Dimensional Discourse

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?

Tony650I just relocated this little discourse on dimensionality at this site. http://home.pacifier.com/~ppenn/secperc_a.html#time At another site Tesseract -- from Wolfram MathWorld is a java applet that relates the three space "shadow" of a tesseract from different perspectives. I am trying to imagine what it would mean to change the time dimension to a frame of reference in the four space. If t can be be changed to x y or z in 3 space then if the four space is "a" what the frame of reference that gives motion to that space?I know that this will take a great deal of time on my part and perhaps a nervous breakdown or two.

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[W]hen people thought the Earth was flat, they were wrong. When people thought the Earth was spherical they were wrong. But if you think that thinking the Earth is spherical is just as wrong as thinking the Earth is flat, then your view is wronger than both of them put together

edit to kill double postThis message has been edited by sidelined, 10-23-2004 10:08 PMThis message has been edited by sidelined, 10-24-2004 11:24 AM

sidelined writes:

quote:

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At another site Tesseract -- from Wolfram MathWorld is a java applet that relates the three space "shadow" of a tesseract from different perspectives.

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Here's a question about observational abilities:Do you see the cube in the link (and you need to put your mouse over it to get it to move) as a view from the top or from the front? When I see it as from the top, it looks kinda like a boxing ring where the poles and the ropes between them are moving up and down while the whole thing rotates anti-clockwise. But from the front, it looks like a flanged tube rolling over on its side as the internal lines writhe in and out.And thus you see the difficulty of trying to describe an N-dimensional object in less-than-N dimensions.

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Rrhain
WWJD? JWRTFM!

RrhainYes I see the movement in two different directions depending on your point of concentration.Even more dificult is relating motion in a 4 space as a consequence of events happening in 4 space.If our eyes that are used to 3 space were to find themselves occupying 4 space would these motions appear to operate simultaneously or would we be incapable of resolving both motions at once?
What a surrealstic sensation when you stare at the the cube long enough to be able to switch it back and forth between perspectives from one rotation to the next.

There is indeed a lost thought or two in here. I have an off line response written but not typed up just yet that specifies what I need programatically to VISUALIZE 4D. No, I have not done this yet. It is possible that we are mistaking such visuals with with temporal assymetries spatialized. Regardless one needs OBJECTS yes. But no I have not rotated "em". You see if it TIME that is NOT SPLIT it might be space here and not the maths of "4d". The VRML program I am contemplating will use the ability to change VIEWS (relative place of USER with respect to the 3D object) to gain this dimension relative scaling levels of organization. There does seem to be somepossiblity if dataformnormalization was to exclude bifurcations numerically that one might find such rotations you proposed as normal intutions but rotating an OOP object method programmatically need not be a natural 4D turn.I think such twists might exist even if one did not have to have it perverted provided that temperature CREATES the assymetry but this would not be sensible, being understandable nontheless. A VRML simulation of the kinetic theory of gases might help. The physical nexus may be POTENTIAL CHANGES CURRENT WITH virial on phase transitions related to gas exchange during life. If so I would suspect any exisiting claim for such abilities to review 4D motion without necessary connections to living objects. I can be wrong. It could also be that the physics thus assertable speaks to preventable reductionism however.I have been thinking I wouldn't post this long response to you directly but rather contain it in a web site of my own trappings still to be decided if it is underconstruction or not.I can abbreviate the conditioning if you like.

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.

Yes the object fasinates me too. I sometimes cant distinguish the word translated in Kant's Critique of Reason, Bertrand Russel's use of the word "Chain" (in relation to Dedekind), a liberal interpreatation of Wilson's biological diversifiability, OOP logic, and my mouse. I had hoped my first love would have reciprocated so I am back to the drawing booorred again. Yes, I havent rotated them but I am begiNing to think why Einstein might have not finished his work. It was not because he couldnt park a car!The first thing it to be clear what "straight lines" one uses are but with an OBJECT they need not be straight and that gets one into mental space very quickly.

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?

Oh,
for me it's because I still refuse to accept Necker Cubism as discussed between Dawkins and Gould and thus it causes me to illegitamately psychologically block such geometric rearrangements.Yes, Einstein was on about being in the center of a circle or not but the reason I suspected his work on might not be able to be finished was because I am coming to the realization that with the addition of the logical manipulations computers afford man the FORMULAIC inuitionn that Einstein exemplified more than any other mortal might be being CONTINUOUSLY being replaced by human centered computer assisted computations. It would have to be continously, not discontinously happnening for me to be correct on that, else one need merely take the position of a Wolfram or a Feynman or some linear business extrapolation economically of a future in quantum computers.I have this thought, ONE out of rejecting Einstein's reasons for ejecting Kant and TWO because I can ONLY set up the conditions of your 4D reality by a haptic interface with a VRML browser able to turn a macrothermodynmaic equilibrium (that WOULD contain the FORMULAIC approach interms of recursiveness programmatically no matter how the Einstein like Intuition thinged itself).Thinking in terms of rotation is not necessary if one THOUGHT in terms of Poincare RETURN POINTS
http://www.math.montana.edu/~pernarow/M455/ComputerLab2.html
but
http://faculty.stritch.edu/...olds/mt322_02/mt322sgExam.html
if?
Class OdesReturnMap
??
group rotations WERE added to strech of some distance one might coherently build OUT OF 3D by linking database tables a representation of 4D that I would think could be graphically replayed in virtual reality as projection from a calculated 4D, could it not?It always seems "intuitive" with me that 5 axes might be condtionally existant in higher order catastrophe sets or at least this is the respect and credit I give a mind like Rene Thom if it came to the senses actually used to achieve these results. Rotation would have meaning interms of crossing control parameter space on this noddle of lack of coherence that is not incoherent nonetheless.Your not "talking" to yourself. That was the GDong show preempted by you.

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.

It's not often that I come across something completely new to me in this subject, so this really took me by surprise. Not only is it a dimensional concept I hadn't encountered before, it's one that, for the life of me, I can't understand. Any help would be appreciated.I'm currently reading through Sagan's Cosmos (yeah, I know...took me long enough to get around to it ) and the other night, I finished chapter X, towards the end of which he describes the concept of higher dimensions using the familiar Flatland examples. All pretty standard stuff...nothing I didn't know there. Then in one of the footnotes, I came across this. I can't say it appeals to me, either. But it does fascinate me.Now, I've tried to liken this to lower dimensions, as is the method when trying to comprehend these things, but I just can't wrap my mind around this one (really bad pun ).I can't figure out how I could take a two-dimensional creature, say a square, and rotate him, move him, manipulate him, etc in such a way that he is turned inside out, and surrounds an infinite (or at least, much larger than he) two-dimensional plane. Does this concept actually have an analogue in two dimensions that I can visualize?I understand how a higher dimensional being can "reverse" us and insert us back into our universe as a "mirror image," but how does one get turned inside out? Does this simply not apply at lower (less than three) dimensions or is there some way I can picture it?Little help?