The Big Bang Bamma

This is something I've always had trouble getting my head around. I have no problem with the concept of our familiar space being curved back on itself within a four (or higher)-dimensional manifold. Well, that is to say, I can't picture it (despite many, many years of trying ) but I understand it.What I don't understand is how that curvature can exist without a higher manifold within which to exist. In the same way that I don't see how a sphere can exist without three spatial dimensions to contain it, I don't see how our three dimensions of space can curve without, themselves, existing within a manifold of at least one dimension higher.My problem with these things always seems to come back to relativity. In this case, what is the curvature relative to? I can only think of curvature in terms of spatial relationship, and always using more dimensions than the thing doing the curving. I can picture a line curving, but only by going through the second dimension. I can picture a plane curving, but only by going through the third dimension.That's where I'm stuck when it comes to three-space. I can't picture curvature in any term but that of a higher embedding space. How exactly can something with dimension n curve without an embedding space of dimension > n through which to do the curving? How can it curve without... well... curving?Sorry, for piling on another reply but you've touched on my favourite subject again.

My understanding is that our three dimensions of space have no center (or edge), just as the surface of the balloon has none. As with the balloon, if our universe does have a center it will be along an axis that exists outside of our familiar three dimensions.However, I think the balloon analogy falls a little short here. I may be wrong so don't quote me, but I believe the analogy is only meant to illustrate how our three-space can be curved, and not meant to indicate that our space actually does exist as a four-dimensionally curved geometric within a higher-dimensional manifold.This is what I've never been able to wrap my mind around... how such curvature can exist when the only way I can think to define "curvature" is in terms of the axes it takes place within. I'm hoping cavediver can explain this to me because it's one of the things that has always driven me nuts.

Hi cavediver. Let me quickly echo CS's appreciation for the time you take explaining this stuff to us poor uneducated plebs.Ok, I think I understand your point regarding intrinsic vs. extrinsic curvature. My problem is that I simply don't understand how intrinsic curvature can exist without that other dimension.In fact, your example is a perfect illustration of the mind jolt that I get when trying to follow this to its logical conclusion. No matter what you do, you will never get any piece of the football to lie flat. So, if we then say that the third dimension simply doesn't exist, then I can't see how the football can possibly exist. By its very nature, it can't exist in two dimensions, can it? This is kind of what I'm getting at. I was under the impression that, without a fourth spatial dimension in which to exist, a true Klein bottle simply can't exist. The best we can do is to create three-dimensional "shadows" of it. Are you saying that a Klein bottle could in fact exist, even if there is no fourth dimension to contain it? It is purely a matter of the object having the necessary intrinsic curvature, even if we could never observe its extrinsic curvature?Incidentally, I'm familiar with the Klein bottle, but what is the RP2? Can't say I've heard of that one. I believe I understand the principle you're referring to. What I can't figure out is what it means without another dimension. I know there's no need to mention another dimension, but, in the example of the Earth's surface, it does exist.I'm having trouble finding the words for this. I think I know what I'm trying to ask, but ugh...How about this? I understand that the curvature of a body can be shown without referencing any dimension outside of the body, but how can the body possess that characteristic without the other dimension?This is one of those things that, ultimately, you can really only understand mathematically, isn't it?P.S. On re-reading my post, I think I have a better way of expressing it. Simply put, how can a body have an intrinsic curvature but no extrinsic curvature? How can one property exist without the other?

Well, thanks again. You clearly have a lot to teach about these subjects and you work hard to help anyone that's interested. You seem to be getting something of a pile on in this thread, and I wanted you to know that your efforts are genuinely appreciated. I've spent years trying to do that. I fear I'm veering even further from the topic but I must ask... can you visualize four dimensions? That is, truly four dimensions. Can you picture four perpendicular axes? The true form of a 4D simplex? A tesseract? A glome? And so on? Well, with your help, I hope to rectify that.Ok, so... a distance-less surface. Whew! I have to admit, that one hurts my brain. Is there any explanation you can give that I won't require a PhD in advanced mathematics to understand... or am I asking the impossible?If you can't, no problem. I'd love to understand this but I get the feeling that we're rapidly approaching some concepts that there are no sufficient analogies for. That's so frustrating. When you say that you can only understand it mathematically, are we talking... you know... really complex math? As in, the kind that you can spend your entire life studying? Do I, as a layman with no higher education in mathematics, have any hope of ever understanding this, or are several decades of university level study the only real chance I have? Heh... yes, I realize that.What I meant was that I can't see how it can have intrinsic curvature without expressing it as extrinsic curvature. Make sense?It seems to me that I'm connecting the two while you're saying they're not connected. You seem to be saying that it can be intrinsically curved yet completely flat. I didn't word that well but I think you get my meaning. I hope.The problem, for me, seems to go back to the football analogy. You can cut it, bend it, twist it, squash it, but it will never settle into a flat, two-dimensional plane. That's what I'm getting snagged on. If intrinsic curvature renders it unable to be flat then how can it have intrinsic curvature in a 2D universe? Doesn't 2D space preclude the very existence of such a property?I apologize if I appear to be repeating myself. To be clear, I'm not doing it to be argumentative. I don't doubt the accuracy of what you're telling me... I'm sincerely trying to understand it.Also, thanks for the info on RP2. Right now, though, I need sleep. I'm pretty tired and that whole last paragraph flew straight over my head. Will recharge the batteries and then check it out. Thanks again.

Heh, that's what Rrhain said. More and more, I'm getting the impression that this is something you simply have to "see" for yourself. To try and clarify a little, when I say "visualize" I mean can you conjure, in your mind, that which a four-dimensional creature in a 4D hyperspace would see? For instance, can you picture 4D primitives as they would appear to an indigenous creature? Or can you picture four mutually perpendicular axes with the four-space perspective such a being would have?Or, to try something slightly different, can you imagine a three-dimensional solid embedded in four-dimensional hyperspace, and then rotate the solid such that its three-dimensional "surface" is exposed and you can see the flat 3D cross-section of the entire solid... every point within its three-dimensional structure simultaneously?To use an analogy (sorry for my rambling style - please bear with me )...Most people in Flatland can only "understand" higher-dimensional concepts like "hyper-circles" and "hyper-squares" by picturing their edge-on cross sections as they pass through 2D space. Some scientists/mathematicians, however, are adept enough in the concepts to be able to actually visualize the constructs known as "spheres" and "cubes" in the way that we 3D creatures actually see them.That is, despite its physical impossibility in their space, they can picture, in their mind, what we see. The image they visualize is a two-dimensional surface projection with three-dimensional perspective i.e. exactly what our eyes receive. There is no way, of course, that they could physically display or describe it to others in their world, but they can imagine it.In essence, what I'm asking is are you one of the scientists in this scenario?Do you believe that what you are able to visualize is actually a true three-dimensional "surface" projection with four-dimensional perspective? Or does your ability to visualize 4D still come up short of what a 4D creature possessing retinas with 3D "surfaces" would see? Frankly, I don't know. I'm not familiar with the underlying mathematical theory of higher-dimensional topology, or whatever the field is called. If I knew what it was I guess I wouldn't need you to explain all this to me.Since you mention it, though, I have seen the kinds of equations to which you refer and, honestly, it always looks more like hieroglyphics than mathematics to me. I've seen some equations in which I am hard pressed to recognize anything. Sounds like my kind of math.Seriously, though, it does sound like my kind of math. I have (I think) a reasonably well developed visual imagination, but abstract mathematics tends to throw me.I've been doing more reading on dimensional concepts in the time we've been discussing this, and when I'm reading the specifics of certain concepts where I'm being presented with measurements or calculations, I'm always looking for images. In a previous discussion on this subject, I once said that I imagined the ability to visualize higher dimensions would require you to virtually re-train your mind in how to think. Living the entirety of our lives in three dimensions ingrains a 3D way of thinking onto our psyche. Clearly, that just gets in the way when trying to conceptualize dimensions higher than three.Thanks for the Wiki link, by the way. I read the page and I believe I got some of it... but, again, I'm reading (and writing) when I should be sleeping. I plan to read it again anyway.Once more, sorry for my prattling... I think I'm even worse when I'm tired. Thanks for your patience, cavediver.

Sorry cavediver... I just realized that my reply isn't showing in your topic index. Probably because I didn't reply to your most recent post. My apologies.Just letting you know that I did reply.

Hi cavediver.I think I understand where you're coming from, but the thing is I already have a reasonable idea of the principles of their construction (at least I think I do)... I understand the logic of how one becomes another as we add dimensions.And I have been pondering their structure for years, yet I still can't visualize them as anything more than moving, pulsating 3D solids. No matter how hard I try, I just can't seem to make myself see all four dimensions at the same time.I'm thinking... and I may be wrong... but I'm thinking that what I want to do is visualize the fourth dimension as a perspective view, if you get me. The same way we experience the third. Technically, we don't see in three dimensions... we see a two-dimensional image focussed on our retina. We don't actually view all three dimensions in their entirety... we perceive the third dimension as depth.So, I'm guessing that what I currently experience as duration, that is, the time that elapses as I picture a hypersolid's 3D "shadow" changing shape, needs to become that perspective. If I could visualize the fourth dimension as a perspective view I would have a way of fitting all of the "shadow's" phases together into one coherent figure... just as you can use our 3D perspective to put all of those slices together to form the complete cube. Does that make sense? Well that's just it... my brain has been cooking for far longer than that and I haven't been able to do it yet. I think that perhaps you underestimate your own ability, CD. We aren't all as smart as you. Yes, but we already have an intuitive feel for that, as our eyes see that way to begin with. That's where I'm having trouble... to visualize true 4D requires you to picture something you haven't seen and have no instinctive feeling for.A 2D image can still give the illusion of 3D depth, but I've never been able to get a sense of 4D depth from any of those 4D analogues. The problem is that it's a different scenario. Trying to grasp the overall shape of a hypersolid by studying its analogue in three dimensions is like trying to figure out what a cube looks like when all you know it from is the edge-on view of its 2D shadow. Hmm... I don't know. I'm not sure how these things are calculated to begin with, but I'll give it a think.EDIT: Amended the structuralized gramaticality of a specific word string whose beginning was in the finding of thine own commencement regarding such required velocity as to necessitate a measure of pseudo-desirability.Sorry... [/Brad mode]... uh yeah... grammar.This message has been edited by Tony650, 11-26-2005 12:47 PM

Damn you spotting that! I was so looking forward to correcting you on it!

Just to clarify, I haven't done any polytope generation, as such. That is to say, I haven't actually built them, myself. I've mostly just played with interactive polytopes online. Over the years, I've probably tried just about every 4D applet on the internet. Yes, but I always found the more complex ones hard to follow. I have a hard enough time keeping track of my perspective when playing with a tesseract. I find it progressively more difficult with the addition of more cells, vertices, etc. It is? Cool! What exactly is unique about four dimensions? Please elaborate! Well, if it's something you can explain relatively easily that is. If it's one of those "decades of study" things that's ok. Ok, I've been looking around and I've come across a few things. In truth, I have entirely too much to scour. I have dozens of old links to pages on the topic, and checking them all, working through all the links... ugh... it can result in a bit of a sensory overload.In any case, I found a fairly decent looking applet you might want to check. Not sure if it will help you illustrate your point about the 24-cell to me, but for what it's worth...HyperSpace Polytope SlicerIt starts with the 600-cell by default (at least on my PC) but let the applet load and then click the "Controls" button. You should be able to get a drop-down menu in the "Object" box where you can select the 24-cell (or whatever else you want).Is that any good for what you wanted to show me?As I happen to have the link handy, here's another one I found...4D 24PolytopeThough you don't appear to be able to do anything with that one.Slightly off-topic, have you ever seen the 4D version of the Rubik's Cube? There are several different versions but this was the first one I ever came across...http://www.hadron.org/%7Ehatch/MagicCube4dAppletI've had the downloadable version on my computer for years and haven't been able to solve the damn thing yet. Perhaps you'll have better luck.And, so long as I'm giving you links, take a look at these ones that I found in my travels...4DHyperbolaKleinsBottleHyperPlaneCompletely irrelevant, but I thought they were nice.

My pleasure, pal! I have a stack of them if you want. I appreciate the time you're putting in here with me and I'm happy to give something back. You mean the fact that it intersects itself? It's an artefact of representing its 3D cross-section? I'm not sure if I read you right. Here's hoping I understood. Ah, ok. Speaking of which, I just want you to know that I didn't ignore your homework assignment. I did try to work it out... just not with much success. The infinite number of regular 2D polygons is quite obvious and needs no explanation, but I must be honest... from three dimensions up I really wasn't sure how to proceed.I may have eventually figured out the number of Platonic solids simply by physically constructing them in my mind or on paper, but I would assume there is a way to actually calculate it. And that's just for 3D! Honestly, I don't think I have the necessary grasp of higher dimensions to have worked out any more than that... if I'd even been able to figure out that much. The Platonic solids... maybe. Polytopes in 4D and above... I seriously doubt it. Oh, well I didn't know that about it. So, basically, it has no analogue in any dimension? I couldn't figure out why I wasn't seeing them and then I realized you can change the figure's orientation, and I believe I found it under "cell first." The octahedron was obvious enough but I was having trouble finding the truncated cube. I think I'm with you now, but, just to make sure, take a look at this page...YahooIf you scroll down just a little less than half way, there's a figure there which, as far as I can tell, is exactly what I was seeing in the center of the 24-cell. It's number six on the page and listed as the "Cuboctahedron." If you scroll down a little further to number nine, the "Truncated cube"... that's what I was looking for.But the middle slice of the 24-cell matched the former, not the latter, with the squares only just meeting at their vertices before retreating again. I realize you already made the point about the square faces "just touching" and I assume that's what you meant. I don't mean to be so analytical... I just want to be sure we're on the same page here. Ugh! Sorry to be such a pain, but which "three" is that? Do you mean the three regular polytopes in 5D and above? Have I thought about them? Yep. Do I know how to build one? Um, sure... start with a point, expand it spherically to its maximum volume, and then return it to a point, all the while stretching it perpendicular to itself along the fourth spatial axis.To be honest, I find this the hardest one of all because there are no straight lines to work with. In the case of, say, a tesseract, I can somewhat comprehend its structure, even if I can't picture it. Its cells are connected by these nice flat (square) faces.Now, I may not be able to visualize how these faces can be connected while their respective cubes are perpendicular to each other, but at least I can get some general sense of the figure's structure. I can see, to a limited degree, what goes where.With a hypersphere, though, I'm kind of lost. Its curvature gives me nothing to work with. I have no problem with the concept of an "unfolded" tesseract, and can picture it folding up (though I can't picture where the folded pieces go... only their distorted "shadows" as they fold out of 3D space). But, for the life of me, I can't imagine what an unfolded hypersphere would look like.

Hey CD. So is a Klein bottle an actual cross-section or a "shadow"? With a tesseract, for example, there isn't really a small cube sitting inside a larger one... the "small" cube is simply further away on the fourth axis, right?Is that the case here? Is what we see as the self-intersection of a Klein bottle in fact the "shadow" of two separate sections where one merely passes "behind" the other on the fourth axis? Oh! Uh, I mean, yeah... of course it was! I knew that!Heh, that's quite a relief, actually. I was thinking I must have been quite dumb not to be able to work out the "simple" problem you gave me. And to think, I was resisting the urge to look the answers up! I don't really know. I would guess that it's because those more complex are just different combinations of the fundamental polygons? That's just a guess, though. I can't say I'd heard of them. So I looked them up...Dual Polyhedron -- from Wolfram MathWorld...and I think I understand, more or less. I know it mentioned certain specifics which I'm missing, but, as far as I can make out, the essence of it is that where one has a face the other has a vertex, and vice versa. That's what I was thinking. In the 24-cell animation, the square faces extend until they touch vertices and then retreat. If, instead, they continued on into each other you would end up with... a truncated cube. Hmm... ok. Well, much as I wish I could, I still can't picture it. And, worse still, this doesn't even have a 3D analogue I can look at. With, say, a tesseract I can look at a square... look at a cube... and get some kind of understanding of the principles behind its construction.Is there anything at all that can give me a similar feel for the 24-cell? Or is it doomed to forever remain an abstract figure to me? Well, assuming I'm not eventually successful in picturing four dimensions, that is. Heh, I know. I was being facetious. That is, I can tell you what's required to do it, but I'll be buggered if I can actually tell you how to accomplish that. And, again, I'm with you all the way up to the final step. That last one just keeps eluding me, it seems.

Yes, I was referring to the latter. I'm still having a hard time understanding intrinsic curvature of a flat 2D surface, so in cases like this I will generally be referring to the three-dimensional figures. In the case of the Klein bottle, I usually imagine the common 3D depictions of it. Something along these lines.I am thinking that, if this kind of depiction is not an actual cross-section but a perspective shadow (in the same way that you can have a perspective shadow of a hypercube, showing cubical "surfaces" with distorted sizes and shapes), then the segment that appears to pass inside the bottle is actually just passing "behind" (along the fourth axis) what we see as the outside of the bottle.That is, the narrow segment inside the 3D figure is just farther away along the fourth axis. It passes "behind" the other so that our edge-on view of its 3D shadow appears to be self-intersecting. But if we could just look "up" at the whole thing, we would see that there is no intersection... it simply loops around and connects back to itself like the Mobius strip, just with an extra dimension.Am I on the right track? Yes, that's what I meant. I think I'm confusing my terms. I always thought that "tesseract" was the name given to the actual four-dimensional figure, just as "cube" is the name given to the three-dimensional one. I thought that the term "hypercube" referred merely to its 3D shadow.You're saying it's the other way around? "Hypercube" refers to the hypothetical 4D construct, and "tesseract" refers to its 3D shadow. Is that correct?In other words, this is a tesseract, and the thing I've spent all these years trying to picture is a hypercube. Correct?Just making sure my terminology is accurate.Thanks for your explanation of the Platonic solids. It would appear that I actually started out correctly (trying to work out how many ways you can surround each regular polygon) but I went off on the wrong track. Anyway, your explanation makes things quite a bit clearer. Thank you!So, four-space has the analogues to all of these plus the 24-cell. From this page...24-Cell -- from Wolfram MathWorld Now, help me out here. "Self-dual"? Does this refer to the same concept as dual solids? If so, is this a quirk of 4D or the 24-cell itself... or does it simply mean that it is its own dual when appropriately rotated?Unless I'm greatly mistaken, this is the case with the tetrahedron, is it not? You take a tetrahedron, flip it upside-down, and that gives you the dual of the solid you just had, yes? Is that the same concept that the mathworld page is referring to in the case of the 24-cell? Always, my friend. Thanks again for all of your help. I'm really enjoying this. I've learned a lot in this relatively short time, and I greatly appreciate the patience you display in your efforts to teach me.