quickfur wrote: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.
gonegahgah wrote: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.
quickfur wrote: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.
quickfur wrote: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.
ac2000 wrote:gonegahgah wrote: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 ).
[...] 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?
[...]
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. [...]
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.
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.
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;
ac2000 wrote: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.
[...]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.
[...] 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 wrote: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).
quickfur wrote: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.
quickfur wrote:This may take a while to sink in.
4Dspace wrote:quickfur wrote: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
quickfur wrote: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.
[...] 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 wrote:4Dspace wrote:quickfur wrote: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.
quickfur wrote: 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.
quickfur wrote: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?
quickfur wrote:All I said was that the entire 3D hyperplane is visible to the 4Der simultaneously.
quickfur wrote: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,
quickfur wrote:... because the line-of-sight is parallel to the vector (0,0,0,-1), and all 6 faces lie perpendicular to this vector simultaneously.
quickfur wrote: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 wrote: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.
quickfur wrote: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.
quickfur wrote: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.
4Dspace wrote: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.
4Dspace wrote:[...] 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").
[...]
gonegahgah wrote:However, neither of us can see the inside edges as they are inseparable from the 2D 'volume'.
gonegahgah wrote:We can only see those outside edges albeit from just less than 360deg of sideways viewing but they are still outside even to us.
gonegahgah wrote:At no point can we see the inside of the edges. That's worth remembering.
gonegahgah wrote: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.
gonegahgah wrote: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.
gonegahgah wrote: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'.
gonegahgah wrote:That 'face' will be described by the 6 faces of our cube as its edges.
gonegahgah wrote: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.
gonegahgah wrote: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.
gonegahgah wrote: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.
quickfur wrote:4Dspace wrote:[...] 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 wrote:[...]gonegahgah wrote: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.
gonegahgah wrote: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 wrote: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".
quickfur wrote:4Dspace wrote:[...] 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.
4Dspace wrote: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.
quickfur wrote: 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.
quickfur wrote: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)?
quickfur wrote: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 wrote: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.
4Dspace wrote: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 wrote: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?
[...]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.
quickfur wrote: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 [...]
quickfur wrote: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).
quickfur wrote: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.
4Dspace wrote: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.
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