Geodesic Curves Covering Surfaces

[maze]

Are there surfaces that have a geodesic curve which completely covers the surface, or (if that's not possible) is dense in the surface?

In other words, if you were standing on the surface and started walking in a straight line, eventually you would walk over (or arbitrarily close to) every point on the surface.

My current thinking is of a torus where you start walking at a wonky angle so that your path never repeats, like so, but I'm not sure if it quite works:
http://img517.imageshack.us/img517/4487/torusgeodesicap9.png

 

[Hurkyl]

It's actually pretty easy to see that it does work. Pick any horizontal line: where does the geodesic pierce it?

 

[maze]

It's actually pretty easy to see that it does work. Pick any horizontal line: where does the geodesic pierce it?

Ahh yes so simple, thanks.

Now actually, thinking about it a bit, this is a very roundabout way of showing that the cardinality of RxR is equal to the cardinality of R. This torus construction is directly analogous to the diagonal counting used when you show ZxZ = Z. EDIT: Scratch that, I can only show it is dense...

 

[adriank]

While a geodesic on a torus might not do it, you can construct a continuous, surjective function from [0, 1] to [0, 1]², so that |R| = |[0, 1]| ≥ |[0, 1]²| = |R²|.

 

[SW VandeCarr]

Ahh yes so simple, thanks.

Now actually, thinking about it a bit, this is a very roundabout way of showing that the cardinality of RxR is equal to the cardinality of R. This torus construction is directly analogous to the diagonal counting used when you show ZxZ = Z. EDIT: Scratch that, I can only show it is dense...

I question whether the cardinality of RxR is equal to the cardinality of R in terms of a "space filling" curve (geodesic or not) because the fractal dimension is always less than 2. The curve always has the cardinality of R, but not R^2. The fractal dimension of the curve (C) lies on the interval 1< C < 2 , so the interval is closed at the lower bound, but open on the upper bound.

 

[adriank]

The idea of the space-filling curve is that you can construct a sequence of continuous functions f_n: I → I² (with I = [0, 1]) that converges uniformly to another, also continuous function f: I → I² that is surjective (which can be proven by showing that every point in I² is in the closure of f(I), which is just f(I) because I is compact). Munkres' Topology (§44 in the second edition) describes it better than I.

(If you're still not convinced that |R| = |R²|, it's easy to find a surjection [0, 1) → [0, 1)² using decimal expansions.)

 

[gel]

you can't have a geodesic completely covering a surface though (not if its a differentiable manifold), just dense. The geodesic would have zero area.

 

[maze]

you can't have a geodesic completely covering a surface though (not if its a differentiable manifold), just dense. The geodesic would have zero area.

Could you elaborate a little bit? Maybe I'm just being dense (ha ha), but I can't see it.

 

[gel]

take a coordinate patch on the manifold and a segment of the geodesic lying in this patch. Using the coordinates, this gives a differentiable curve in R², which must have zero area. Adding up all the segments of the geodesic lying in this coord patch will give a set with zero area and can't cover the whole coordinate patch.

 

[maze]

take a coordinate patch on the manifold and a segment of the geodesic lying in this patch. Using the coordinates, this gives a differentiable curve in R², which must have zero area. Adding up all the segments of the geodesic lying in this coord patch will give a set with zero area and can't cover the whole coordinate patch.

So the patch contains only countably many segments, each with zero measure, and so the union of all segments has zero measure. OK

 

[Hurkyl]

the fractal dimension is always less than 2.

That's not true.

 

[Doodle Bob]

So the patch contains only countably many segments, each with zero measure, and so the union of all segments has zero measure. OK

For some reason, I am not convinced that there would countably many segments. But a non-measure theoretic argument that the geodesic will not fill the torus is simply finding a point that won't be on the geodesic: if this is the standard unit square torus and the geodesic is given as a line through (0,0) with an irrational slope, then any point of the form (rational, rational) except (0,0) will not be on the geodesic.

 

[gel]

The point of the measure theoretic argument is that it applies to any manifold, not just the torus which was used as an example. You could also use the Baire category theorem.

and a geodesic will intersect a coordinate patch in countably many segments simply because any open subset of the real line is a union of countably many connected segments

 

[SW VandeCarr]

That's not true.

Hurkyl

This quote is in response to my statement that the fractal dimension is always less than two in this particular example.

Yes, the Peano curve passes through every point in the plane, but how do you get there? This is the limiting condition. How many iterations of the relevant (non intersecting) fractal generator does it take to get there from some arbitrary starting point? The fractal dimension is a real number (n), in this case on the interval D1 < n < D2 where D1 and D2 are whole number fractal dimensions. The upper bound of the interval is open. In what way can it be closed? The Peano curve (sometimes called a 'monster') is either a limit value or identical with all the points on a plane by definition.

 

[Hurkyl]

Hurkyl

This quote is in response to my statement that the fractal dimension is always less than two in this particular example.

My original comment was under the assumption you were talking about fractal dimensions of cuves in general.

If not, then what exactly are you trying to say? The Peano curve, being space-filling, is obviously of fractal dimension 2. Each of the chosen approximations to it, being a polygonal chain, are obviously of fractal dimension 1.

(This is not a problem, because fractal dimension is not continuous)

 

[SW VandeCarr]

My original comment was under the assumption you were talking about fractal dimensions of cuves in general.

If not, then what exactly are you trying to say? The Peano curve, being space-filling, is obviously of fractal dimension 2. Each of the chosen approximations to it, being a polygonal chain, are obviously of fractal dimension 1.

(This is not a problem, because fractal dimension is not continuous)

I'm questioning Maze's assertion that the cardinality of RxR is equal to R by virtue of a space filling curve described in his first post where he walks arbitrarily close to every point on a surface. This suggests approximations to the Peano curve (P) not P itself. All approximations to P would have a fractal dimension less than 'two'. P itself has fractal dimension 'two' (but zero area). The cardinality of all approximations to P is the cardinality of R (or aleph 0).

What is the cardinality of P itself?

What is the cardinality of R^2?

Why is the fractal dimension of approximations to P 'one' instead of a real number 'Dn' (actually a fraction) on the interval D1 < Dn < D2? Is P necessarily a polygonal chain?,

 

[adriank]

I think you might getting a bit confused between the two curves discussed: the dense geodesic (which is not a space-filling curve and does not cover the entire surface, but is only dense), and the Peano curve (which does cover the entire unit square). The geodesic, as gel said, has zero area, while the Peano curve has area 1.

The cardinality of the Peano curve P, being an image of [0, 1], has cardinality at most that of [0, 1], which is the same as that of R. But P is the entire unit square, so it has the cardinality of [0, 1]², which is that of R². Thus, |P| = |R| = |R²|.

P is the limit of a (uniformly converging) sequence of polygonal chains, but is not itself a polygonal chain. (Indeed, if it were, it would be the union of countably many lines, but each line has measure 0, so the entire curve must have measure 0. But P has measure 1.)

Minor note: You say "The cardinality of all approximations to P is the cardinality of R (or aleph 0)", but \aleph_0 is the cardinality of the natural numbers, which is different from the cardinality of R, which is 2^{\aleph_0}.

 

[Hurkyl]

P itself has fractal dimension 'two' (but zero area).

How are you defining area? If you do it in what seems to me to be the most obvious way by taking the area of its image, P would have area equal to the square it fills.

I'm questioning Maze's assertion that the cardinality of RxR is equal to R by virtue of a space filling curve

The thing a space-filling curve does is, by definition, give a surjective function

[0,1] --> [0,1] x [0,1]​

(at least in the case where we consider curves that merely fill the unit square), thus proving

|RxR| <= |R|​

 

[Hurkyl]

Is P necessarily a polygonal chain?,

P is not. But the usually-used sequences of approximations to P are. For example, this image shows the first three approximations in one particular sequence.

 

[SW VandeCarr]

Thanks Hurkyl, adriank, maze et al. My remaining question is if |RxR|= |R|, then why confine P to the plane? Can |RxRx...xR| = |R|? If so, what do the transfinite numbers represent?

 

[adriank]

It is true that |Rⁿ| = |R| for all positive integers n. In fact, even R^ω (the set of all sequences of real numbers) has the same cardinality as R. (This can be shown using the Hahn-Mazurkiewicz theorem.)

 

[SW VandeCarr]

It is true that |Rⁿ| = |R| for all positive integers n. In fact, even R^ω (the set of all sequences of real numbers) has the same cardinality as R. (This can be shown using the Hahn-Mazurkiewicz theorem.)

Thanks adriank. I was unable to find Hahn-Mazurkiewicz on the net, but I found Hahn-Banach and the separation theorem of Mazur (which might be the same thing or similar). I guess the term "transfinite" is somewhat dated and such numbers are simply characterized as infinite.

It seems there are no infinite cardinals between aleph null and aleph one (the continuum hypothesis), but this idea depends on the Axiom of Choice. An alternative view which does not depend on the Axiom of Choice is based on the Dedekind infinite cardinal 'm' such that m+1=m, and alpha null < m. However, it seems that for any infinite cardinal there must be a power set which is larger such as 2^(aleph null) that Hurkyl alluded to. Doesn't this number lie between aleph null and aleph one?

 

[Hurkyl]

It seems there are no infinite cardinals between aleph null and aleph one (the continuum hypothesis), but this idea depends on the Axiom of Choice.

Actually, that's the definition of aleph one. The continuum hypothesis is that |R| is aleph one.

2^(aleph null) that Hurkyl alluded to. Doesn't this number lie between aleph null and aleph one?

I don't recall saying anything related to cardinality other than the inequation |RxR|<=|R|.

 

[SW VandeCarr]

Hurkyl

I'm sorry. It was adriank in post 17. You clarified this by stating that the cardinality of R is aleph one.

 

[adriank]

That \lvert \mathbb{R} \rvert = \aleph_1 is only a hypothesis (called the continuum hypothesis). In fact, it has been shown by Gödel and Cohen that the continuum hypothesis can be neither proven nor disproven from the usual axioms of set theory (ZFC).

\aleph_0 is defined to be the cardinality of the natural numbers. \aleph_1 is defined to be the next larger cardinality. Since \lvert \mathbb{R} \rvert = 2^{\aleph_0}, the continuum hypothesis states that \aleph_1 = 2^{\aleph_0}. Again, I must emphasize that it is impossible to prove or disprove this.

 

[maze]

Since \lvert \mathbb{R} \rvert = 2^{\aleph_0}, the continuum hypothesis states that \aleph_1 = 2^{\aleph_0}. Again, I must emphasize that it is impossible to prove or disprove this.

This is more a statement about the deficiency of ZFC than it is about the continuum hypothesis. CH really ought to be false.

 

[adriank]

Whether it ought to be true or not, as long as we're using ZFC we shouldn't go around asserting that CH is true or false; that it is true seems to be the impression SW VandeCarr had.

 

[maze]

yes, certainly

 

[SW VandeCarr]

Whether it ought to be true or not, as long as we're using ZFC we shouldn't go around asserting that CH is true or false; that it is true seems to be the impression SW VandeCarr had.

Actually, I never said that I thought ZFC or certainly the continuum hypothesis was either true or false. I did say that the continuum hypothesis depends on the Axiom of Choice. The alternative, without this axiom, is Dedekind's infinite cardinal that I described (post 22). It's true that the continuum hypothesis is consistent with ZFC. In any case, the "truth" of any axiomatic system mean trusting that deductions from the axioms don't lead to contradictions.

 

[morphism]

It is true that |Rⁿ| = |R| for all positive integers n. In fact, even R^ω (the set of all sequences of real numbers) has the same cardinality as R. (This can be shown using the Hahn-Mazurkiewicz theorem.)

I'm curious, how can you get this from Hahn-Mazurkiwicz? (Perhaps you meant Shroder-Bernstein?)

I think the usual (and perhaps easiest) way would be via cardinal arithmetic:
|\mathbb{R}^\omega| = |\mathbb{R}|^{|\mathbb{N}|} = (2^{\aleph_0})^{\aleph_0} = 2^{\aleph_0} = |\mathbb{R}|.