Dimensional Baby Steps

[4Dspace]

Oh quickfur! Thank you for your beautiful post

Very interesting about the terms! Wow, who could have known. I will read the reference later. But.... how can it be? The terms are everything in math, in logic. How you define them, that's your bricks with which you build up your models. If they are flimsily defined... no wonder that our models are a mess.

... 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.

Yes, this is a similar thought that gonegahgah expressed in a neighboring thread, that n-object must exists in N-space and no less. Still, there must be a minimal requirement for an nD-object being able to exist in (n+x)D universe. I have a hunch that 3d is such a minimal requirement.

Still, it is still interesting, as an exercise in geometry, what sort of projections to a POV are possible for us, humans, in N-space looking at a n-object (with n ≤ N). Our latest discussion helped me see the details of a cube that is made of stacked 27 cubes (3 per edge), colored red outside, green inside.

From our 3D, we see a red cube made of smaller red cubes. We can't see that they are green inside. Now, move to 4D and what do we see? From that (0,0,0,5) POV, we see a hexagon, filled to the rim with 27 cubes, we see them all. And half of each those cubes 6 faces we see red and half, green. Sort of like a chess board.

You or gonegahgah may object, what about the faces of the main cube? Would not the front face obscure at least some of the small cubes? And then the small cubes would obscure the far face of the large cube. -? But.. the faces of the large cube are made of the small cubes. Which means, that, indeed, instead of a large cube, what is seen in 4D is a hexagon broken up into 27... hexagons, each of which is broken into 6 segments still, 3 red, 3 green...

As you change you POV from (0,0,0,5) to, say (5,0,0,0), 2 of the 6 segments of the 27 hexagons will flip their color, while the other 4 will stay the same. As you change your POV to (0,5,0,0), another 2 of 6 of each of those 27 hexagons will flip their color, and yet another 2 of 6 will do the same from (0,0,5,0) POV... Have you noticed?

The colors of the 162 segments (27*6) this hexagon is broken into will undulate, as we walk around the cube, 54 of them (one third) flipping their colors with each change of direction of our POV. But it takes only 3 out of 4 possible directions in 4D to see them turn "full circle" (= meaning that we see the same thing as from the first POV at (0,0,0,5).

-? in other words, there are 4 distinct POVs in 4D, yet the cube "turns" to its original view after 3 changes in direction -?

...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.

I don't see how this follows. A mere analogy with 3D does not strike me as a valid enough argument to make such a conclusion.

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).

Now, that's an interesting thought!

So in a sense, 4Ders can't see "real" 3D objects at all.

Why, he can certainly draw them on his hypersheets tacked to his hyperdesk, lol.

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.

Really? I daresay that 3d is a valid object in 4D. I can't prove it yet, it's just a strong hunch at this stage.

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.

And I absolutely adore how well you described the robots below. You see what I see. And what we see is physics, how it should be. I believe we are those robots. We live in 4D, we always have.

We are attached to the 3D display, which we call "empty space". But is is not empty, of course. It is a super-rigid solid, on the surface of whose hyperplane we the robots "crawl" (well, actually, glide).

Yes, it is a rigid solid. How else could it support transverse EM waves going at such phenomenal speeds? So, according to Maxwell, space is a solid. According to us, robots, it is empty. But what is indeed empty is the 4th spatial dimension, in which we glide just above the 3d-hyperplane, thinking all the time that the EM radiation that delivers all the info about our world to our senses (warmth and light, for starters) traverses the emptiness, through which we seem to move with such ease.

But the atoms that make up our bodies actually exist in 4D. Does it mean they are 4d objects? I am not sure yet.

All the info we get, is via this EM radiation. And that is confined to the solid we call space. I don't like the word bulk. It is a display. It is very much like a touch screen of a computer. All the atoms "touch" it and send forth disturbances through it, which arrive to other collections of atoms attached to this display elsewhere, and those atoms say, wow, I see things!

And so, everything we "see" about our world, is actually a projection onto 3D from the 4th.

That's physics that I see.

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..)

That was just beautiful. Thank you for it.

[4Dspace]

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.

It's because every point of space inside the cube is visible from 4D. But points are dimensionless, and => are invisible. What is seen are the aspects of the structure inside a 3d object, determined by your POV. So, only the 'front' side of any structure is seen, not all around .

[4Dspace]

-? in other words, there are 4 distinct POVs in 4D, yet the cube "turns" to its original view after 3 changes in direction -?

Ah! I got it: the 4th direction is the "edge" gonegahgah was talking about. So, from the 4th direction, we see our familiar cube, like in 3D, so, we see only 3 red faces, and the other 3 are obscured.

So, yeah, that hexagon we see in 4D will turn all red from the 4th direction, the one where w=0. Which means that I messed up the definitions of POVs (x,y,z,w) above. -?

[quickfur]

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.

That reminds me of a story I read once, about a person who is becoming increasingly suspicious and paranoid that something about his otherwise mundane environment is desperately wrong. He eventually becomes convinced that "aliens" have invaded his city, disguised as humans, and he engages on a long convoluted scheme to outsmart the "aliens". His mental state deteriorates, and he becomes increasingly desperate and obsessed about these "aliens", to the point of mania. Eventually, some mishap (I forget what) befalls him and he falls unconscious, and the scene shifts to a hospital, where the doctor examines the unconscious person, and laments to the nurse that it's such a pity that such a bright, intelligent young man would end up in such a tragic state. The nurse agrees, and the two exit the scene, and in the last sentence is the one-word giveaway that changes the meaning of the entire scene: the doctor burbled goodbye as they left.

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.

I read somewhere that the best prose is one that, instead of bombarding the reader with verbose descriptions of everything down to every last shirt button and blade of grass, only explicitly describes the most salient points, and leaves the details to the imagination of the reader. Readers will fill in the blanks with details that cause them to love the story in a way that dictating every last blade of grass could never do.

Because many people prefer to believe what they themselves have imagined and concluded to what they have just read.

Unfortunately this also works too well in a wrong way -- sometimes we read and think we understand, but actually we completely missed the point.

[...] 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 .

I don't think I understand that either.

But I did read about a very fascinating discovery, that if one were to take Maxwell's equations for electromagnetism and to recast them in terms of 4 spatial dimensions, then one would obtain equations that are equivalent to Einstein's general relativity. This suggests a deep connection between electromagnetism and gravity, although the nature of this connection turns out to be more mysterious than first thought, because even though the forms of the equations match, when you actually plug in the numbers and work through the calculations, the values you get don't match up with experimental evidence. It is still too unlikely a coincidence, so some connection must be there, but obviously there's more going on than is apparent to us presently.

[...]
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. [...]

When I first investigated this matter, I also came to the same conclusion, that visualizing 4D is something beyond our limited mental reach. Without eyes that can see 3D images in their full glory, and without a brain wired for interpreting these images, how could we hope to ever visualize the 4D world?

But one day I had an epiphany. If 3D beings had 2D retinas and see the 3D world through projections of 3D to 2D images in this retina, then 4D beings would have 3D retinas and see the 4D world through projections of 4D to 3D images in this retina. But -- and here was the inspiration that struck me -- if the images in the 4D being's retina are 3D, then, at least in theory, we should be able to visualize them, because even though our eyes only see in 2D, our brain is so adept at constructing 3D models of things through these 2D images, that most of the time we are quite unconscious of the fact that we only see in 2D! So though 4D itself may be beyond our reach, the 3D images that form in the 4D being's eye lies within our grasp. We may not be able to instantly see the entire 3D image as they can, but given some time and a way to probe these 3D images, perhaps by studying them from different 3D viewpoints, we should be able to reproduce these 3D images in our mind. Even if we are unable to fully visualize every single voxel in the 4D being's retina in full color, we can at least visualize the most salient points: the polyhedral images of projected 4D surfaces, their edges and ridges, how they are fitted together, etc..

But if we can form mental models of these 3D images, albeit indirectly, then we too, in a limited sense, can "see" the same 3D images that the 4D beings see when they look at 4D objects. And since the 4D beings themselves do not directly see 4D, but infer 4D depth through various clues in the 3D images, such as foreshortening, light and shade, parallax, etc., just as we also infer 3D depth from the same clues in the 2D images that our eyes capture, then in principle we too, through conscious application of the same inferences, should be able to infer 4D depth, and thereby "see 4D". We will not have the help of a natively-wired brain to do all these inferences for us, of course, but the kinds of visual effects produced by the 4D->3D projection have direct analogues in the 3D->2D projections that we are so familiar with. So, with proper study and training, we should be able to learn how to consciously and manually apply the 4D depth inferences that the native 4D being unconsciously and instinctively applies. Thus, through this kind of "surrogate vision", we should be able to "see" 4D just as a native 4D being would.

The only thing that remains, then, is for us to obtain "samples" of 3D images that form on the 4D beings' eyes, so that we can study them and hopefully learn the principles of 4D depth inference. Now, obviously, hand-drawn (or otherwise hand-made) images will fall far short of our goal, because they are bound to have inaccuracies and perhaps fatal flaws, either through mechanical error or through our own mistaken conceptions of the 4D world. What is needed is to build accurate mathematical models of 4D objects and have a computer perform the 4D->3D projections for us in an automatic way, free of human error. We can then study the resulting images, with the understanding that they will exhibit visual effects analogous to what happens when we project 3D objects to 2D -- things such as foreshortening, light and shade, parallax, etc., and by applying dimensional analogy to these effects, hopefully learn how to infer 4D depth from them.

Now, light and shade is a difficult issue, one that I have not fully resolved, because while a computer will happily represent 3D arrays of voxels with full color lighting and shade, it is rather difficult to convey this information to us through our limited 2D sight (and through the 2D computer screen) in a way that our brain will be able to reconstruct into a full-color 3D image in its full glory. However, foreshortening is something quite easy to convey -- at least, its effects are readily apparent, and parallax can be done by doing the 4D->3D projection from two different 4D viewpoints, one slightly displaced from the other in some horizontal direction. Foreshortening is most readily represented with objects that have well-defined edges and corners, so this means that the simplest 4D shapes for starting out with 4D visualization the 4D analogues of our 3D polyhedra, which are generally known as 4D polytopes or polychora (singular: polychoron). Some of the simpler curved shapes, such as various cylinder-like 4D shapes, should also be within our reach.

It is with this in view, that I made my website. It has not been an easy task, and thus far I rely mainly on foreshortening as the method of 4D depth inference, but on the site you will see images that were made from various 4D->3D projections of 4D objects, and in my 4D visualization document (recommended starting point) I describe some of the 4D depth inference principles that I have discovered that may be applied to infer 4D depth from these images.

I don't claim that my method is the "best" or even of any help -- some people on this forum have found my approach rather unhelpful -- but it has worked wonderfully for me, and I have been able to independently rediscover a number of 4D objects purely through "visualizing" them, which I later confirmed matched what mathematicians have proven about them. So you may find it helpful.

[4Dspace]

My Asperger syndrome commands me to address this tinny bit of your fascinating discussion, guys, and that is:

But I did read about a very fascinating discovery, that if one were to take Maxwell's equations for electromagnetism and to recast them in terms of 4 spatial dimensions, then one would obtain equations that are equivalent to Einstein's general relativity.

It was the other way around: Einstein's general relativity was set in 4 spatial dimensions by Kaluza and Maxwell's equations popped out by themselves. This was the beginning of the String theory.

[gonegahgah]

From our 3D, we see a red cube made of smaller red cubes. We can't see that they are green inside. Now, move to 4D and what do we see? From that (0,0,0,5) POV, we see a hexagon, filled to the rim with 27 cubes, we see them all. And half of each those cubes 6 faces we see red and half, green. Sort of like a chess board.

Let's say we have these 27 cubes arranged as a single cube and that they are made of red wood and that we have painted their outsides green. So what colour will our 4D friends see our cube?

Well, it could be say blue. The tricky 4Der could paint the two 'faces', that they see, as blue and we wouldn't even know about it. We would still see green painted cubes and when we cut into them they would still be red inside. But the 4Der would laugh at us and say they are mainly blue because that is what he painted them.

The same goes for a 2Der. If they have a square made of red wood and they paint the outside, that they can see and touch, green and we then go and paint the face that we see blue; they will not be able to see our blue paint and as far as we will be concerned the square will be mostly blue with green edges. The 2Der won't be able to see the blue paint at all and will tell us that we're lying and that we haven't painted the square blue at all. They will still see a green square and even when they chop into the square to see if maybe we are telling the truth they will still see the red wood without a trace of blue.

Ah! I got it: the 4th direction is the "edge" gonegahgah was talking about. So, from the 4th direction, we see our familiar cube, like in 3D, so, we see only 3 red faces, and the other 3 are obscured.

Not quite but almost. I'll talk more about the 4th direction soon. You are right that edge on the 4Der will only see the 3 (I thought it was green) painted faces as the 3Der painted them.

But, you're making me detour, 4DSpace. I know there may seem to be no point to my going on about rings having an inside edge and circles not having an inside edge but it is important. Can we start there please? Is the notion, that a ring has an inside edge and a circle doesn't, acceptable? Or, do you view both circles and rings as identically have the same seeable inside edge of their rim? Can we start there?

You're trying to think ahead which is admirable and a good sign of intelligence but just for the moment can we take it by these partial steps?

[4Dspace]

Let's say we have these 27 cubes arranged as a single cube and that they are made of red wood and that we have painted their outsides green. So what colour will our 4D friends see our cube?

Well, it could be blue. The tricky 4Der could paint the two 'faces', that they see, as blue and we wouldn't even know about it. We would still see green painted cubes and when we cut into them they would still be red inside. But the 4Der would laugh at us and say they are mainly blue because that is what he painted them.

What 2 'faces'? The cube is a cube is a cube. If it is a tesseract in disguise instead, as quickfur suggested above, then yes, they could do such a trick. But if it is just a cube, then it has only 6 faces, and we get to see them all as we rotate the cube in our 3D.

The same goes for a 2Der. If they have a square made of red wood and they paint the outside they can see green, and we then go and paint the face that we see blue; they will not be able to see our blue paint and as far as we will be concerned the square will be mostly blue with green edges. The 2Der won't be able to see the blue paint at all and will tell us that we're lying that we have painted the square blue at all. The will still see a green square and even when they chop into the square to see if maybe we are telling the truth they will still see the red wood without a trace of blue.

Agree. But this implies again the same thing, namely, that in 3D we only have 3d objects disguising themselves as 2d.

When I started the discussion about the color of the faces of the cube, what I meant was that 2d-planes had direction, which is expressed as their chirality. And to see it easier, I thought a color would be better than trying to see, which way that thing is turning, which can be shown on the face as a circle with an arrow. Rather than looking at which way the arrow is turning on each face, I thought that color would be quicker to judge. But in actuality, a 2d plane is a 2d plane and you cannot color it's "sides" differently, because it would imply layers of paint in the direction a 2d object does not have. But it still has direction of its chirality. That's all.

Ah! I got it: the 4th direction is the "edge" gonegahgah was talking about. So, from the 4th direction, we see our familiar cube, like in 3D, so, we see only 3 red faces, and the other 3 are obscured.

Not quite but almost. I'll talk more about the 4th direction soon. You are right that edge on the 4Der will only see the 3 (I thought it was green) painted faces as the 3Der painted them.

In the case above, the outside of the cube was red and inside green. The POV is outside of the space the cube is bounding. So, what's seen is the outside color, which is red.

But, you're making me detour, 4DSpace. I know there may seem to be no point to my going on about rings having an inside edge and circles not having an inside edge but it is important. Can we start there please? Is the notion, that a ring has an inside edge and a circle doesn't, acceptable? Or, do you view both circles and rings as identically have the same seeable inside edge of their rim? Can we start there?

I'm not making you anything. You're right about rings and circles. Only, when you go from one D to another, you gotta keep analogies separate from what is. Analogies are helpful, but they can also be the source of confusion.

No, I do not view both circles and rings identically having the same inside edges to their rims. This was part of the analogy.

[gonegahgah]

I'm not making you anything. You're right about rings and circles. Only, when you go from one D to another, you gotta keep analogies separate from what is. Analogies are helpful, but they can also be the source of confusion.
No, I do not view both circles and rings identically having the same inside edges to their rims. This was part of the analogy.

Cool. I'll go onto the next bit now but I'm going out apparently. I'll type up more when I can...

[quickfur]

[...]

...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.

I don't see how this follows. A mere analogy with 3D does not strike me as a valid enough argument to make such a conclusion.

It follows for the simple reason that for anything to exist in N dimensions, it must occupy a non-zero amount of N-dimensional bulk in that space, for if it occupies zero N-dimensional bulk, then it means that it occupies zero space, which is equal to not existing at all.

Furthermore, for something to exist in that space, it must be made of the N-dimensional equivalent of an atom -- it stands to reason that you can't cut something into an infinite number of pieces, as the Greeks realized. But in order for an N-dimensional atom to be the stuff that N-dimensional objects are made of, then it follows that it must occupy non-zero N-dimensional bulk, since otherwise you couldn't possibly build an N-dimensional object out of them and have it fill a non-zero amount of space. To make an object of non-zero bulk from atoms of zero bulk would require an infinite number of atoms, which is equivalent to cutting an object into an infinite number of pieces -- it's not plausible.

[...]

So in a sense, 4Ders can't see "real" 3D objects at all.

Why, he can certainly draw them on his hypersheets tacked to his hyperdesk, lol.

Certainly. But what would he draw them with? One would assume some kind of ink, or chalk, or whatever it is that can make a contrast with the background color of the hypersheets. Then one must ask, what kind of stuff is the ink or chalk made of? Ultimately, it must be made of 4D atoms that fill non-zero 4D bulk, since otherwise the 4Der would have no way to handle such an ink. A liquid of zero bulk would occupy no space, and therefore the inkpot is effectively empty, and there is no ink at all!

So the supposedly 3D diagrams that the 4Der draws on these hypersheets actually are made of 4D ink of non-zero 4D bulk. Which means they are not "real" 3D objects at all; they must be at least one 4D atom thick (however thick the layer of ink is on the hypersheet). They are actually very thin 4D objects in disguise.

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.

Really? I daresay that 3d is a valid object in 4D. I can't prove it yet, it's just a strong hunch at this stage. [...]

But if a true 3D object exists in 4D, then it must have zero 4D bulk, since, by definition, it has only 3 measurements (i.e., dimensions); its fourth measurement is zero. Which means that it occupies no space in 4D. Would you say that something exists if it occupies no space at all?

[quickfur]

My Asperger syndrome commands me to address this tinny bit of your fascinating discussion, guys, and that is:

But I did read about a very fascinating discovery, that if one were to take Maxwell's equations for electromagnetism and to recast them in terms of 4 spatial dimensions, then one would obtain equations that are equivalent to Einstein's general relativity.

It was the other way around: Einstein's general relativity was set in 4 spatial dimensions by Kaluza and Maxwell's equations popped out by themselves. This was the beginning of the String theory.

Ahh, so I got it the wrong way round. My apologies.

[4Dspace]

... for anything to exist in N dimensions, it must occupy a non-zero amount of N-dimensional bulk in that space, for if it occupies zero N-dimensional bulk, then it means that it occupies zero space, which is equal to not existing at all.

Yes, this is logical, but... That anything existing in N dimensions would be qualified as an N-d object in that space, in other words, "N-matter". But certainly, matter is not the only thing that exists in our 3D. There are structures, like for example, the before-mentioned invisible lines of force that Maxwell described, and they pertain to the electric and magnetic planes, which are perpendicular to each other. The E plane is always perpendicular to our POV, and M plane we always "see" edge on (no matter which way we turn, in our homey 3D). We can't see nor feel those structures, but know of them only indirectly, in the way they interact with material objects in our space. And we can make them manifest, as Maxwell did, when he studied them.

Our space is 3D, but the interesting thing about those planes is that one of them completely belongs to our space, as a 2d object proper, while another only intersects it, also as a 2d object, which, however shares only 1 of its dimensions with our 3D, while its other dimension undoubtedly exists, even though it is not accessible to us directly. Those 2-d planes do exist and they are not 3d objects in disguise.

Furthermore, for something to exist in that space, it must be made of the N-dimensional equivalent of an atom -- it stands to reason that you can't cut something into an infinite number of pieces, as the Greeks realized. But in order for an N-dimensional atom to be the stuff that N-dimensional objects are made of, then it follows that it must occupy non-zero N-dimensional bulk, since otherwise you couldn't possibly build an N-dimensional object out of them and have it fill a non-zero amount of space. To make an object of non-zero bulk from atoms of zero bulk would require an infinite number of atoms, which is equivalent to cutting an object into an infinite number of pieces -- it's not plausible. ...

Logical. and well said

But if a true 3D object exists in 4D, then it must have zero 4D bulk, since, by definition, it has only 3 measurements (i.e., dimensions); its fourth measurement is zero. Which means that it occupies no space in 4D. Would you say that something exists if it occupies no space at all?

Agree, but then, there are different kinds of objects. Some are "material" and, you're right, they should have non-zero bulk, but others and not.

PS

Ahh, so I got it the wrong way round. My apologies.

Apologies? Why, you do take yourself seriously

[quickfur]

... for anything to exist in N dimensions, it must occupy a non-zero amount of N-dimensional bulk in that space, for if it occupies zero N-dimensional bulk, then it means that it occupies zero space, which is equal to not existing at all.

Yes, this is logical, but... That anything existing in N dimensions would be qualified as an N-d object in that space, in other words, "N-matter". But certainly, matter is not the only thing that exists in our 3D.

Quite so. But if we now speak of immaterial things, the question of how they would be "seen" (by us or by hypothetical 4D beings, from some hypothetical viewpoint) is no longer a question of actual vision as such, but rather, of how we represent these immaterial things, perhaps as diagrams or mathematical structures of some sort. And perhaps this is what tripped us up in our various miscommunications: there are many ways of representing the same thing, and they are not all compatible with each other.

Furthermore, since there are many ways of representing something, it seems fruitless to argue over which representation is the "right" one, for that will depend on what our goal is, and what we wish to be able to do with it. For example, I can represent a cube by the four letters C-U-B-E. If I'm merely describing a bunch of objects in the room, one of which is a cube, then this representation is good enough for my purposes. But if I want to speak of its symmetries, say, then these four letters are clearly inadequate -- I need some kind of representation that expresses the symmetric elements of the cube, perhaps a list of its vertices, edges, and faces, and how they are connected to each other. This representation may take the form of a written list of elements, 8 lines describing the 8 vertices, 12 lines describing its edges, and 6 lines describing its faces. I can then number these lines, and describe the symmetries of the cube as various rearrangements of these line numbers that preserve the structure of the cube. But then, if I want to be able to visualize the cube, this list of 26 elements would not be very helpful. I would need some kind of visual representation of the cube, perhaps an image on the computer screen produced by projecting the coordinates of the cube's vertices from 3D to 2D from some arbitrarily chosen viewpoint (each of which would produce a different representation of the cube!), and drawing lines and perhaps painting the projected polygons in some arbitrarily chosen color.

And the question of representations gets compounded even more when we speak of 4D, because now there is not only the question of what to use to represent the target object (lines, or polygons, opaque or transparent, color, etc.), but also what kind of projection to use (4D->3D, 4D->2D, parallel or perspective, etc., etc.) -- assuming, of course, that we're trying to achieve some kind of visual representation. And then if we want to "view" the target object from a 4D vantage point, there's the question of what our assumptions about this "view" are. Would it be a 4D->3D projection? Or a 4D->2D projection? Or a cross-section? Do objects obscure each other? How is this obscuring determined (since these are not material objects that have well-known visual properties)? What parameters are we assuming? How do we compute this representation, and in what form (coordinates, 3D models, 2D pictures, etc.) will the result take? And since we're talking about planes that have orientation -- how will we represent this orientation? Not to mention, whether or not the resulting representation will even have an orientation depends on how we choose to represent these abstract, immaterial things. (Remember, in the four-letter representation of the cube, there aren't any vertices, edges, or faces at all, even though the cube itself certainly possesses these elements.) Moreover, since we are no longer talking about material things, what constitutes "vision" needs to be made clear, as one person will have a different definition from the other, there being no unifying common notion of "sight" in the realm of the immaterial.

Unless these assumptions are all made clear, it should be no surprise that miscommunications and misunderstandings happen.

[gonegahgah]

I think somehow we are crossing a little bit of string theory with 4-space theory here.
String theory would, somehow, seek to constrain our atoms to 3-only of their many dimensions for the purposes of spatial occupation.
Their many extra dimensions are not spatial at all otherwise we could rotate freely into them.
Under string theory it is perfectly legitimate for an object to occupy only 3 dimensions while those extra dimensions exist in which other things are happening.
So in string theory a cube made of three dimensional atoms in an extra-dimensional universe is perfectly legitimate.

But, this section of the forum is about spatial dimensions; not string theory.
If it is string theory that you are trying to elucidate 4DSpace then you should please take the discussion to the right area of this forum.
The discussion in this forum area is 'what if we had 4 spatial dimensions instead of 3' not 'how does string theory and 4-space mix'?

String theory and 4-space are two completely different topics and should not be jumbled together as proof of anything.
Discussions about fields - which aren't spatial - to somehow refute the need for 4D atoms is not relevant here.

[quickfur]

[...]
The discussion in this forum area is 'what if we had 4 spatial dimensions instead of 3' not 'how does string theory and 4-space mix'?

String theory and 4-space are two completely different topics and should not be jumbled together as proof of anything.
Discussions about fields - which aren't spatial - to somehow refute the need for 4D atoms is not relevant here.

Thank you, gonegahgah, I think you've hit the mark on the source of the miscommunication / misunderstandings between 4DSpace and myself. I'm thinking about spatial dimensions, whereas 4DSpace is thinking about string theory. No wonder we could never seem to agree on anything.

[gonegahgah]

It would have been nice to help someone else into the 4-space fold but it may be that 4DSpace is pursuing a different type of animal than ours.
None-the-less, this has been an interesting exercise so, with everyone's forbearance, I'll continue to explore it...

It is important to remember that nobody from any dimension can look at things from inside them when they are filled.
If a 2Der can not look inside his square than we can also not look inside his square from anywhere in his perspective.
We can look at their inside from the extra dimension but we can not look at it from their inside.

If we look at a 2Der and their square this will help us to progress to an understanding of how a 4Der looks at our insides.

Step 1. Looking at a 2D square from the extra direction:



Here the 2Der sees only a green line. The 3Der also can not look at the square from inside the square, as shown in the left picture. They will only see the inside immediately in front of their eye; and won't see the whole side that we expect to see. Instead the 3Der has to step out into our extra dimension, as per the right picture, to see the square side on and see that all the insides are red. Again however we could paint the sides blue and the 2Der would be none-the-wiser unless our paint got onto the edges. They could dig into the square that we've painted and would still only find red inside.

[quickfur]

[...]

Here the 2Der sees only a green line. The 3Der also can not look at the square from inside the square, as shown in the left picture. They will only see the inside immediately in front of their eye; and won't see the whole side that we expect to see. Instead the 3Der has to step out into our extra dimension, as per the right picture, to see the square side on and see that all the insides are red. Again however we could paint the sides blue and the 2Der would be none-the-wiser unless our paint got onto the edges. They could dig into the square that we've painted and would still only find red inside.

This jives with my recent realization that in order for the 3Der to see the 2D square at all, the 2D square must really be a very thin cube in disguise. Two of its faces have macroscopic dimensions, and the other 6 faces are macroscopically long but microscopically thin, so that they effectively behave like "edges".

Seen in this light (pardon the pun), this means that our red square with green edges is really a very thin cube made of red material (say red wood) with 4 thin faces ("edges") painted green. The 2Der only ever sees the microscopically-thin "edges", and while the 3Der in principle could also see these edges, he usually just sees the two red faces (because the "edges" are microscopically thin). If the 3Der were to paint one of the red faces blue, the 2Der would still just see the green edge, and if the 2Der digs into the square, he would see red. But the 3Der would see blue. It seems strange that a square can be both red and blue simultaneously, but that's because it's really a square prism in disguise. What the 2Der sees are the "side faces" which microscopically thin; the 2Der can't see the two macroscopic faces that the 3Der can.

This has even more implications. Suppose the atoms in the 2Der's world aren't just spheres, but cylinders confined to the plane. From the 2D point of view, a confined sphere and a confined cylinder looks exactly the same -- a "circle". But cylindrical 2D atoms have far-reaching consequences, because it means that what the 2Der sees is never the same as what the 3Der sees. For example, the 2D atom can reflect red light from its curved sides but reflect blue light from its circular lids. Then what the 2Der sees as a red atom never appears red to the 3Der, but blue. So the 2Der looking at his world may see a scenery with lots of green grass, bluish water, etc., but a 3Der looking at the same scene from the 3D point of view will see a completely different set of colors. (Not to mention the 3Der can see the innards of every 2D being, so the scene that the 3Der witnesses is dramatically different from what the 2Der might imagine!)

Similarly, if 3D atoms in our world are actually confined 4D spherinders, then what the 4Der sees can be dramatically different from what we imagine -- a beautiful scenery with green grass and blue snow-peaked mountains will appear nothing like that to the 4Der -- for all we know, the 4Der could be seeing purple grass and neon pink mountains with black streaks (iron ores buried inside the rocks, say) and holes (caves) filled with green liquid (actually water, except the transparent atoms in the water appear green on the 4D sides). For sure, what we imagine as our own handsome visage appears nothing like that to the 4Der -- who sees a thin layer of skin with muscles, veins, arteries, organs, half-digested food, circulating blood, etc., possibly in totally foreign colors from what we normally associate with these things.

[ac2000]

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!

Hmm, but what about the following scenario:
It is possible in our 3D world to project an image of a, say triangle, from a slide projector onto a screen. We can see the triangle on the screen. The screen has to have some thickness, but the triangle projected onto the screen is not a part of the screen itself. I suppose the triangle is only made of photons. Since photons have zero mass, they probably have no thickness either. That would be an example for a 2D Polygon in our 3D space, wouldn't it?

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.

I think the possibility that different spaces are sort of "overlapping" each other with some tiny parts into the next higher dimension quite possible.
And if this was not the case, all those analogies with creatures of 2D wouldn't properly work, I assume.
I had this idea today, that for a 2D creature to see anything at all of its 2D world, it must have an eye (retina or whatever) that is positioned at an angle to the edge of those supposed polygons (which it perceives as a line or 'wall'). It wouldn't matter if that angle was 90 degrees or 40 degrees or whatever. But there must be an angle between the thing to be perceived (say the edge of a square) and the retina of the 2D creature. If the eye lay on the same plane (without any angle) than nothing would be perceived. So if that angle is necessary, there must be some kind of rotation to get that angle. But this kind of rotation would ever only be possible if there was a bit of 3D space to rotate into. (Hmm, does this make any sense, what I wrote? I’m not sure. It seems to, but maybe only for the moment until one sees things differently again.)

He eventually becomes convinced that "aliens" have invaded his city, disguised as humans, and he engages on a long convoluted scheme to outsmart the "aliens". [...]
The nurse agrees, and the two exit the scene, and in the last sentence is the one-word giveaway that changes the meaning of the entire scene: the doctor burbled goodbye as they left.

That sounds like an interesting novel. Although I don't really get the point why one knows that the doctor is an alien because of the word "burbled". The reason for this is probably that I'm no native speaker of English. Because I only knew the word "burble" meaning something like "making a gurgling noise", I looked it up and it says it could also mean "to speak in a confused or silly way that is difficult to understand". So is the doctor an alien, because he speaks confused or silly? Or rather because the aliens have a strange kind of speech which sounds like gurgling? I would have rather concluded that he's an alien from the fact that it's so uncommon for a doctor to say "goodbye" to an unconscious patient.

It is with this in view, that I made my website. It has not been an easy task, and thus far I rely mainly on foreshortening as the method of 4D depth inference, but on the site you will see images that were made from various 4D->3D projections of 4D objects, and in my 4D visualization document (recommended starting point) I describe some of the 4D depth inference principles that I have discovered that may be applied to infer 4D depth from these images.

I already know your website and I think the document about 4D visualisation is the best that can be found on the web. I already read it, I think about two years ago, and then later I read some parts again. But maybe I should have a look at it again, because I forget so many things and now I know whom to ask when I have a question about it .
The images of the polychora are also very beautiful!

... but it has worked wonderfully for me, and I have been able to independently rediscover a number of 4D objects purely through "visualizing" them, which I later confirmed matched what mathematicians have proven about them.

That's great! I have no clue how one could do something like that.

I think I will still need a couple of years to be able to visualize even such a simple thing as a hypercube properly. And I even doubt, if that will work out. As a matter of fact, I have a pile of about 120 little wooden cubes lying on my living room table waiting to be painted in the manner C.H. Hinton describes in his book about the 4th dimension. But the cubes have already been there for half a year or so, since I always postpone the painting and reading of the book (of which some pages also still have to be printed from .pdf). I fear that it might not be successful and that's possibly the reason for procrastinating all the time:(.

[gonegahgah]

[...]
Similarly, if 3D atoms in our world are actually confined 4D spherinders, then what the 4Der sees can be dramatically different from what we imagine -- a beautiful scenery with green grass and blue snow-peaked mountains will appear nothing like that to the 4Der -- for all we know, the 4Der could be seeing purple grass and neon pink mountains with black streaks (iron ores buried inside the rocks, say) and holes (caves) filled with green liquid (actually water, except the transparent atoms in the water appear green on the 4D sides). For sure, what we imagine as our own handsome visage appears nothing like that to the 4Der -- who sees a thin layer of skin with muscles, veins, arteries, organs, half-digested food, circulating blood, etc., possibly in totally foreign colors from what we normally associate with these things.

Exactly. It would be like this if we had 4D beings. They would have a much better picture of us than even the best MRI could produce.

Continuing with my explanation in steps here is a picture showing the view on the back of the 2Der and 3Der's eyes.

Step 2. The 2Der's and 3Der's retinal image of the square:



You can see the 2Der's retina consists of a line whereas a 3Der's retina consists of something like an array of red, blue, and green cones (with rods as well but not shown).
In my picture our 2Der can only see red and green and whatever colours they combine to; whereas the 3Der can see red, green and blue and whatever colours they combine to.

Pretty much as you were saying earlier in this thread I think QuickFur.

[gonegahgah]

It is possible in our 3D world to project an image of a, say triangle, from a slide projector onto a screen. We can see the triangle on the screen. The screen has to have some thickness, but the triangle projected onto the screen is not a part of the screen itself. I suppose the triangle is only made of photons. Since photons have zero mass, they probably have no thickness either. That would be an example for a 2D Polygon in our 3D space, wouldn't it?

I personally wouldn't agree to any real particle being a point but that is not the discussion here. To be completely accurate there is no such thing as a true flat surface. Even then light interacts with the surface in different ways depending upon the light's orientation and the state and position of the atoms and their sub-parts in the molecules that they meet. Light will also be reflected from different levels within the surface. Colour is primarily about reflection and re-emission less absorption but other things can affect it slightly such as direction change or scattering. But, I do really like your example. It is about as close to true 2D that we will ever get. Good one!

I think the possibility that different spaces are sort of "overlapping" each other with some tiny parts into the next higher dimension quite possible.
And if this was not the case, all those analogies with creatures of 2D wouldn't properly work, I assume.
I had this idea today, that for a 2D creature to see anything at all of its 2D world, it must have an eye (retina or whatever) that is positioned at an angle to the edge of those supposed polygons (which it perceives as a line or 'wall'). It wouldn't matter if that angle was 90 degrees or 40 degrees or whatever. But there must be an angle between the thing to be perceived (say the edge of a square) and the retina of the 2D creature. If the eye lay on the same plane (without any angle) than nothing would be perceived. So if that angle is necessary, there must be some kind of rotation to get that angle. But this kind of rotation would ever only be possible if there was a bit of 3D space to rotate into. (Hmm, does this make any sense, what I wrote? I’m not sure. It seems to, but maybe only for the moment until one sees things differently again.)

The 2Der certainly has to look at their square from somewhere but I believe that is from within their plane at any angle inside the plane. I'm getting the impression from your words that you think that the 3D world needs to somehow come into play for the 2Der to see? If that is what you mean; I'm not quite seeing your explanation for why this should be so? If you don't mind could you clarify this for me?

[4Dspace]

[...]
The discussion in this forum area is 'what if we had 4 spatial dimensions instead of 3' not 'how does string theory and 4-space mix'?

String theory and 4-space are two completely different topics and should not be jumbled together as proof of anything.
Discussions about fields - which aren't spatial - to somehow refute the need for 4D atoms is not relevant here.

Thank you, gonegahgah, I think you've hit the mark on the source of the miscommunication / misunderstandings between 4DSpace and myself. I'm thinking about spatial dimensions, whereas 4DSpace is thinking about string theory. No wonder we could never seem to agree on anything.

Miscommunication? You, quicifur, spent years persisting in your error, where you confused the 3 directions in 3D with one direction of a POV in 4D and called it "a 4Der vision". I spent quite an amount of my time and energy on getting you to see your error, not to mention all the emotional distress that inevitably comes with any conflict, and all you did in gratitude was to merely to stop have-a-nice-day me. You did not even say Ah! Thank you, now I see where I was wrong! Why, you can't be wrong by definition. You merely stopped hissing.

And stop putting stupid words into my mouth, both you and gongahgah. I never said what you are ascribing to me, don't misplace your personal misunderstandings of both topology and physics to me. You can continue on your inept discussions about mythical beings in mythical spaces, unaware yourselves that you live in 4D world, just like ancient people could not believe that Earth was round and hurling through space at mindboggling speed. You are the Flatlanders.

So long!

[gonegahgah]

Step 3. Our 2Der imagining what the 3Der sees:



For the 2Der, from their limited perspective, they will imagine that we must see the insides of the square as a front column of green pixels, followed by a second column of mainly red pixels, followed by further columns of mainly red pixels, finally finished by a column of green pixels. And the poor 2Der questions how we can possibly see things that are behind each other. It is hard for them to wrap their mind around there being anywhere else for the pixels to be.

They assume that we can see the inside; as we assume a 4Der sees the inside of our 3D objects. So they may ponder if we look at the square from the inside somewhere and see the green walls from the inside admist a '2-volume' of red somehow looking in every direction at once, or if we look at the square from their outside and look into it seeing through two green front edges from that outside, through the red '2-volume' and then see the rear walls as two inside green edges (just as 4DSpace is describing for a 4D viewer of our cube).

We know that that is not true and that we are actually perpendicular to their plane. To us the pixels are all side by side; but it is hard for the 2Der to understand this as they can only mentally picture them as columns of pixels one by one behind each other. They may think that we are somewhere inside the square - as the first diagram depicted - which we are not really; as we know. Otherwise they might picture it as 4DSpace was describing; except for a 3D viewer of their 2D square; instead of the 4D viewer of our 3D cube referred to by 4DSpace.

[quickfur]

[...]
The discussion in this forum area is 'what if we had 4 spatial dimensions instead of 3' not 'how does string theory and 4-space mix'?

String theory and 4-space are two completely different topics and should not be jumbled together as proof of anything.
Discussions about fields - which aren't spatial - to somehow refute the need for 4D atoms is not relevant here.

Thank you, gonegahgah, I think you've hit the mark on the source of the miscommunication / misunderstandings between 4DSpace and myself. I'm thinking about spatial dimensions, whereas 4DSpace is thinking about string theory. No wonder we could never seem to agree on anything.

Miscommunication? You, quicifur, spent years persisting in your error, where you confused the 3 directions in 3D with one direction of a POV in 4D and called it "a 4Der vision". I spent quite an amount of my time and energy on getting you to see your error, not to mention all the emotional distress that inevitably comes with any conflict, and all you did in gratitude was to merely to stop have-a-nice-day me. You did not even say Ah! Thank you, now I see where I was wrong! Why, you can't be wrong by definition. You merely stopped hissing.

And stop putting stupid words into my mouth, both you and gongahgah. I never said what you are ascribing to me, don't misplace your personal misunderstandings of both topology and physics to me. You can continue on your inept discussions about mythical beings in mythical spaces, unaware yourselves that you live in 4D world, just like ancient people could not believe that Earth was round and hurling through space at mindboggling speed. You are the Flatlanders.

So long!

You see, this is exactly what I meant when I said your definition of geometry is not the same as the one that we're talking about in this forum. You have a particular definition of "4D world", and it's not the same definition as the one we're using. This is the source of the miscommunications. From your definition of "4D world" you draw certain conclusions that don't match ours, not because either of us is "wrong", but because we're starting from different definitions!

We keep getting into conflicts because you insist that your definition is the "correct" one. But you see, whether your definition is the "correct" one is kinda beside the point, because the whole premise of this forum is "what if space has 4 spatial dimensions?" -- by which we mean that the 4 mutually perpendicular directions that are completely interchangeable and equivalent, unlike the situation in physics where 3 of the directions are special (because we live in the space of these 3 directions and perceive all things from that perspective). So the very basis of this forum is the definition that you regard as "incorrect" -- is it any surprise then that we can never agree on anything?

And since the whole purpose of this forum is to discuss the definition of "4D world" that is different from your definition, is it any surprise that some of us are unhappy with the way you barge in here and tell us that we're all wrong? Garrett Jones, the original maintainer of this forum, very specifically stated that the purpose of this forum was not to talk about (3+1)D spacetime, or the physics version of 4-space, etc., but to speculate on what the world would look like in the hypothetical situation where we have an extra dimension of space where we can move around in freely. This being the case, wouldn't you regard it as rude if someone came in here and demanded that we all discuss another definition of "4D world"? That is what you have done, yet now you blame us for being annoyed.

As you see, gonegahgah has identified the problem here: what you have in mind is to discuss how how physics relates to 4-dimensional space, but this is the wrong place for that discussion, because the topic of discussion here isn't physics but Garrett Jones' hypothetical scenario as I described above. (And lest you think I am ascribing to him what he did not say, you can read it for yourself here.) Your point of view would be much better received in the "string theory" section of this board, which was created precisely for this reason, that sometimes people show up here wanting to discuss how physics relates to higher dimensions, but cause our discussions of Garrett Jones' hypothetical scenario to be derailed because the two simply aren't the same subject at all. That section of this board dedicated to physics-related discussions is where gonegahgah is trying to point you to, but you apparently take offense to the fact that our topic of discussion here happens to not match what you want to discuss.

And this isn't the first time you take offense to views contrary to your own, either. From the time I started replying to your posts, you have been offended every time I suggested something that contradicts your definition of "4D world". You appear to believe that the only possible definition of "4D world" is the one you have, and you took offense when I suggested that perhaps our definitions are not the same. The way you rail against "the physicists" shows that you are unable to accept that others may not be using the same definitions as you are. It seems that the very fact we're talking about another definition of "4D world" offends you so much that you either have to convert us to your point of view or dismiss us as hopelessly misled or just plain stupid.

Here's my suggestion. Take a deep breath and relax, and don't get so worked up about converting everyone to your viewpoint. Yelling at people generally will not convince them about anything. Barging into a discussion about a different subject from what you want to discuss and insisting they change their views usually doesn't work very well either. People will be more willing to listen to you if you would listen to them and understand where they're coming from. It doesn't mean you have to give up what you believe is true. If you explain yourself clearly, in the right place for that discussion, without imposing your views on others, you are much more likely to get a sympathetic audience.

[4Dspace]

Again you attempt to misrepresent the situation here. It is you who until couple of days ago did not know how a POV relates to what is seen in what D, nor how it differs from directions in a given N-space.

This fact --i.e. that you did not know, nor understood it-- is clear to anyone with a functioning brain who reads this thread and many others here. For years, you have been giving people WRONG information. Not wrong simply "according to me" as you are trying to misrepresent it now, but simply wrong.

I could have let you continue to BS to people for another number of years, perhaps to the end of your life. But I actually showed you where and how you were wrong, and what's the difference between a POV and a direction in N-space, the simple things which you obviously did not understand. So have decency to at least stop trying to continue to misrepresent what's going on. People are not stupid.

And I asked you twice already to stop putting stupid words to my mouth and paint me as dense as you paint your "2Ders".

Rather than keeping spreading them around, kindly keep your personal misunderstandings to yourself. And stop misrepresenting what's going on.

[gonegahgah]

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

From the tone of this forum, ever since I've joined, I would not doubt that it was the original builder's intent to mainly focus on a free form of 4D-space; rather than a real 4D-space that somehow logically constrains us to just 3 of its dimensions - just as QuickFur says. So, from our model what happens in one dimensional depth is a clue to what happens at higher dimensions which is why we keep modelling similarities from one depth to a deeper depth. We are not trying to match our models to real world phenomena.

I am a little familiar with Aspergers having two foster sisters who's children all have the condition (4 kids for the oldest and 1 for the youngest). The oldest of the children, Gus, has 'high performing aspergers' and it is obvious that you are similarly gifted. I don't think that QuickFur, nor I, have ever attacked you personally. You seem to think we are, I'm getting that vibe, because you are attributing characteristics to both of us that are just in neither of our personalities. We are both generally as a rule friendly, helpful and - both of us would like to think - fairly intelligent people. Otherwise I don't think either of us would bother. I know for myself that I can readily and openly say that I'm wrong when I realise I am and I have seen QuickFur do the same. Being wrong is not an error in character; it is simply a learning moment and is also human. But, QuickFur has been doing this a long time and I think both of us can sift, compare and visualise and both of us do try to actively listen.

So I remain convinced that you're not talking about a free form 4D space. You seem to have rejected 2D creatures as not worth talking about. I agree they are fictitious. However I would still use them as an example of a free form 2D-universe to help visualise what I also regard as a ficticious free form 4D-universe. Whether others here regard a free form 4D universe as ficticious or not it doesn't stop us working together because our models are similar enough and allow for free logical translation between the various dimensional depths.

If you want to target something, it shouldn't be us personally as people. Instead it should be our conviction that you are not working on a free form 4-space; one where the lower dimensions seamlessly step up geometrically to the higher dimensions in a consistently similar manner. For us no new rules are added by the extra dimensions except for the greater freedom of movement and placement that is afforded and what this would imply. For us 2D and 1D people are perfectly acceptable even though they could never hope to actually function.

But as we have both mentioned, there is another place in this forum to talk about a 'real' physical phenomena based 4D space if that is what it is that you are wanting to talk about.

[ac2000]

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

Thank you .

The 2Der certainly has to look at their square from somewhere but I believe that is from within their plane at any angle inside the plane. I'm getting the impression from your words that you think that the 3D world needs to somehow come into play for the 2Der to see?

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

[Hugh]

I hope there will be a sequel to the nice drawings with 4D examples ? Maybe then I can understand better what you wrote about the 4Ders that can't see some edges of a cube (and/or a hypercube?) and how this goes together with quickfurs model of 4D seeing, where they can see every little bit at once. Somehow it sounds both true but somehow also a little contradictory, in any case, quite difficult to comprehend.

This has been a fascinating thread to read through. I am looking forward to see how it develops.

My own feeling is that a 2D being couldn't actually see anything around itself, or even itself at all, because it is looking at everything edge on, along an infinitely thin plane that has zero thickness... So it sees Zero dimensions, 2 less than what it is.

We are supposed to be only one dimension up from that, 3D, well at least that's what almost everyone believes, yet we see a full 3D world all around us.

From what I understand, a 4D being would only see a 3D "slice" of the 4D world around itself. It wouln't actually see in 4D.

Aale de Winkel made an interesting point in the "Flaw in Flatland" thread from this Tetraspace forum on Dec. 4, 2003 when he said: "I doubt very much that tetra-vision would be the same as x-ray-vision.
Tetronians will not be able to see within a trionian body, they see the lightrays reflecting of a body just in a direction more then we trionians do!"

Food for thought.

[gonegahgah]

Hi Hugh, what is your particular interest in 4D?

[Hugh]

Hi Hugh, what is your particular interest in 4D?

Hi gonegahgah. I'm particularly interested in how a 4D being would see itself, and its 4D world around itself.

[gonegahgah]

Hi Hugh. I mean how did you come to be thinking about 4D; or 4D creatures?

[Hugh]

Hi Hugh. I mean how did you come to be thinking about 4D; or 4D creatures?

I experience Visual Reorientation Illusions (VRIs), and the way that they allow people to see their perceived 3D surrounding environment from different directions led me to think that it could possibly be related to the experience a 4D being might have of its perceived 3D world around itself, being able to see the same view from different orthogonal directions.

If you're interested in VRIs, we can discuss them in this thread: http://teamikaria.com/hddb/forum/viewtopic.php?t=401.

As far as this thread goes, i'd be interested in hearing how you think a 4D being would see itself and it's surroundings.