Question about a flat torus topology

[Jimster41]

I'm trying to get comfortable with the idea of a flat torus topology that is also an everywhere a smooth manifold like the video game screen where you got off the screen to the right and pop out on the left (because as I understand it this topology could be a model of space) I can't get how this kind of surface can be everywhere smooth (differentiable?).

If I assign any coordinate system (I can think of) to the grid and start marching from the left edge of the screen toward the right (or vice versus) at some point isn't that counting going to have to be discontinuous. How can a step from 10 to 1 be considered continuous in an ordinal set form 1 to 10. Or a sequence that goes 8,9,10,-9,-8 (turns around, or starts counting down - if that's the coordinate system).

I don't understand exactly what constraint such a topology would place on the geometry of space, or how the topology and geometry specifically have to relate. But why it's considered a candidate topology has been bugging me... forever. Every since I heard this chestnut of how counter-intuitive the universe is.

I can imagine the answer is obvious, but it totally eludes me.

It occurs to me that a radial angle goes from 360deg to 0deg. But that just adds to my confusion - as to why a circle is everywhere smooth. Or why the tangent between those two points isn't busted, compared to all the other tangents.

 

[Orodruin]

Every local change of coordinates is smooth, there is nothing saying that your coordinate system has to cover the full manifold (in fact, very few manifolds allow a global coordinate chart).

On the circle, you can cover it with two charts, one with $-\pi < \theta < \pi$ and one with $0 < \theta' < 2\pi$. On any of the overlaps, the coordinate transformations are smooth.

 

[WWGD]

Don't mean to nitpick, OP, but it is usually the geometry that is (my be )described as being flat, not the topology. If you want an example of relations that exist between geometry and topology, there is the Gauss-Bonnet theorem:

https://en.wikipedia.org/wiki/Gauss–Bonnet_theorem

 

[Jimster41]

Don't mean to nitpick, OP, but it is usually the geometry that is (my be )described as being flat, not the topology. If you want an example of relations that exist between geometry and topology, there is the Gauss-Bonnet theorem:

https://en.wikipedia.org/wiki/Gauss–Bonnet_theorem

Helpful link. Maybe the part that was confusing me was the idea that the "total curvature must be zero". That makes sense, but I can't see how you can have a topology that "identifies opposite sides of a square" but has zero curvature at all points.

 

[Orodruin]

Helpful link. Maybe the part that was confusing me was the idea that the "total curvature must be zero". That makes sense, but I can't see how you can have a topology that "identifies opposite sides of a square" but has zero curvature at all points.

Do you understand why a cylinder has zero curvature? (Note that we are talking about intrinsic curvature, which is a property of the manifold itself, not of the extrinsic curvature of an embedding of the manifold in a higher-dimensional space.)

 

[Jimster41]

I think I can understand why it has zero total curvature, but no I guess I don't understand how a manifold can be cyclic (comes back to where it started) and still be everywhere flat?

 

[Orodruin]

I think I can understand why it has zero total curvature, but no I guess I don't understand how a manifold can be cyclic (comes back to where it started) and still be everywhere flat?

Then I really suspect that you are thinking of extrinsic and not intrinsic curvature.

 

[Jimster41]

So after reading up on intrinsic and extrinsic curvature... An ant living on the surface of a cylinder would not be able to detect a departure from flat Euclidean geometry. So she could not detect the extrinsic curvature. But doesn't the cyclic nature of motion, goes straight and winds up back where she started, act as a give-away that the surface is embedded with extrinsic curvature?

 

[Orodruin]

So after reading up on intrinsic and extrinsic curvature... An ant living on the surface of a cylinder would not be able to detect a departure from flat Euclidean geometry. So she could not detect the extrinsic curvature. But doesn't the cyclic nature of motion, goes straight and winds up back where she started, act as a give-away that the surface is embedded with extrinsic curvature?

No, there is no need to make an embedding in a higher-dimensional space to properly describe the surface. All you would get would be a notion of the fact that the topology of your surface is not the same as that of Euclidean space.

 

[lavinia]

I think I can understand why it has zero total curvature, but no I guess I don't understand how a manifold can be cyclic (comes back to where it started) and still be everywhere flat?

So after reading up on intrinsic and extrinsic curvature... An ant living on the surface of a cylinder would not be able to detect a departure from flat Euclidean geometry. So she could not detect the extrinsic curvature. But doesn't the cyclic nature of motion, goes straight and winds up back where she started, act as a give-away that the surface is embedded with extrinsic curvature?

It does tell you that you are not on a flat sheet of paper. But it does not tell you that you are embedded in a higher dimensional space.

 

[lavinia]

I think I can understand why it has zero total curvature, but no I guess I don't understand how a manifold can be cyclic (comes back to where it started) and still be everywhere flat?

A surface has zero curvature if after finitely many cuts, it can be rolled out onto a flat plane. A cylinder cut along a vertical line flattens out onto a flat sheet of paper.

In order to get the cylinder back, one must bend the sheet of paper until the opposite edges touch. But bending does not cause curvature. Bending does not stretch or warp the paper. Locally angles and distance are preserved. A small right triangle remains a right triangle and the Pythagorean theorem still holds.

Paper is a geometrically amazing substance since it cannot be warped. There is no way to make it have curvature. It can only be bent. If you try to wrap it around a curved surface, for instance a sphere, it will crease an crinkle.

Other flat surfaces are the Mobius band, the flat torus, and the flat Klein bottle. The Mobius band can be made from a piece of paper just as a cylinder.
The flat torus can not be made from a piece of paper in 3 dimensions and any attempt to bend a paper cylinder into a torus will crinkle the paper. This means that any torus in 3 dimensions must have some non-zero curvature. So a flat torus has a geometry that can not be realized in 3 dimensions. But in 4 dimensions it can. The same is true that the flat Klein bottle cannot me made in 3 dimensions though I am not sure if it can be realized in 4 dimensions or not. ( No Klein bottle regardless of its shape can be made in 3 dimensions.)

Here is a parametrization of a flat torus in 4 dimensions.

$$ (x,y) \rightarrow (1/\sqrt(2)) (cos(2πx),sin(2πx),cos(2πy),sin(2πy))$$

 

[Jimster41]

Thanks, I've been thinking about this. I remember hearing the whole flat toroidal space analogy embellished with this idea that we might be staring through our telescope one day and all of a sudden there appears the back of our head staring through the telescope, or something like that. Or more like we would see ourselves at some point in the past. Even though we might not be able to deduce a specific extrinsic topology/geometry, we would surely be able to eliminate all topologies/geometries without a dimension of curvature. No path that was straight in all dimensions could explain that result could it? Seems like an understatement to think our reaction would be "that's not very Euclidean".

 

[Jimster41]

A surface has zero curvature if after finitely many cuts, it can be rolled out onto a flat plane. A cylinder cut along a vertical line flattens out onto a flat sheet of paper.

In order to get the cylinder back, one must bend the sheet of paper until the opposite edges touch. But bending does not cause curvature. Bending does not stretch or warp the paper.. Locally angles and distance are preserved. A small right triangle remains a right triangle and the Pythagorean theorem still holds.

Paper is a geometrically amazing substance since it cannot be warped. There is no way to make it have curvature. It can only be bent. If you try to wrap it around a curved surface, for instance a sphere, it will crease an crinkle.

That helps. As I picture that I want to say that if you look into the dimension of paper thickness, you would see bending stretching, crinkling as it were, but of course the paper is assumed to have zero thickness, as far as the flat ant knows. I guess if the flat ant was smart she would be able to imagine that there was a hidden dimension, that contained curvature, even if it was only observable indirectly, as a transport phenomenon like recursion, which seems to logically require it.

 

[Jimster41]

After thinking about it and trying a bit of googling I can't convince myself that recurrence logically implies or requires curvature. I thought maybe they were from the same Latin roots, but that's wrong. One comes from the verb "to run", the other from the adjective for "bent". I can't find anything specifically definitive. I've learned topologies can be discrete, while geometries have the notion of "infinitesimal" So I guess I can imagine a cyclic topology, that has no notion of curvature. But I have a hard time imagining a topology that could be geometric, that supports recurrence without curvature.

Just trying to think through this long abiding confusion.

 

[Orodruin]

After thinking about it and trying a bit of googling I can't convince myself that recurrence logically implies or requires curvature. I thought maybe they were from the same Latin roots, but that's wrong. One comes from the verb "to run", the other from the adjective for "bent". I can't find anything specifically definitive. I've learned topologies can be discrete, while geometries have the notion of "infinitesimal" So I guess I can imagine a cyclic topology, that has no notion of curvature. But I have a hard time imagining a topology that could be geometric, that supports recurrence without curvature.

Just trying to think through this long abiding confusion.

Imagine you are the spaceship in the game Asteroids! Its world is flat (for example, all triangles have angle sums of 180 degrees, parallel lines remain parallel, etc) and still recurrent (and for some reason contains cosmic strings that seem to spawn asteroids ...). In fact, its world is a flat torus.

As already noted, another example is the cylinder embedded in R^3.

 

[lavinia]

After thinking about it and trying a bit of googling I can't convince myself that recurrence logically implies or requires curvature. I thought maybe they were from the same Latin roots, but that's wrong. One comes from the verb "to run", the other from the adjective for "bent". I can't find anything specifically definitive. I've learned topologies can be discrete, while geometries have the notion of "infinitesimal" So I guess I can imagine a cyclic topology, that has no notion of curvature. But I have a hard time imagining a topology that could be geometric, that supports recurrence without curvature.

Just trying to think through this long abiding confusion.

I suggest that you learn the mathematical idea of curvature starting with Gauss curvature of a surface.
In general surfaces that have a straight-line through each point have zero Gauss curvature. There are many examples.

 

[mathwonk]

i like the simple definition that a surface is flat if on every small circle the circumference is 2π times the radius. this holds on any surface that can be rolled out flat, like a torus. so in my view, your problem is that your intuitive meaning of "flat" is different from the mathematical definition, which i just gave you a simple version of.

 

[Jimster41]

This is all helpful, and I appreciate it. The straight line through each point. The relationship between r and circumference, I can picture these definitions. And I just got a book on Elementary Differential Geometry. I think my problem is really just with the idea someone would describe a "flat universe", where a straight line passes through each point, but where you can see the back of your head by looking through a telescope. End of story.

Surely they would have also meant, "and even though it seems (in the strongest possible sense of seems) flat as can be (in the everyday sense of flat) to us ants and all circles seem like normal circles, and all points carry only straight lines, something really hard to explain is going on. The topology and apparent (intrinsic?) geometry are at odds because there are either these things out there that do something like absorb, move and re-spawn our asteroids and light rays and everything, and how could they do that given our iron clad unbreakable line and circle rules? Or, our apparently 2d world is embedded in an additional hidden dimension (the R3 holding the cylinder) and has extrinsic curvature. One of these must be so or else we would not be able to do such a weird thing as stare at the backs of our heads through a telescope"

What if we looked into a microscope and saw ourselves looking up! [edit] I'm not remotely suggesting such a thing, just trying to grasp principles and terms.

Is a thing like that or the back of the head through the telescope a failure of symmetry expectation? [edit] they seem lie pretty different cases, not equally bizzare.

Maybe my problem is I was only any good at Galaga.

 

[Jimster41]

i like the simple definition that a surface is flat if on every small circle the circumference is 2π times the radius. this holds on any surface that can be rolled out flat, like a torus. so in my view, your problem is that your intuitive meaning of "flat" is different from the mathematical definition, which i just gave you a simple version of.

I think you said it. I think sometimes popular science books etc do try to say shockingly confusing things. Which is fine (I enjoy that) as long as they go on to give a real lesson on how to make sense of it.

 

[lavinia]

That helps. As I picture that I want to say that if you look into the dimension of paper thickness, you would see bending stretching, crinkling as it were, but of course the paper is assumed to have zero thickness, as far as the flat ant knows. I guess if the flat ant was smart she would be able to imagine that there was a hidden dimension, that contained curvature, even if it was only observable indirectly, as a transport phenomenon like recursion, which seems to logically require it.

No. There is no hidden dimension.

 

[Jimster41]

No. There is no hidden dimension.

So how does the ant see the back of it's head?

 

[mathwonk]

there is an axiom for euclid's geometry that says that any finite segment can be arbitrarily extended. He does not say however that the extension might not someday come back on itself and form a loop instead of an infinite line. If we allow such loops then when you look along such a looping back line you see the back of your head.

We must understand a "line" to be a shortest possible path along our surface between any two nearby points. So the geometry of the euclidean plane, thought of say as a piece of paper, is not changed if we make waves in the paper. Or if we allow loops, even if we roll up the paper, say into a cylinder.

it turns out in this sense that all locally euclidean geometries (allowing loops) occur in this way by rolling up the usual euclidean plane. I.e. (if I recall correctly), they are all either obtained by rolling up the plane into a cylinder, or by twisting the plane before rolling it up to form a twisted cyclinder (infinite mobius strip), or by sliding the cylinder back over itself to form a doughnut, or by (possible in 4 dimensions) slipping the cylinder through itself and then sliding it over itself in the wrong direction from inside, to form a klein bottle.

In particular those are essentially all the flat surface geometries, and on some of them, you can see the back of your head in certain directions.

The best possible reference for this theory of flat surface geometry is the book "Geometries and Groups" by Nikulin and Shafarevich. available here for under $20:

http://www.abebooks.com/servlet/SearchResults?an=nikulin,+shafarevich&sts=t