Dimensional Discourse

Time for another (hopefully) non-controversial thread. By "non-controversial" I just mean that I don't want to incite any arguments if I can help it. I know this is a debate forum, and that's cool, but as was the case with my first thread, my purpose here is not to argue, it's just to learn.Incidentally, my first thread, Relative Motion (A Thought Experiment), is of course still available for anyone to reply to, if they wish. It kind of died about half way through, I'm afraid. This was partly my own fault for getting so far off topic. Still, if anyone thinks they can help me with it, by all means, feel free. Also, I may still be uncertain about the overall scenario but I am a little clearer on certain specifics so thank you to those who helped me out, it is greatly appreciated.Anyway...Of all the concepts I've pondered over the years, the one that stands out in my mind as having fascinated me the most is that of higher dimensions. Why is this? I honestly don't have a clue. Perhaps it's because the basic principles of it are so unimaginably simple, yet the consequences of it are so paralyzingly inconceivable. Who knows?Whatever the reason, I spent years trying to grasp the reality of higher dimensions. Not to merely "understand" them in terms of extrapolated principles but to actually comprehend their true nature; to visualize four dimensional space and geometrics (naturally I am in no way averse to visualizing indefinitely more than four dimensions but let's not get ahead of ourselves ).I long ago gave up on this pursuit, concluding that it was simply impossible. Then for a long time I didn't look into it very much, until my interest was resurrected by a comment from our resident mathman, Rrhain. I nearly fell out of my chair when I read that...Seriously!A brief exchange between us ensued in these three posts and Rrhain's replies. Unfortunately I am no closer to achieving my goal than I was before, but in fairness, I didn't really expect to be; I don't think this is something that can simply be taught, I think you have to "get it" by having a certain depth of understanding regarding the principles and probably the math, too.To that end, I thought it would be educational to open a thread on the subject. I don't really have any specific topic in mind, except the principles of dimensionality and concepts relating to higher dimensions in particular.I hope Admin is ok with this; the point of this thread is simply for my own (and anyone else's) education on the subject. I apologize for being so vague and not really making any opening statements or asking any specific questions; I was hoping to leave the floor somewhat open for others to comment on and discuss more specific points under the general topic, and just see where it goes.So, Admins permitting, I propose a generic dialogue on the topics of higher dimensions, dimensional constructs, properties of dimensionality, etc. If however we do need something to get us started, I might be inclined to ask if anyone else has ever been able to comprehend higher dimensions, and if so, how did you achieve this? What led to your understanding? Is there any way you can describe it (without using three dimensional analogues)? Do you think it requires a strong proficiency in the underlying math?Incidentally, just to be clear, I'm not so naive as to think that I will learn enough from this thread to start visualizing higher dimensions. I really should amend the third paragraph above this one, to read; "To that eventual end..."Indeed, if the mathematical fluency necessary to visualize higher dimensions is anything approaching Rrhain's then I will likely never achieve it. However it's a topic that I have always found absolutely mesmerizing regardless of my incomplete comprehension, and if I learn something new from this or gain even the smallest insight I didn't have before, then the thread has served its purpose.Ok, having finished my ramble...Dimensions...Discuss!

Thank you, Ned.So would anyone like to contribute? I realize the opening post was somewhat ambiguous but I didn't really know what to narrow it down to. If I were to simply ask how to visualize higher dimensions, I think that would end the conversation pretty quickly.That's why I gave the thread such a wide scope; I just want to learn more about the general subject. I know I won't be able to comprehend 4D space based on the small amount of tuition this thread will give me, but every little bit counts.Also, don't assume that you have to be knowledgeable about this to post; feel free to ask your own questions if you have any. Would anyone like to ask me anything? I can't guarantee I'll be able to answer (I am by no means an expert; my knowledge of this subject is all "self-taught", so to speak) but I'll see what I can do. And if I can't give you an answer, I'm sure there will be others capable of doing so. In any case, it's a way to kick off the discussion.So...Questions? Comments? Anything? Anyone?Feel free!

Yup, that's about right!Actually it's up and down, left and right, forwards and backwards, ana and kata (although, you'll have to forgive me, I don't remember who coined those terms). Yes, Rrhain suggested something like this to me (temperature gradient, as I recall), but as you said, it is not a true spatial dimension. It still requires what I believe Rrhain referred to as a "visual leap". Ack! How do you make that leap? Tell me about it! Did you read Rrhain's quote where he said that when he really concentrates, he can even catch glimpses of five dimensions? At the risk of sounding unscientific, how bloody cool is that? Yes, so I understand. They are there but curled up so tightly that they're too small to detect, at least with current technology. Actually that may be a good question to discuss as part of this thread; can we ever detect such "micro-dimensions"?Mind you, I don't think there's any reason (in principle) that more dimensions couldn't exist on a macroscopic scale. It's just that it's something we've never seen, so visualizing it is tricky (to say the least).I think the problem is that our minds are trained to think in three dimensions and we can't help but do the same when trying to imagine 4D (or higher) concepts. So it would seem that if we're ever to comprehend them we need to change the way we think about them. How we do that, though, I have no idea.It's frustrating that I always find myself falling into this trap. No matter how hard I try, I can't seem to get away from visualizing 3D analogues. It's all my mind knows, it's all the world has taught me.I'll attempt to visualize the fourth axis and then realize that I'm trying to align it, perpendicular to the other three, in three dimensions of space. Argh! That won't work!The problem here is that I'm trying to visualize the fourth axis so that I can find the fourth dimension, but to place it correctly I first have to visualize the fourth dimension; it's a vicious circle.Anyway, I'm getting a little carried away now. [ / rant mode ] Heh, are you kidding? Thank you just for replying! At least my other thread went a few pages before throwing it in; I thought this one was going to die after the first post.To be honest, I was quite surprised when I came back and there were no replies; I find this subject virtually off the scale as far as fascinating concepts go. Perhaps I'm the only nut who thinks so. I haven't read that one, but I know of it. I'll have to look for it some time.I can't say I'm very familiar with the author but I've read quite a few others; Stephen Hawking, Richard Dawkins, a whole bunch of Paul Davies. And I just bought Carl Sagan's Cosmos...Woohoo!

Ok here are a couple of questions to (hopefully) give things a bit of a push along.The first one hit me during my last post. It's open to everyone, of course, but it might be interesting to hear from someone in the appropriate fields; particle physics or quantum theory, perhaps?As far as I'm aware, we currently have no way to directly detect the "extra" dimensions of our universe (the non-macroscopic dimensions). They are curled up so tightly that they're smaller than the smallest elementary particles. So is there any way (even in principle) that we could ever detect them?The clincher for me (although I may be misinformed about this) is that I understand they are believed to be on the scale of, or close to, Planck length. If this is true, doesn't it somewhat seal our fate? Considering we have nothing on that scale to work with, how would we go about "looking" for them?Or am I on the wrong track? Is there another way we could test for their presence? Perhaps we could do away with matter altogether and "scan" for them with fields somehow? Still, even then, I can't see how we could get a meaningful result at that level. Mind you, we're starting to get into the realm of quantum physics and I'm not as well versed in that field. Hell, is anybody well versed in it?The second topic regards one of the more "exotic" ideas I've read about; the concept of a space-time universe with more than one temporal dimension.Ok, obviously we aren't talking about our universe here but it's an interesting concept, none the less.To keep things "relatively" simple, let's say that this universe is the same as ours, in spatial dimensions. However, whereas our universe is four dimensional space-time with three of space and one of time, this universe will be five dimensional space-time with three of space and two of time.Well, I guess my first question has to be; is this even possible? Or does the nature of time simply forbid such a combination?If it is possible, what would this mean? What kinds of properties would its geometry have? What would it be like for beings living in this universe? Could anything even exist in such a realm? Or would the entire concept of cause and effect collapse into chaos?This is another one I'm having a hard time picturing. Anybody think they can clarify it a little for me? Or is this concept simply not possible?Incidentally, the original question about visualizing dimensions still stands; if anyone can give me any pointers, I'll be forever in your debt (and I mean that literally because I can't pay you anything ).Ack! I need to sleep. I hope I was coherent enough. I have a hard enough time collecting my thoughts on these topics when I'm not tired.P.S. I've passed 100 posts! What do I win?

Rrhain, just the mathematical dynamo I was hoping to hear from! Damn! Are you sure you're not psychic? I read this exact same analogy, like, two days ago! Heh, spooky. But yes, I understand the principle.One thing I'm not clear on, though, is whether or not the "wire analogy" is supposed to actually represent the reality of the situation or if it is merely a lower dimensional analogue, such as the three dimensional analogues of 4D primitives.My understanding is that superstrings are only one dimensional anyway. So how does this actually constitute "extra" dimensions? Even if we allow the "clockwise/anti-clockwise" axis to count as a dimension, doesn't that just give us a two dimensional object in three dimensional space?As I understand it, one of the properties of dimensions is that an object with dimension x cannot be contained by a space of any dimension less than x. Hence my confusion; if these "curled up" dimensions contain more than three spatial dimensions, they cannot be all there (in our space), can they? Most of their volume (hyper-volume?) must be poking out of our three dimensions along their other axes, right? Or are they so tiny that they literally squeeze all of their "extra" dimensions "flat"?Even in this case, I'm not sure I see the distinction. If I take a cube and flatten it into a square, could the inhabitants of Flatland really say that it was three dimensional on the microscopic level? Haven't I simply turned it into a two dimensional object?If 3D space truly has zero size along the fourth axis, the only way these other dimensions could be contained within it would be if they too were squashed down to zero size. That's fair enough but if they have zero size, how can they be a dimension? Isn't zero size dimensionless, by definition?Or perhaps our space isn't really three dimensional at all; perhaps our space is actually three and a bit dimensional (that is to say, three and the teeniest, tiniest bit dimensional).If it has the most minuscule (yet non-zero) size along the fourth axis then I could understand how these can be higher dimensions curled up on the smallest conceivable scale. But if it truly is without any size at all along the fourth axis, I really can't see how these dimensions, however small, can "fit" in our space. Unless they are completely flat, in which case, aren't they simply three dimensional?Hmm...Looking back, I'm not too sure how that came across. Just to be clear, I'm not arguing with you; I'm just asking questions. I know what you're telling me will be correct and I'm just explaining what I'm having trouble understanding. Just in case I seemed like I was being confrontational; I wasn't. Yes, this is one of my problems, no doubt.Rrhain if you don't mind telling me (because I am very curious about this), just how well can you (could you) visualize 4D? I understand that there's no way you can describe what you see but can you perhaps give me some very general examples?For instance, you mention thinking in a linear way; can you actually picture four perpendicular axes? Can you rotate a 3D object in 4D such that you can see all of its points simultaneously?Or 4D primitives? Can you visualize a tesseract? Not the 3D analogue but the real thing? Can you see its true form? To use another analogy, Flatland's resident mathematician has such a grasp on the properties of the "hyper-square" that he is actually able to picture, in his mind, a true-to-life, three dimensional "cube". Can you make this kind of leap from a hypercube to a true tesseract?Ugh! I'll leave it at that for now. I greatly appreciate your reply; you're the person I really wanted to hear from in this topic. Thank you!And I apologize for rambling so much; this is just something I love. You really picked the wrong guy to reveal your special skill in front of, didn't you? I think you've awoken a sleeping giant.

Hmm...My thread isn't that boring, is it? My apologies guys, I really thought this topic would generate more interest. It fascinates me almost to the point of obsession; I guess I just assumed it would be popular with others, too.Well, a little bump to see if anyone's still out there. Also, I tend to ramble a bit between applicable points, so here's a brief summation of my main questions, so far.1. Does anyone have any suggestions on visualizing/perceiving 4D, or any of its properties, beyond mere 3D analogues?2. Is there any way (at least, in principle) that we could test for, and detect, the "extra" dimensions of our universe, curled up at the smallest level?3. Is the concept of a spacetime (not ours; just hypothetically) with more than one temporal dimension possible? If so, what would be the properties of such a universe? What would life there be like? Could anything even exist in it or would causality fragment into chaos?If any of these interest you, or something else under the general topic that I haven't mentioned, please feel free!

Heh, it sure is! I've been looking around the internet and have actually come across a few others who say they can do it. The impression that I get from them, generally, is that you have to virtually retrain your mind in how to think about things.Since our sensory input is entirely in three dimensions from the time we're born, that is how our minds are trained to codify everything. This can take quite some undoing, apparently. I've heard that even excluding learning the math and physical concepts, it can take a lot of time and patience just to get your mind thinking the right way. I don't doubt it. Yes, the problem is that the models can only show us three dimensional "slices" of 4D objects. As I understand it, if a true 4D space/object is ever to be seen, the only place it can happen is in the mind; there is simply no way to do it in our spatial dimensions.It would be like someone in Flatland trying to build a model of an actual cube; he couldn't do it because there is simply no "space" to do it in. The only place he could ever conceivably "see" a true cube is in his mind. Well, thanks but it was Rrhain that brought it up. Hmm...I'm afraid you lost me here. Could you elaborate on this? In particular, what do you mean by "the 2D present and horizon then the 3D around?" Really? None at all? Bummer. Is there no way to even test the mathematical models to see if they're accurate?I guess the question at this point is; are there any qualitative differences between what we should see if our model of tightly curled up dimensions is correct and what we should see if it's incorrect (even if present technology doesn't have the means to test for those differences)? Heh, good answer. Neither do I.I think this question, in particular, really delves into the realm of speculation, as it's a concept that we have not only never experienced, but quite possibly, never could experience. As beings whose psyche is so intimately linked to a single, mono-directional time, I have my doubts as to whether or not we would have the capacity to make any sense of a "multi-temporal" universe, even if we were thrown into one.That's not to suggest, of course, that we can't create models to determine the properties it ought to have, but as far as experiencing it, I really don't know if we could.At any rate, does anyone want to take a shot at it? Any ideas on what a universe with multiple dimensions of time would be like?

Hi Brad. I apologize for taking so long to reply to this.Well, as so often seems to happen, I'm afraid that much of what you said went right over my head again. I think I managed to follow some of it, but man, that was an effort. You obviously go to a lot of trouble and I apologize for getting so lost.Perhaps we can back right up and start with my original question, as it applies to you. Are you able to visualize, in your mind, more than three spatial dimensions? Can you visualize 4D primitives or "surfaces"? Can you mentally "rotate" a 3D object in 4D space such that all of its points are visible simultaneously? Can you "see" four perpendicular axes in your mind?These are some of the things I would eventually like to be able to do. It's not going to happen any time soon...it may never happen at all...but this is the direction I'm trying to head in. I just wanted to explain this so that you know where I'm coming from.I was going to type more but I think I'm starting to lose coherence, myself. I've been awake for around 30 hours now and I can feel myself struggling to maintain my own clarity. Ugh! Why am I doing this when I should be sleeping?Well, hopefully this will suffice for now. Sleepy time.

Ok, here's a little more. Sorry I didn't get to it last time. Incidentally, these (and the above questions) are for anyone who is interested, not just Brad.I think I understand where the wire analogy is coming from but I'm not sure I can see how to apply it to the next level. This may not be quite what Rrhain was getting at but I had a thought the other day which I'll try to explain. This is hard to put into words so please bear with me.Each time you go up a dimension, the properties of spaces and bodies are altered accordingly. So with that in mind, let's take a look at how boundaries are affected by changes in dimension.As I understand it, one of the properties of dimension is that x-dimensional space is bounded by (x - 1) dimensions and its boundaries are connected by (x - 2) dimensions. In other words, a square (x = 2) is bounded by lines (x - 1 = 1) which are connected to each other by points (x - 2 = 0). A cube (x = 3) is bounded by squares (x - 1 = 2) which are connected to each other by lines (x - 2 = 1). Finally, a tesseract (x = 4) is bounded by cubes (x - 1 = 3) which are connected to each other by squares (x - 2 = 2).Now, you can unfold the six segments of a three-dimensional cube into a two-dimensional plane. To remake the cube you must fold all but one of its boundaries (squares) up and out of the plane it sits in. The edges (lines) of the connected squares never stop touching each other, however they may be seen to rotate in the 2D universe as the squares are folded up and out of the plane. Of course, they rotate in a direction that beings in the 2D universe have no concept of.Rrhain, if you happen to read this, is this close enough to the wire analogy? Similar to the way an ant on an apparently one-dimensional wire may, at its level, have a concept of forwards and backwards, as well as clockwise and anti-clockwise...perhaps two-dimensional creatures, though unable to see a third macroscopic axis, could see the way the boundaries (lines) of a square, which itself is one of the boundaries of an unfolded cube, rotate as the cube is folded up into three dimensions. It would rotate in a direction that doesn't exist (at least, on a large scale) within their universe, but they could only observe this at the lowest levels. Am I on the right track here?If this is correct so far, could we do the same thing (at least, in principle)? A tesseract is bounded by cubes which connect to each other at their own boundaries (squares). When an unfolded tesseract gets folded up, those bounding squares stay in complete contact with each other despite the cubes being rotated "around" them. In essence, in four dimensions of space, you can rotate a cube "around" one of its faces without the face itself moving, just as you can rotate a square around one of its edges without the edge itself moving.Obviously, we can't see the "direction" of this rotation at our everyday levels. However, if we could go low enough, could we conceivably see which "direction" the cubes move in, by observing the tiniest scale "rotation" at the squares connecting the bounding cubes of a tesseract as they get folded "up" and out of our three-dimensional space?

Hi sidelined. You relocated it? I'm not sure what you mean. You mean you moved it there? Or are you just referring to the piece on time? Sorry about my confusion...I have a bad habit of reading these things when I'm way too tired to be doing so.In any case, thanks for the link. I read the bit about time and had a look around at some of the other stuff as well. Ah yes, I've been there a few times. Thanks for the link, though. Yep, this is my ultimate goal. The problem with that, I think, is that you really can't visualize space without time. At least, I don't see how. Even if you visualize an unchanging void, unless you can picture it for an infinitely short length of time, there will always be duration. None, as far as I've been able to tell. Unless of course we introduce another temporal dimension to take its place. As I said, I don't think we can visualize space alone. As time is such an integral part of our existence, our minds will tend to impose it on anything we imagine, so technically, what we visualize will always be some variety of space-time, and as we are already quite familiar with four-dimensional space-time, we will be looking at the next step.I generally just ignore time for simplicity's sake. In actuality, when I say that I want to comprehend "four-dimensional space" what I mean is, of course, by necessity, "five-dimensional space-time."I've actually encountered confusion on this before. Just so that what I'm talking about is clear to everyone reading this, I'll paste an explanation from another thread in which I think I described it better.I said... So in short, I'm aware of the relevance of time in any discussion on dimensions, but I just take it for granted that whatever the dimensionality of a space/body I am visualizing, I will always be picturing it with duration. I can't think of any other way I could do so.I hope none of this seemed patronizing, by the way. The reason I elaborated so much is just to be clear about what I mean. In my searches for more info on the internet, I've come across a number of people asking similar questions to mine, wanting to understand the nature of higher spatial dimensions, and you'd be surprised how many people simply reply with, "The fourth dimension is time." Well, I certainly hope that doesn't happen.But yes, it really does contort your mind, trying to comprehend this. It's extremely frustrating, too. I know what I have to do (in theory), I just can't make it happen in my head.

Hi Rrhain.Yes, I understand what you're saying. I've played with a heap of four-dimensional applets online. Some are better than others, but perspective is a constant problem.Some get around this to a point by have a double image which you focus on cross-eyed, some use different colours to indicate what is "near" and what is "far," I even recall trying something which showed two real-time 3D graphs simultaneously, allowing you to see the motion of plotted points in all four dimensions at the same time (but which obviously couldn't be shown in the same graph, all at once).In any case, please feel free to comment on anything here, Rrhain. I greatly value your input on this.

This is a very good question. I've long wondered what we, as three-dimensional creatures, would "see" if we were to be thrown into four-dimensional space. Here's what I think.I doubt that we could actually see all four dimensions simultaneously even if we were suddenly cast into 4D space. Our eyes relay visual data to our brain when light is cast onto the two-dimensional surface of our retina. As such, we can see all the points of a 2D image simultaneously (by looking down on its cross-section), but not 3D objects (we have to rotate a cube, for example, to see the other side).Now, look at it from the perspective of a flatlander. His visual data would come from the one-dimensional image cast on his "retina." By its nature, the only way his retina could transfer data to his brain would be as a one-dimensional image. I would also imagine (though I'm not certain about this) that his eyes, being infinitely flat, could only collect light travelling directly into them from whatever 2D plane they happen to be aligned with.If he were to suddenly be lifted out of his universe into the third dimension, would he then be able to see 3D space/bodies as we do? I seriously doubt it. Even if you were to show him a three-dimensional object, say a sphere, I really don't think he would see it as we do. I believe the nature of his eyes would still only allow him to see it in two-dimensional "slices."If, for example, you were to move him such that his view started at the top of the sphere and scanned down to the bottom, it could still only cast a 1D image on his retina, and as such, his brain would interpret the data as an object appearing from a point, growing to a maximum size, then shrinking back to a point and vanishing...essentially the same thing he "sees" when he witnesses a sphere passing through his 2D universe.So, taking this to the next step, I believe that if we were to find ourselves in a realm with four spatial dimensions, we would probably only be able to see much the same as we see now when viewing 3D analogues of 4D figures. We would probably see all sorts of apparently solid objects appearing and vanishing, changing size, shape, colour, and so on.In other words, I don't believe it's simply that we don't have access to a 4D space to look at; I believe it's a fundamental limitation of our dimensionality. In order to see true 4D the way 4D beings would see it, we would need eyes with the same dimensional properties.A 4D being's eyes would have a retina with a three-dimensional "surface" upon which would be cast a three-dimensional image. Just as a 2D image on our retina is capable of showing us all the points of a circle simultaneously by looking down on its cross-section, the 3D image on the "surface" of a 4D being's retina would show him all the points of a sphere simultaneously.Damn, I ramble! The point of all this is that I don't think we could truly see a four-dimensional object, even if we had one shoved in our face (at least, not the way a 4D creature would see it).It seems to me that what we "see" is analogous to the dimensional "shadows" we talk about. A cube casts a two-dimensional "shadow" on 2D space, and similarly, a two-dimensional image on our retina. As such, a 2D creature with a 1D retina is simply ill-equipped to see a cube the way we can.It is conceivable, though, that with the proper understanding, perhaps...just perhaps, he could perceive its true form in his mind, which isn't bound by the physical limitations of his eyes. Indeed, according to Rrhain, it can be done. I realize, of course, that his mathematical fluency is "slightly" superior to mine, but if there's even the tiniest chance that I could ever understand this, I truly want to.Argh! Apologies for the ramble. I just can't help myself when it comes to this topic. Anyway, this is just my take on it. I am, of course, completely amenable to change if anything I've said here is in error. I'm interested in hearing other people's (especially Rrhain's) thoughts.

Hi Brad. Please feel free to take whatever time you need to prepare it. There is no rush. If you are referring to visualizing higher dimensions, don't worry, neither have I.Seriously, I don't imagine there are a great many people who can, but if it is indeed humanly possible, I have to try. It may end up being beyond me but it's too fascinating a concept for me not to at least try to understand it.I actually gave up trying years ago because everything I read basically said, "Don't bother...humans can't comprehend it." Then I read Rrhain's comment, which gave me a whole new lease on life regarding the subject. Since then, I've searched around for others who can do the same, and while they do appear to exist, they always seem to be either physics or math majors (or both), so I'm not sure how well that bodes for my chances. This is one of the many things about this subject that fascinates me. The notion that you can "rotate" a three-dimensional solid, such that all of its points are visible simultaneously, is merely the logical conclusion of adding a dimension but still...it just sends shivers up my spine.

Hi Brad. It's nice to know I'm not the only loony around here who finds this stuff interesting. Sometimes it does feel like I'm kind of talking to myself in this thread. Heh, just kidding.Ah but to "rotate" a cube (for example) such that its orientation presents me with its "flat" three-dimensional "surface"...grr...why can't I picture it?The nearest I can get to the concept is to imagine myself inside a cube, at its centre. A two-dimensional being, for example, could picture himself at the geometric centre of a square. Now, in some sense, he would have a view of the square somewhat analogous to what we do. When we view 2D shapes we aren't concerned with viewing their perimeters edge on, the vast majority of what we see is their "inside." The difference being, of course, that our 2D friend would have to rotate 360 to view all of the square's inside, while we could place it all in simultaneous view by pulling back and up along the third axis.Now, mathematically, there shouldn't be any difference between this and me sitting at the centre of a cube. The problem is that every direction I can visualize myself moving will take me out through the cube's surface.The "direction" I need to go will take me "away" from the cube (inside and outside) without ever passing me through its surface. As soon as I make the slightest move in that "direction" I will instantly be "outside" of the cube, yet simultaneously perpendicular to its inside. Ok, you've got me. I'm afraid I can't figure this one out. Rrhain suggested that I try thinking a little less linearly and more in terms of rotation. I think that's a good idea but I still can't really resolve the rotations, in my mind. Oh I can do it with the familiar three, but no more.Actually, when discussing dimensions, rotational axes are an interesting concept, in their own right. I always assumed that they were simply equal to the number of dimensions. Indeed, in our three-dimensional space there are three rotational axes. But in two-dimensional space there is only one. In one-dimensional space there are none.I don't know the mathematical theory so perhaps someone who does can fill me in on this...but if there are less rotational axes than dimensional axes in lower dimensions (than ours), could there be more rotational axes than dimensional axes in higher dimensions? Did that make sense?For example, could four-dimensional space actually house five rotational axes? It seems to me that "rotation" doesn't have any real meaning until you reach two dimensions in which there is one rotational axis. Then when you step up a dimension that number leaps to three. So what happens when you step up another dimension?

Hi Brad. I'm not familiar with Dawkin's and Gould's discussion on the Necker cube but I've always assumed that it should theoretically apply to any number of dimensions. Of course, this makes things even more complicated than they were before. I can't even conceive what a 4D object looks like, if we add the concept of four-dimensional perspective to the mix, I'm even more confused.I've long wondered how to get around this, and it occurs to me that I may be able to break it down a bit further.Rather than trying to visualize the complete structure of a 4D object, perhaps I should try to visualize a 3D object from a four-space perspective. Much like I discussed last time, if I could learn to "rotate" a familiar solid around its fourth axis, in my mind, perhaps I could get a handle on three-dimensional "surfaces"...this would be a very good start. In fact, as four-space is bounded by these "surfaces," understanding them would seem to be a necessary step in understanding four-space itself. Whew! I'm trying to stay with you.I think that, in principle, we (humans) should be able to perform any calculation a computer can. It's just that a computer can perform far more, much faster than we can. Mathematically speaking, there is no difference between solving an equation with a pen and paper, and doing it on a computer. But it's just not practical (or necessary) to perform enormous and complex calculations this way.I've come across some impressive mathematicians and I'm sure that, given enough time, they could perform the same complex calculations a computer does. As far as I'm aware, computers follow all the same mathematical rules that we do; they just follow them more quickly.Of course, this is based on my limited understanding of mathematics (and computers). Rrhain, Lam, or anyone else, please feel free to correct me, if I'm wrong.And that's also assuming I understood what you meant. I'm not sure if I followed your train of thought, but I hope I did. Do I understand you correctly, that you're actually creating a 4D program, of some sort? Or are you just speaking hypothetically? Just so that you know, I've used many interactive 4D applets online and spent quite some time trying to acquaint myself with the physical constructs.I don't wish to stop you, though. If you are doing something along these lines, I would love to see it. But again, I'm not sure I read you correctly. *head explodes*Heh, but seriously, thanks for the links.Once again, I somewhat dropped by the wayside, but let's see if I'm anywhere in the ballpark.You mentioned "linking database tables"...I may be way off, but I once saw a representation of 4D which used two separate graphs, each showing a different combination of three of the four dimensional axes. That is, between the two graphs, all four axes were shown. I don't recall precisely but for clarity's sake, let's say graph 1 displayed axes w, x and y, and graph 2 displayed axes x, y and z. Does that make sense?Now, by manipulating these, you could see how plotted points move, relative to each other in all four dimensions, simultaneously. Unfortunately, it is still impossible to combine the graphs in three dimensions without "distorting" the image to make it "fit" three dimensions. You could see the points' actual relative positions using both graphs, but only their apparent relative positions with a single graph. As far as I can tell, true 4D perspective simply can't be displayed in 3D; only its "shadow."Incidentally, I don't think Rrhain was saying that it is necessary to think in terms of rotation in order to comprehend higher dimensions, just that it is one method of going about it. But I don't want to put words in his mouth (if I misunderstood you, Rrhain, by all means, please correct me). I'm not familiar with higher order catastrophe sets, but just to be clear, my reasoning was based entirely on an extension of what is known about the lower dimensions; if 2D has one rotational axis, and 3D has three, might 4D have five? So, there was no real mathematical theory behind my idea; it was pure speculation on my part. My logic may be completely flawed.I just wanted to make that clear in case you thought I had performed actual math to get the idea. On the contrary, I very much encourage the real mathematicians to correct me on this, if I'm wrong. Heh, it just feels like it sometimes. Don't get me wrong, I'm very grateful for your contributions, I just find it hard to believe that this doesn't generate more interest. Personally, I can't get enough of it (this topic).And when I really think about it, I can't even explain why. It does, after all, stem from some of the most basic of concepts. Why, then, do I feel so compelled to understand it?Perhaps, it's precisely because it's a concept that is so fundamental to reality. Maybe I desire to understand higher dimensions so that I might better understand our own. Certainly, if I could visualize four dimensions I would be able to see our familiar world in a much different light.And then, perhaps it's simply stubbornness; my mind's refusal to accept the apparent "impossibility" of it. It would certainly be consistent with other interests of mine. I have the annoying tendency to be drawn to concepts I have little chance of ever understanding. For some reason, I just find them irresistible.Indeed, the concept of higher dimensions is undoubtedly the most difficult to truly comprehend that I've ever encountered. This is probably the primary reason I find it so compelling.Oh, and what did I pre-empt? The "GDong show"? I may have to Google that one. Something tells me I won't find it in the dictionary.