Original Post Date: May 21, 2012
Original Post: http://bit.ly/1rJFNKa
Far fetched asking about topology on here, but there might be someone that knows. Posting this onto a specialized group might get specialized attention, though I doubt.there are topologists on here…
For a given n-space, does n have to be an integer? I can see how an n-manifold doesn’t necessarily have to be an integer (as is in the case of fractals), but can a fractal-dimension k (in essence, a non integer, positive or negative, dimension) be expressed as a k-space? If a k-manifold “Flatlander” were to exist in a k-space, would it perceive itself on a flat plane with no discontinuity? How would k-manifolds behave in other non integer m-spaces? If k>m, then what would happen? Would it be like in Flatland, how a three-sphere passing through two-space would look like a circle with changing diameter, except without a smooth transition? What does the m-manifold look like to a k-manifold in k-space? In m-space? Is there even a branch of topology concerning fractal manifolds in fractal dimensions? If not, I must create it. But… ahhh so many questions that I want answered!
Well, NOW I know what a manifold is. Stupid me. Most fractals aren’t manifolds. Most fractals are anything but locally Euclidean! But I mean, at the time, I was just learning about topology. It’s fun to reply to the “Ask me anything” questions with “Define a mathematical manifold using only two words.” No one is yet to answer with “locally Euclidean.”
Again, this was posted at a time when I’d given up on MIT and aimed for Princeton. Blah. At this point, I’m just desperate to get into MIT. And I’m compiling this now mainly for admission into MIT. But I will still update this even after I get a reply back, regardless if I get accepted or not. Because by then, I’d have gained some followers, and hopefully they’d be more erudite than the ones I get on EP. I’ll post to EP and to here. That way I can target the more lay audience to educate them, and also the more erudite audience to get erudite responses. Because I want to get feedback! I want to learn, too! Occasionally, I run into some geniuses on EP, and I learn, like in this case 😀
This is also when EP had a bug where it randomly inserted HTML tags in the middle of words or sentences. Ignore them.
User 13 [Jun 15, 2014]: You should have a look at “Alexander’s Horned Sphere” as it fails to be a “tame” 2-manifold at some point even though there is a continuous map from the unit sphere.
Me [Jun 15, 2014]: It looks fractalesque. Perhaps that’s why it isn’t a two-manifold. It likely has fractal dimensionality just slightly under two.
User 13 [Jun 15, 2014]: also, I think that you’re not quite clear on separating a space from a manifold. A manifold IS a topological space, but I suppose that you might be meaning a k-manifold as a subspace of an m-dimensional euclidean space. For example, the shell of a ball, known as the 2-sphere, can be viewed as a subspace of 3-dimensional euclidean space, but you have to realize that the space is also a manifold. There are plenty of topological spaces that aren’t manifolds, too. For a simple example, just take a figure-eight curve “8”. It fails to be a 1-manifold because of the point where it intersects itself. Sometimes we make particular reference to immersion of manifolds in higher dimensional space because they don’t fit well in other places. For example, take the Klein Bottle, which is just a 2-manifold, like a sphere or a mobius band, so you might think that you could view it as a “submanifold” of R^3, but you can’t. You need to put the bottle into at least 4-dimensional euclidean space before you can get a good picture of it. <br />
Off the top of my head, I would say that the Koch snowflake is not a manifold even though it is a continuous image of a straight line (getting bent up). Straight lines are 1-manifolds, but I don’t think that the snowflake is one because my feeling is that you’re not going to get a continuous inverse image. It is also clearly not a 2-manifold because there is no area enclosed. I might be wrong, though. Also, you want to think that a big part of the study of manifolds is differential geometry, and really you can’t very easily apply the same techniques to fractals. For example, you talk about your “Flatlander” people, and you want them to be able to walk around on your 2-manifold, and to do so you need to be able to give them coordinates like 5 degrees north by 20 degrees east. You can’t really have nice charts and or an atlas with fractals, so you might lose a lot of the good machinery that goes with the study of manifolds if you try to develop a parallel theory for fractal dimension.
Me [Jul 15, 2014]: I thought manifolds were topological surfaces, and the space is where it resides in. I know what a two-sphere is, and from what I learned, a two-sphere is a two-manifold that resides in three-space. But how is the figure eight not but a “one-lemniscate” of sorts? I know what a Klein Bottle is (and in fact, I have visualized it in its native two-manifold four-space form. Obviously the Koch Snowflake is not a one-manifold; the Koch Curve is 1.26-dimensional. But wouldn’t that make it a 1.26-manifold? Yes, you can’t have an atlas with fractals, but we also talk about four-space as a legitimate theory without us being able to physically construct an atlas of it.
User 13 [Jun 16, 2014]: Calling something a manifold means that around each of its points there is a bit of space that looks like euclidean space. For example, if you look at the space up close around any point on a sphere shell, it kinda looks flat, like it was just a piece of the euclidean plane where we drew our circles and lines in high school geometry class. You know, the earth is round, but a small piece of it like the ground where your house is seems basically flat. In this sense, being a manifold is a local quality, yet there is the condition that all the little neighborhoods about all the points have to look like the same kind of euclidean space. It can’t look like the flat plane on one side of the manifold and look like a solid volume on the other. A balloon with a string attached to it wouldn’t be a manifold because the string is like a line, but the balloon itself is more like a planar surface or a volume, depending on if you count the interior. In this way, the lemniscate fails to be a manifold because even though it looks like a line in most of the places, it doesn’t look like a line at the one point where it intersects itself.
User 13 [Jun 16, 2014]: You might find something called a “CW Complex” to be a looser definition to incorporate some of your exotic, geometric ambitions.
Me [Jun 16, 2014]: It sort of does from what I’ve read so far. It does mention Hausdorff dimension, so there’s a good chance that this is what I’m thinking of… or relates to it. On a random but still math-related note, I’m writing a paper on a particular aspect of phi. Would you be interested in reading it once I’m done?
Me [Jun 16, 2014]: Oh I see ^_^ so if it essentially looks like a two-space locally, then it’s a manifold? In that case, fractals aren’t manifolds?
User 13 [Jun 16, 2014]: yeah basically, if it’s like a plane locally it’s a 2-manifold. If it’s like a line, then it’s a 1-manifold. If it’s like a volume it’s a 3-manifold, and so on. As for fractals, my gut feeling is that often they are not manifolds, especially if they aren’t uniformly dimensional. I guess what confuses me is that Hausdorff dimension is computed in a way that is very different from how we just sort of feel that something is one-dimensional or two-dimensional; hence, I think that there may not be a definite picture of how 1.6 dimensional euclidean space should look up close, at least not to the degree of familiarity that we feel about basic lines and planes.
Me [Jun 16, 2014]: Hausdorff dimensions are calculated by (log m)/-(log r) where m is the number of copies made of itself and r is the scaling factor. Technically, when we talk about dimensions in general, we do refer to its Hausdorff dimension. That’s why I’m still skeptical about how fractals aren’t manifolds. I mean, wouldn’t one 1.26D fractal be classed among other 1.26D fractals? That is to say, wouldn’t they be manifolds in the same class as other 1.26-manifolds of sorts? They will locally appear similar to one another, so why wouldn’t they be manifolds?
User 13 [Jun 16, 2014]: the problem is that you don’t have a 1.26-dimensional euclidean space to serve as a reference for all the other spaces to which you want to compare it. If you try to put copies of the whole euclidean plane back into itself in a suitable way, you just end up with the whole plane that you had to start.
User 13 [Jun 16, 2014]: Also, Hausdorff dimension doesn’t really tell you enough about a shape to decide if it’s a manifold. Think again about the balloon with the string, ignoring the air inside. The Hausdorff dimension for the whole thing is going to be 2, but it is still not a 2-manifold because the string part isn’t 2-dimensional.
Me [Jun 16, 2014]: Oh I see. Hmm… do you know of any good topology books I should read? Something a high school kid with some advanced mathematical knowledge would understand? All the books I find are either too simple or too complicated (for my current level of understanding).
User 13 [Jun 17, 2014]: No, I think most books target grad students and beyond. I didn’t really understand much of it until grad school, but you seem to be eager about it in ways that I wasn’t when I was young. You might just go to your favorite university web site and find out what books they’re using for their undergraduate topology course..
Me [Jun 17, 2014]: I’m aiming for Princeton for a Topology and Cosmology degree. They’re #1 and #3 in one or the other.
User 13 [Jun 17, 2014]: If I could do it over again, I would have studied more algebraic geometry.
Me [Jun 16, 2014]: But what if you put a different non-integer n-manifold into k-space such that n<k or n>k? Wouldn’t it differ depending on how it’s placed in there, sort of like trying to shove a cube through a plane?
User 13 [Jun 17, 2014]: Putting a higher-dimensional space into a lower one is called a projection. Often it ruins structure because it’s like compressing a thickness down to a point. Putting a lower-dimensional space into a higher one is an immersion. It doesn’t really change anything…a sphere is a sphere whether you’re thinking of it immersed in R^3 or or larger spaces. If you project a cube into a plane you might end up with just a square, but I suppose you could project it a different angles.
Me [Jun 17, 2014]: I know that; but what would its shadow look like? How could one take a non-integer dimension shadow? A sphere will look like a circle when projected onto R^2. In a similar way, what would, say, the Menger Sponge look like when projected into a Sierpinski Triangle space?
User 13 [Jun 17, 2014]: OK so the naive way of doing something like this would be to view the sponge as a subspace of R^3 and the triangle as a subspace of R^2, then you really just focus on projecting R^3 onto R^2 so that whatever information that you’re putting inside the space gets projected along with it. This actually seems more like what computational geometers call–I think–constructive solid geometry (CSG) where you are basically just flattening the sponge and taking its intersection with the triangle. You actually might be able to compute it with a program like Open-SCAD, which is free. If you ask me though, it seems like an ugly mess.
User 14 [Jun 17, 2014]: i don’t think I could answer this question more elegantly than [User 13] answered, though I don’t agree with his assertion that rendering a fractal-like manifold would remove the complexities that make fractals so interesting, it’s just a matter of figuring out HOW to do it.
Me [Jun 17, 2014]: That’s what I was wondering, boy. I was wondering HOW it was possible and what it would look like.
Well, NOW I know what User 13 was talking about. In fact, there’s a page on it in my copy of 1990 edition of Kenneth Falconer’s Fractal Geometry: Mathematical Foundations and Applications book. Page 84, to be exact. User 14 is not my boyfriend… but he is a sort of FWB. My heart belongs to math, though.