[The Polymath]

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.

[Qfwfq]

The Polymath, on 27 January 2012 - 08:04 AM, said:

Intuitively, it would make sense that other dimensions would be simply "perpendicular" to each of the others.

Locally.

[sigurdV]

The Polymath, on 27 January 2012 - 08:04 AM, 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?

[maddog]

sigurdV, on 29 January 2012 - 09:27 AM, 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 1917
during WW I. This theory later generalized by Felix Klein to become the Kaluza-Klein
theory which allowed a "fifth" dimension that had the radius of curvature of that
dimension to be very small... This makes it "roll up" and not "seen".

Even Einstein admitted that it did effectively merge the EM theory exemplified by
Maxwell's Equations with General Relativity (GR).

sigurdV, on 29 January 2012 - 09:27 AM, 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 a
coordinate 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 coordinate
system is one that represents our view of our surroundings.

maddog

[lawcat]

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.

[sigurdV]

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 Königsberg
introduces an extra dimension thereby unifying Maxwell and Einstein
Oskar 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.

[Qfwfq]

lawcat, on 10 February 2012 - 11:01 AM, 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.

[freeztar]

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...

[lawcat]

Qfwfq, on 11 February 2012 - 03:23 PM, 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.

[maddog]

sigurdV, on 10 February 2012 - 11:21 AM, said:

This Felix Klein: He is the one with the Klein Bottle right?

Yes this was Felix Klein.

sigurdV, on 10 February 2012 - 11:21 AM, 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.

sigurdV, on 10 February 2012 - 11:21 AM, said:

Theodor Kaluza University of Königsberg
introduces an extra dimension thereby unifying Maxwell and Einstein
Oskar Klein (Swede)helped.(Wasnt forgotten!)

I stand corrected. I was not aware of a second Klein. Good Job!

maddog

[maddog]

Qfwfq, on 11 February 2012 - 03:23 PM, 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 be
described as "radius of curvature" in Differential Geometry terms, wouldn't it.

maddog

[Qfwfq]

maddog, on 15 February 2012 - 02:11 PM, 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.