joekgamer Posted January 27, 2012 Report Posted January 27, 2012 When spacial dimensions other than the "basic" three are being discussed, I often hear that the other dimensions are "curled up" inside of the "normal" ones. I don't quite get where this came from. Intuitively, it would make sense that other dimensions would be simply "perpendicular" to each of the others. Analogously, if one had a 2-d plane, the third dimension would be found by looking at the cube defined by an axis perpendicular to the plane. Thus, the cube would be composed of an infinite number of 2-d "slices". Similarly, one would think, a 4-d "hypercube" would be composed of an infinite number of 3-d "blocks", all aligned next to each other on a four-dimensional axis. As this is a fairly intuitive view, I find it rather highly unlikely that it has simply never been thought of before, so I would like to ask: what, exactly, is the reasoning behind the common "curled up" view? Of course, I could simply have made an error somewhere, and the view described above may not make any sense at all. Quote
Qfwfq Posted January 29, 2012 Report Posted January 29, 2012 On 1/27/2012 at 5:40 PM, The Polymath said: Intuitively, it would make sense that other dimensions would be simply "perpendicular" to each of the others.Locally. Quote
sigurdV Posted January 29, 2012 Report Posted January 29, 2012 On 1/27/2012 at 5:40 PM, The Polymath said: When spacial dimensions other than the "basic" three are being discussed, I often hear that the other dimensions are "curled up" inside of the "normal" ones. I don't quite get where this came from. Intuitively, it would make sense that other dimensions would be simply "perpendicular" to each of the others. Analogously, if one had a 2-d plane, the third dimension would be found by looking at the cube defined by an axis perpendicular to the plane. Thus, the cube would be composed of an infinite number of 2-d "slices". Similarly, one would think, a 4-d "hypercube" would be composed of an infinite number of 3-d "blocks", all aligned next to each other on a four-dimensional axis. As this is a fairly intuitive view, I find it rather highly unlikely that it has simply never been thought of before, so I would like to ask: what, exactly, is the reasoning behind the common "curled up" view? Of course, I could simply have made an error somewhere, and the view described above may not make any sense at all.The idea is from early 20th century ...His name eludes me at the moment,but I think he was German or Swedish.And, business as usual,his revolutionary idea was ignored until it found some use.I think it is, in String Theory. So now there seems to be more than one way to add dimensions: The ninety degree, and the curled up extensions. So IF we add an extra time dimension,should we (following trends) curl it up? Quote
maddog Posted February 10, 2012 Report Posted February 10, 2012 On 1/29/2012 at 7:03 PM, sigurdV said: The idea is from early 20th century ...His name eludes me at the moment,but I think he was German or Swedish.And, business as usual,his revolutionary idea was ignored until it found some use.I think it is, in String Theory.Kaluza (I forget his name - maybe someone else knows) -- He is German by the way.Einstein accepted his paper in 1915. Kaluza later died at the Eastern front in 1917during WW I. This theory later generalized by Felix Klein to become the Kaluza-Kleintheory which allowed a "fifth" dimension that had the radius of curvature of thatdimension to be very small... This makes it "roll up" and not "seen". Even Einstein admitted that it did effectively merge the EM theory exemplified byMaxwell's Equations with General Relativity (GR). On 1/29/2012 at 7:03 PM, sigurdV said: So now there seems to be more than one way to add dimensions: The ninety degree, and the curled up extensions. So IF we add an extra time dimension,should we (following trends) curl it up?These are distinct things orthoganlity and non-Cartesian Geometry. In acoordinate system you can one or the other or both on any axis or all. In fact you are not limited there. You can have complex coordinates (x = a + ib),or over any field or ring you wish to compose. You can even do all three. The devil is in the details though when you attempt to consider this coordinatesystem is one that represents our view of our surroundings. maddog Quote
lawcat Posted February 10, 2012 Report Posted February 10, 2012 I think the reasoning behind the curled up view is graphic. One dimension with respect to another creates a gradient, a curl. So graphically we see any one of the 4 dimensions we deal with curled due another which we cannot even visualize. All we can visualize of that other unseen dimension is its effect on the 4 we deal with. So we say, it's curled up in there. Quote
sigurdV Posted February 10, 2012 Report Posted February 10, 2012 This Felix Klein: He is the one with the Klein Bottle right?Was he perhaps Swedish? Im convinced there was a Swede in on this somewhere. Perhaps it is as in the case of The Hubble Discovery... Swedes almost never get credit for their work, I suppose the sad story of forgetting their involvement began with the discovery of oxygen. PS done some reading...Theodor Kaluza University of Königsbergintroduces an extra dimension thereby unifying Maxwell and EinsteinOskar Klein (Swede)helped.(Wasnt forgotten!)The driving idea behind adding dimensions seems to be to unify separated things.I think we at the moment are up to around twelve or so dimensions...I suppose there are lots of separate pieces in need of unification. Quote
Qfwfq Posted February 12, 2012 Report Posted February 12, 2012 On 2/10/2012 at 8:37 PM, lawcat said: One dimension with respect to another creates a gradient, a curl.By this, I would take it that you refer to the differential operator known as curl. Given it is a differential operator it cannot have anything to do with the same word curl as describing global topological trait of the manifold, which is the topic of discussion. Quote
freeztar Posted February 12, 2012 Report Posted February 12, 2012 The fifth dimension can perhaps be best visualized with the hypercube. No curling necessary. :) But of course, we run into another problem when we add a sixth dimension. Then, the idea of combining hypercubes and being able to take 5d slices of them to amount to a six dimensional shape starts to get a little hairy. Some mathematical constructs just do not have any real world representation. Imaginary numbers are a good example...though my bank seems to disagree...:lol: Quote
lawcat Posted February 12, 2012 Report Posted February 12, 2012 On 2/12/2012 at 12:59 AM, Qfwfq said: By this, I would take it that you refer to the differential operator known as curl. Given it is a differential operator it cannot have anything to do with the same word curl as describing global topological trait of the manifold, which is the topic of discussion. Yes, that is what I mean by curl. I do not recognize any other curl, I recognize flat dynamic universe. Quote
maddog Posted February 15, 2012 Report Posted February 15, 2012 On 2/10/2012 at 8:57 PM, sigurdV said: This Felix Klein: He is the one with the Klein Bottle right?Yes this was Felix Klein. On 2/10/2012 at 8:57 PM, sigurdV said: Was he perhaps Swedish? Im convinced there was a Swede in on this somewhere.It was my understanding that he (Felix) was German. You might be thinking of Sophus Lie who was Norwegian, also a contemporary of Klein. On 2/10/2012 at 8:57 PM, sigurdV said: Theodor Kaluza University of Königsbergintroduces an extra dimension thereby unifying Maxwell and EinsteinOskar Klein (Swede)helped.(Wasnt forgotten!)I stand corrected. I was not aware of a second Klein. Good Job! :) maddog Quote
maddog Posted February 15, 2012 Report Posted February 15, 2012 On 2/12/2012 at 12:59 AM, Qfwfq said: By this, I would take it that you refer to the differential operator known as curl. Given it is a differential operator it cannot have anything to do with the same word curl as describing global topological trait of the manifold, which is the topic of discussion.I must admit that I was a bit confused by Lawcat's comment. Of course topologically this would bedescribed as "radius of curvature" in Differential Geometry terms, wouldn't it. maddog Quote
Qfwfq Posted February 16, 2012 Report Posted February 16, 2012 On 2/15/2012 at 11:47 PM, maddog said: I must admit that I was a bit confused by Lawcat's comment.No doubt he gets the matter very confused indeed. However, radius of curvature is hardly relevant in differential geometry. Saying that a dimension is curled up isn't really mathematical terminology, it's just a description that follows from embedding such a manifold. The only feature intrinsic to the manifold itself is the topology. Quote
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