with dimensions up to 8 or 12 (which is what wendy's polygloss is targeted at), perhaps we can make do with an agreed-on convention closer to everyday terminology. So here's my proposal:

First of all, we need to understand the difference between dimension-specific terms (words that refer to elements of a fixed dimension, regardless of the dimension of space we're talking about), and dimension-relative terms (words that refer to elements that function in a particular way relative to the dimension of space we're talking about). Dimension-specific terms are important because sometimes we need to talk about how an object of dimension X behaves across different dimensions of space. Dimension-relative terms are important, because sometimes we want to refer to an element in dimension X, that behaves analogously to another element (of a different dimension) in dimension Y.

So, here are the dimension-specific terms:

Vertex = 0D point, fully represented by a single vector in n-dimensional space. (Hopefully this is clear.)
Edge = 1D line segment, which can be thought of as a line drawn between two (and only two) vertices.
Face = 2D polygonal face, consisting of the area bounded by a connected circuit of edges.

Length = distance between two vertices, i.e., the amount of space occupied by an edge.
Area = the amount of space occupied by a face (i.e., polygon). Measured in units of length^2.
Volume = the amount of space occupied by a 3-face (i.e., a polyhedron). Measured in units of length^3.

For higher dimensional elements, or if we want to be ultra-specific in what we're talking about, I propose:

j-face (where j is some integer) = j-dimensional element, for example, a 0-face is a vertex, a 1-face is an edge, a 2-face is a polygon, a 3-face is a polyhedron, etc..
j-bulk (where j is some integer) = j-dimensional "volume", that is, the amount of j-dimensional space occupied by a j-dimensional object. Measured in units of length^j. For example, 1-bulk = length, 2-bulk = area, 3-bulk = volume.

Now the dimension-relative terms, which must always be used in the context of some space of dimension N:

Surface = (N-1)-dimensional boundary of an N-dimensional object.

Facet = (N-1)-dimensional element. In a space of N dimensions, objects of (N-1) dimensions divide space, and N-dimensional objects have surfaces which consists of (N-1)-dimensional elements, so you may think of "facet" as the thing that you can assemble into the surface of something in N dimensions. For example, in 2D, a facet is an edge (because assembling edges together gives you the boundary of a polygon, i.e., the "surface" of a polygon). In 3D, a facet is a polygon (=face), because 3D objects like cubes have a boundary made of polygons.

Ridge = (N-2)-dimensional element. A ridge is where two facets can meet; for example, in 2D, a polygon's ridge = one of its vertices, and that's where two edges (facets) meet. In 3D, a polyhedron's ridge = an edge; that's where two polygons (facets) meet.

Peak = (N-3)-dimensional element. A peak is where two or more ridges meet. 2D objects don't have peaks, unfortunately, because there aren't that many dimensions. 3D objects do, though -- the peaks are the vertices of a polyhedron. That's where the edges of the polyhedron meet.

Bulk = the amount of N-dimensional space occupied by an N-dimensional object, measured in units of length^N.

In the case of dimension-relative terms, there isn't really a compelling need to go past peaks ((N-4)-dimensional elements, etc.), mostly because we use these terms in the context of dimensional analogy, and most of the time we're trying to understand something by a 3D (or lower) analogy, and peaks are as far as it goes in 3D.

The extra face they see is like a whole foreign land to us though it is worth describing and that is what we should try to do... But first, just see how these arguments here sound to you.

Yes, totally agree with you. I find these discussions very helpful in honing my vision of 4D. I do want to learn. Let's call the "extra faces" in 4D... say, hyperfaces. And similarly, we could use "hyperedges" -? I think it would be best to use the proper term. Perhaps quickfur could suggest it to us

Using the terminology I suggested above, you could say a cube in 4D has 2 facets, 6 ridges, 12 peaks, and 8 vertices. Or, to use dimension-specific terminology, the cube has 2 3-faces, 6 faces (precisely, 2-faces), 12 edges (1-faces), and 8 vertices (0-faces).

The 3-faces of the cube are invisible to us, because they can only be seen from 4D. In fact, even if we, 3D beings, are snatched from our 3D world into a location in 4D, we still can't see the 3-faces of the cube, because they occupy 3-bulk (volume), and our eyes can only see 2-bulk (area).

P.S. You don't have to keep bolding my nickname, as if I'm blind or something. That's just silly.

[gonegahgah]

Well, from ~45° POV we do see, sort of, "inside" the edge of the square, even though, of course, this is only if we assume for the benefit of our analogy that the edges are not 1-D lines but, say "tubes", or very long and thin parallelepipeds that make up "walls" bounding the square. Then from a side angle we see the 2 "walls"(=edges) from "inside" as we see the other 2 "walls" from "outside".

You are correct 4DSpace that this is the major area of disparity and if we can clarify it then I will hopefully then be able to show how it applies to a 4Der's view of a cube though I will use a diagram to assist with that. I'll seek to explain the distinction between seeing the inside of an edge better when I get home but I'll just quickly refer to it here.

If you turn a square edge on towards yourself and look across the face of the square you can see clear across. But as soon as your line of vision drops to inline with the edge; the opposite edge is obscured by the front edge in front of your eyes. Just like the 2Der you can not see the back edge because the front edge gets in the way. As far as your eye is concerned the square could go on forever with no back edge. Now if you rotate the square around the edge you begin to be able to see that the square stops at a point and gives way to the view behind it.

We can see a flat layer across the square but we can not get into the topmost layer of the square (the only layer if 2D but we can never make a square that thin). Even if we could get inside we would only see the square where our eye finishes and not the back edge still. We can never see a distinct inside edge. We can only ever look across the face of the square.

Again that is important and applies to the 4Der also which, if I can help you to see that you can not see the inside edge, I hope to then proceed to demonstrating how this also applies to the 4Der.

I did used term 'extra face' deliberately because that is how the 4Der will see it and I wanted to emphasize that.
They will not see it as a volume. Again I'll explain that if we can get through the 'inside edge' distinction.
I also used 'volume' deliberately for the 2Der because that is what they would call it (that's what it would mean to them).

Back Soon!

[gonegahgah]

Okay, this should hopefully help...

Lets take four thin bits of wood. Matches without the red ignition material will do and are nicely square end shaped for simplicity.
We put the matches together to form the parameter of a square.
So essentially we have an empty square.
If you look at this you can see the inside edges of the the square because it is hollow.

But, if you now take some more matches and fill the inside of the square things change.
You can no longer see the inside edge of the outer matches.
This is because they are obscured by the other matches that fill the square.
You can now only see the face of the square and see that it ends at the edges.

This is where I emphasize that we can only see the faces and the faces give way to what we call the edges.
We can not see the inside edges unless we hollow out the square first; but we are dealing with filled objects.

How does that work for you? I'm hoping it does so that I can go onto the next bit...

[4Dspace]

Thank you quickfur, those are very helpful suggestions for terms. Are they "standard" in topology? (just in case I use them elsewhere) or a mixture of standard terms with your personal preferences?

(I bold user names, because that is done automatically on another board I frequent, and to me that always seemed like a thoughtful feature. In case you just skim the posts, then you'd notice that someone is addressing you. Besides, this leaves off the necessity to perhaps uppercase the 'proper name' as a sign of respect.)

gonegahgah! I do understand what you mean. But quickfur will undoubtedly hate it , cause now we have not 2 but 3 opinions as to what is seen instead of our familiar cube in 4D (and we thought that it was such a simple question!)

Here is what I understand: you basically are saying similar thing to quickfur, namely, that all points of 3d subspace, that the cube "carves off" from the 4D it is in, are seen at once. Except that in your case, the 'front' points obscure the 'back' points. lol.

I think this mishap is due to the fact that 4D seeing is indeed a novel thing to us. Cause we generally do not see what's inside solid objects in our homey 3D. But in 4D we can see inside a 3d solid object. So, to remedy this problem, we have to agree in advance, what this 3d object is made of. If it is made of only faces, like our cube, then that's all we shall see. If it has some structures inside (say, it consists of small cubes for simplicity) then that's what it will be. BUT IT CANNOT BE POINTS, lol. Let's just agree on this, and most of our misunderstandings will promptly go away. Ah?

PS
I just saw your last post, I was still replying to the one above. But that's the same thing. Then you have to go to quickfur and he will explain to you that in 4D, indeed, each and every point in space inside the cube is seen from 4D. Except that we can't see points. points are dimensionless. If there is a structure inside, then you will see certain aspects of this structure, determined by your POV.

[4Dspace]

[...] So, we have a cube described in 4D as (±1,±1,±1,0) and our POV is (0,0,0,5). The outside faces of the cube are colored red and inside, green. Question: what sides of the faces of the cube we see?

Answer: From the given POV, the 3 faces that lie on the (+1,+1,+1,0) hyperplane is seen as red (="outside") and the 3 faces of the (-1,-1,-1,0) hyperplane is seen as green (="inside").
[...]

It would have helped greatly if you had bothered to say this earlier.

I always said the same thing:

And, the question is : WHICH SIDE OF EACH 6 FACES DO YOU SEE?
Answer: whatever they are, you see ONLY ONE SIDE AT A TIME.

Now it's a bit clearer where our miscommunication lies. You're still thinking in terms of 4D->2D projections. I'm thinking in terms of 4D->3D projections, as I've stated before (and that you even acknowledged, apparently). You apparently believe that all vision must be 2D, but I've tried to tell you many times that that is a necessity only in 3D. It is not necessarily true for native 4D beings, which is what the discussion was about prior to your rude interruption.

Dearest quickfur, please do not ascribe to me such stupid notions, they do not match my super-high IQ. Do not put words into my mouth that I would never --ever!-- say myself. You are mistaken when you think that you know what I am thinking

And I did not think that my interruption of your thoughtful discussion was rude, 'cause I have a mild form of Asperger syndrome, and speaking up for the truth is more important to me than what is often considered a 'civility'. I saw that you were giving a wrong answer to the person, and thought it was my duty to interfere.

I'm not sure I understand what you mean by "the (+1,+1,+1,0) hyperplane"; I assume you mean the hyperplane perpendicular to (+1,+1,+1,0)?

No, the given POV at (0,0,0,5) is perpendicular to (+1,+1,+1,0). This is the front hyperplane, to which 3 faces of the cube belong. We see them as red. The same POV is also perpendicular to (-1,-1,-1,0) hyperplane, which, however, is seen 'from inside' (green in case of our cube).

If so, then let me ask: why (+1,+1,+1,0) and (-1,-1,-1,0)? Why not (+1,+1,-1,0) and (-1,-1,+1,0)? Or (+1,-1,-1,0) and (-1,+1,+1,0)? Nothing about the viewpoint (0,0,0,5) dictates one over the others, since its first 3 coordinates are 0. So whence this arbitrary choice?

It's not arbitrary, but is determined by the POV at (0,0,0,5), which you set yourself. The only 'convention' I sort of applied was to set the positive direction of the w axis in the same half where positive parts of x, y and z lie. But if you use different convention, then you will get to see those hyperplanes. But the picture will still be the same: 3 faces will be red and 3, green.

[gonegahgah]

Here is what I understand: you basically are saying similar thing to quickfur, namely, that all points of 3d subspace, that the cube "carves off" from the 4D it is in, are seen at once. Except that in your case, the 'front' points obscure the 'back' points. lol.

I'm not quite saying the bolded part but we'll get to that. All in good time...

I just saw your last post, I was still replying to the one above. But that's the same thing. Then you have to go to quickfur and he will explain to you that in 4D, indeed, each and every point in space inside the cube is seen from 4D. Except that we can't see points. points are dimensionless. If there is a structure inside, then you will see certain aspects of this structure, determined by your POV.

I will get to that; don't worry. How does the idea, that you can't see the inside edges of a square, measure up? Is that acceptable?
You can see that the face comes to an end; but you can't see an inside edge...
Just thought I'd add another one... A ring has an inside edge; a circle doesn't.
Just thought I'd add another... A coin has an inside edge but that is because it has a raised edge. Basically a coin is a circle surrounded by a ring.

[quickfur]

Thank you quickfur, those are very helpful suggestions for terms. Are they "standard" in topology? (just in case I use them elsewhere) or a mixture of standard terms with your personal preferences?

Unfortunately, there is no standard terminology. If you want to get a taste for this sad state of affairs, take a look at Grunbaum's paper "Are your polyhedra the same as my polyhedra?". (You may mock the mathematicians now. I will wait. )

The basic problem is that the standard terminology came from the Greeks' study of polyhedra, specifically the Platonic solids and the Archimedean solids. But so much time has passed, and people have moved on to more exotic objects, and so they extended the idea of "polyhedron" to all sorts of things -- non-convex objects like star polygons, higher-dimensional analogues of polyhedra, other more bizarre stuff like (what appears to me to be) random collections of polygons, edges, and vertices, and abstract polytopes which can't even be built in any kind of space. Everyone generalized the basic concepts in different ways, inventing their own terminology as they went. There was no standardization, because usually people aren't studying these things for the sake of studying these things; they often arise as a result of research that is previously thought to be unrelated to the subject. So they didn't even have the thought of standardization.

Sadly, this led to the situation today where nobody can agree on which terms mean what. Wendy's polygloss is an attempt to correct this problem, but it hasn't been widely-adopted. (And besides, it's not very easy for beginners to learn, because the words are constructed from Greek roots with a smattering of other roots invented to fill up where there is no existing word for a concept.)

The terms I propose are what I use personally -- you may choose to follow them or not. But since there isn't any standard terminology either, we might as well adopt one of them, so I chose mine. If you prefer a different set of terms, please speak up -- the important thing is that all of us agree on what the terms mean.

[...]gonegahgah! I do understand what you mean. But quickfur will undoubtedly hate it , cause now we have not 2 but 3 opinions as to what is seen instead of our familiar cube in 4D (and we thought that it was such a simple question!)

On the contrary, I quite agree with gonegahgah's interpretation of things. After I took the time to define my proposed terminology, the thought occurred to me that actually, we (3D beings) can't see points or lines at all, because they have zero area! The supposed "points" that we see actually aren't points; they are tiny dots, which if you magnify them enough, will be (roughly) circular in shape. Such is the case with a dot of ink drawn on paper, for example. If you could draw a dot on paper that has 0 area, it would be invisible. Same thing with lines: a mathematical infinitely-thin line has zero area, and so it can't be seen by us! What we 3D beings perceive as lines are actually very thin rectangles (or some similar thin long shape). The edges of a polyhedron actually are unseen (unless we deliberately constructed thin cylinders to fill in their role) -- we infer them because that's where two faces of the polyhedron meet. It's a mental construct, not a physical one. Only the faces of the polyhedron are actual physical things that we can see with our eyes.

So actually, we will never be able to "see" as a 2D being sees -- because for a native 2D being, the only thing that's visible is length (i.e., lines and line segments, or 1-bulk using my proposed terms). The 2D being can't see area at all, so polygons and other 2D shapes are perceived only by their edges.

In the same vein, a 4Der's vision is quite unlike ours, because they cannot see points, lines, or faces! They can only see in 3-bulk (i.e. "volume"), and so any points or lines they can see must occupy 3-bulk, since otherwise it would be invisible to them. So they actually don't see faces (2-faces) at all; the mathematical 2-faces have zero volume, and are thus invisible to the 4Der's eyes. The only "polygons" visible to them are actually very thin prisms -- what they see as a hexagon is actually a very thin hexagonal prism. And they don't see it in the same way we do -- two hexagons and 6 (very thin) rectangles; they see it as a 3-bulk, which we can't see.

Here is what I understand: you basically are saying similar thing to quickfur, namely, that all points of 3d subspace, that the cube "carves off" from the 4D it is in, are seen at once. Except that in your case, the 'front' points obscure the 'back' points. lol.

Actually, that makes sense. All of this dawned on me when I was working out the definitions for bulk (length, area, volume, etc.). A 3D being sees only 2-bulk -- we can't see 3-bulk, though we certainly can infer that it's there, inside the faces of that cube. Similarly, a 2D being sees only 1-bulk: it sees edges, and from the shape 2D space enclosed by those edges, they infer the 2-bulk enclosed. But they can never see this 2-bulk directly.

But there's more to this. Our universe is 3D, and therefore every object must also be 3D, otherwise it would cease to exist. Even the thinnest polygon we can make has to be made up of atoms, so it must be at least at thick as 1 atom. Otherwise, it couldn't even have existence in our 3D world. So this means that what we think of as polygons in our 3D world are actually very thin prisms, which have non-zero (albeit very small) volume. A true mathematical polygon of zero thickness (and therefore zero volume) cannot possibly exist in our 3D universe! But since it is actually a prism and not a "true" polygon, it only stands to reason that it has two faces -- after all, a hexagonal prism has two hexagonal faces, no matter how thin it may be. And that's why we perceive polygons as having two faces, a front and a back. Our polygons are actually 3D prisms in disguise!

Similarly, in 4D, a true 3D cube of zero 4D thickness cannot possibly exist in 4D, because for anything to exist in the 4D universe, it must be made of 4D atoms, and 4D atoms have non-zero 4D thickness -- they occupy 4-bulk. So for a cube to exist in 4D at all, it cannot be a mere cube; it must be a very thin tesseract, one where two of the 8 cubical facets have macroscopic measurement, and the other 6 are very thin, so as to appear to be polygons (and thus, we have the 6 "faces" of the "cube"!) And so, these two facets with macroscopic measurements are the two "sides" of the "cube" that the 4Der sees -- they are two 3-bulks that form the boundary of the "cube" (which is actually a very thin tesseract in disguise).

So in a sense, 4Ders can't see "real" 3D objects at all. The objects that exist in our 3D world, if indeed space is only 3D, have zero 4-bulk, so they occupy no space in 4D and cannot exist there. Asking how a 3D cube appears to a 4D being is, in a sense, an irrelevant question, because real 3D cubes don't exist in 4D (and can't exist there).

But to stop there seems like such a cop-out. But don't worry, the story doesn't end there. Here is where we get into an immensely interesting topic. Suppose what we imagine is our 3D universe actually isn't merely a 3D universe. Suppose we actually have 4D thickness, albeit so small that it's imperceptible, and we are merely confined to 3 macroscopic dimensions (for whatever reason). In that case, what we perceive as purely 3D objects actually aren't just 3D objects; they are actually very thin 4D prisms of 3D objects. Then, in a very real sense, everything in our world actually has two 3-faces (i.e., 4D facets) which we cannot see. A 4D being observing our world from outside the 3D hyperplane that we're confined to will be able to see one of these 3-faces. A cube, then, is actually a thin tesseract, and the 4D being is able to see one of its two macroscopic facets. It would have two "sides" -- two macroscopic facets which can only be seen one at a time from the 4D point of view, just as gonegahgah said.

But what about us, who are actually 4D yet confined to 3D? What does the existence of another dimension imply for us?

Allow me to use a little illustration that I've used before. Suppose we have a very large desk, with some objects on them -- say hexagonal prisms, pentagonal prisms, cylinders, etc., all of which are rather thin, only 1cm thick, say. Suppose further that the radius of these objects are rather wide compared to their thickness, say their radii measure at least 10cm or more, so there's no chance they can fall over sideways. On top of these objects there's a large glass pane the size of the desk's surface, such that these prisms are confined to the surface of the desk and cannot move off of it, though they are free to slide around on the surface. Now imagine that some of these objects are little machines with some AI that, in a sci-fi sorta way, give them some kind of artifical consciousness. As far as these robots are concerned, their universe is 2D: they cannot access the 3rd dimension (leave the surface of the desk), and their light sensors ("eyes") are built in such a way that they can only receive light travelling horizontally, parallel to the surface of the desk. Any appendages they have are also constructed of 1cm thick joints, and so they can only ever interact with the 1cm high sides of the prisms and cylinders. They have no way of measuring this thickness, since their measuring instruments are also 1cm thick -- so they can't detect any 3D thickness at all. So effectively, these robots are "2D beings" living in a "2D world". Even though they're actually 3D constructs, they can't access the 3rd dimension, and can't perceive anything in their environment that would suggest space has any more dimensions than just two. As far as they can tell, the universe is just 2D and nothing more.

But suppose these robots one day start experimenting with cutting these supposed "polygons" (which are actually 1cm prisms) into smaller pieces. Everything seems fine as long as the pieces are significantly wider than 1cm --- they can't fall over, and so they continue to appear as though they were merely 2D constructs. But one day, the robots manage to cut out a cylinder that's only .1cm in radius. This isn't anything surprising at first -- it just behaves like a very small 2D object. But then, because its radius is so small, the glass pane isn't enough to keep it standing upright: it falls over. Suddenly, its 3D nature starts to show through: whereas before, as far as the robots could tell, polygons cannot occupy the same space, now two of these fallen-over cylinders can be stacked on top of each other. They also exhibit "strange properties" -- like being slanted in the 3rd dimension, which causes them to interact in "strange ways" with the macroscopic objects around them. Now the robots have reason to believe that perhaps there's something more to just 2D -- perhaps there's a 3rd imperceptible dimension at work here. All those "strange properties" of these "microscopic polygons" could be explained in terms of a 3rd dimension that's confined.

As long as these prisms' radius is "large enough" (i.e., macroscopic), everything seems to be well-behaved 2D objects. But once you get them down to very small radii (i.e., "subatomic" level), they start behaving really oddly. But these odd behaviours can be explained in a mundane way once you realize there's a 3rd dimension involved. Does this sound familiar? This is the idea behind string theory -- dimensional analogy style.

(Of course, all of this rests on a very important assumption -- "what if we are actually higher-dimensional beings confined to lower dimensions ...". That's a very big "what if". You are free to disagree with this assumption. I can't say I'm convinced about string theory myself, either. But it does make for nice dimensional analogies, with all sorts of interesting consequences for 4D visualization, etc..)

[ac2000]

I haven't had the time to reply to the older messages of this thread in due time, so pardon me if comment on the earlier ones:

Actually, I wanted to write it in such a way that the 4Dness of the boy is not made explicit until later in the story. It would read like a regular story, but then little things here and there would seem misfitting [...]

I think this is a very good idea. I've once written a short dialogue, where in the course of the dialogue the people seem to be very strange, or even outright crazy, but in the end the attentive reader might have noticed that the dialogue partners are no people whatsoever - but ... flies.
In general I suppose the most effective parts of a story are often those where the reader has to do most of the work of concluding and inferring and imagining. Because many people prefer to believe what they themselves have imagined and concluded to what they have just read.

Similarly, in using dimensional analogy to understand 4D geometry, we're taking the liberty to imagine living 4D beings who interact with their 4D environment in analogous way to our own interactions with the 3D world. So we postulate "4D light" that propagates in 4 dimensions rather than 3 -- something quite foreign to physics as we know it (since if light in our world were to actually propagate in an additional dimension as well, it would fade according to an inverse cube law, rather than an inverse square law as we observe it).

I have this book about Hyperspace (it's an older book by Michio Kaku) and I remembered there was something mentioned about light and additional dimensions there. So I looked this up and in fact light in 4 dimensions seems to be quite a familiar concept at least for those physicists that deal with Kaluza/Klein & string theory.
It says there, that the fact that light can travel through vacuum as a wave is somewhat contradictory , because waves always need some kind of medium to travel in. But if they add another spatial dimension (in addition to the three common spatial dimensions plus one time dimension, so they have 5 dimensions in all) it seems to be possible for the light to travel in vacuum. Why exactly, I didn't really understand, though .

However, we may gain some insight into a 4D being's sight if we understand that their eyes must have a 3D array of light-sensitive cells. A 3D array of cells shouldn't be too hard for us to imagine; now imagine that these cells are lit or not, according to whether they fall near enough to the surface of a sphere and the cube enclosed inside it. So this array of cells would, in effect, have some kind of "pixelated image" of the sphere and the cube (sorta like those old 3D voxel-based games, in which everything is made of little blocks, with a jagged appearance).
Now the key here is to understand that every "voxel" in that 3D array is visible to the 4D being simultaneously. That is to say, they do not merely see the surface of a sphere or the surface of the cube; they see the entire 3D volume of both objects, every point on the surface, and every point the interior, all at the same time. This may take a while to sink in.

I think this is quite plausible. The 4D creature's eye could probably be imagined as a kind of Rubik's Cube just with many more cells. The image of the 3D object with all inside and outside parts would just be filling the threedimensional photoreceptor cells of that cubical eye.
Although I cannot "really" imagine how it would look like, because that would probably require a 4D brain that sorts out all the giant amounts of 4D visual data. . And what a hypercube would look like in 4D for a 4D being is still completely beyond me.

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And I did not think that my interruption of your thoughtful discussion was rude, 'cause I have a mild form of Asperger syndrome, and speaking up for the truth is more important to me than what is often considered a 'civility'. I saw that you were giving a wrong answer to the person, and thought it was my duty to interfere.

Speaking up for the truth is of course always a noble cause .
But I can't see why quickfurs answer should have been wrong. I have looked it up in Rudy Ruckers book about 4th dimensions and he describes it in much in the same way, namely that a 4D creature with a 3D retina would be capable of seeing every part of a human being at the same time (every square centimetre of the skin, organs and everything) from one single point of view.
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