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 Posted: Fri Nov 16, 2007 1:32 pm
What donut hole? And I ask that in all seriousness.
The donut hole needs no physical existence; the only reason we think there is a hole is because the model is flawed.
The problem of models such as this is that it's hard to visualize without the ambient space; all surfaces we see as embedded in three-dimensional Euclidean space, but such an embedding inevitably adds things that we don't want. I personally don't really know how to deal with this problem except by resorting to pure formalistic descriptions, since all visualizations are inevitably embedded in an ambient three-dimensional space.
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Posted: Fri Nov 16, 2007 5:08 pm
Layra-chan What donut hole? And I ask that in all seriousness. The donut hole needs no physical existence; the only reason we think there is a hole is because the model is flawed. The problem of models such as this is that it's hard to visualize without the ambient space; all surfaces we see as embedded in three-dimensional Euclidean space, but such an embedding inevitably adds things that we don't want. I personally don't really know how to deal with this problem except by resorting to pure formalistic descriptions, since all visualizations are inevitably embedded in an ambient three-dimensional space. (omg wow, a post! heart ) I meant that if it could be considered a torus, there'd be that "hole" thing in the middle, like the hole in a donut. I was wondering what would be in that 'empty space' per se, but then it hit me, it's ALL space. So... yeah? What do you mean, formalistic descriptions? Like models? I see that it would be hard to explain a possibly endless dimensional space of vast semi-emptiness as a three dimensional model. I started thinking about something else, on the same note. How would it get a shape? Say it was a torus. Why would it be that shape? It reminds me of a quasar, sorta. Like there's a black hole like thing in the middle sucking it into a revolving disk or something, I'm unsure. I wish I knew what I was talking about... gonk
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Posted: Wed Nov 21, 2007 6:29 pm
The problem is that we are inside of the space that we are considering. Humans in space. Consider humans in torus. If we are actually living in the torus (the surface of the torus, I mean, as 2-dimensional beings) we do not "see" a hole. There is no possible way for us to know that a hole is there; we can't reach up out of the surface and go "wow, there's nothing in the middle".
This is what Layra means by "ambient space." When we embed (visualize) the torus in three dimensions, it has a big gaping hole in the center. But from the intrinsic perspective of the torus itself, from the perspective of people who would live on the torus, there's no such thing. Similarly, we must consider the intrinsic geometry of space, since we, you know...live in it. Even if space has an embedding in a higher-dimensional system, it's not like we can lift ourselves into that system and see what it looks like.
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Posted: Fri Nov 23, 2007 8:22 pm
That's true. But since space is supposed to be infinite (right?), the "surface" area would be infinite, and the space around it infinite, we'd never really know...
I'm probably misunderstanding. D:
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Posted: Sat Nov 24, 2007 6:39 pm
Well, in the language of manifolds (the sense that we live inside something that we can never get a glimpse of the entirety of), there are a few invariants that we should be able to measure. Total volume would be one of them. The problem is that many of these invariants would require us to measure some data at every point in the universe. Others are more local, like curvature, and those we can compute. But in terms of the "total" geometry of the universe...no, we really don't have much of a chance of finding anything out.
This includes (probably) figuring out whether or not space is embedded in another manifold, i.e. whether there is "space outside of space". Mathematically, it's a nonsensical question.
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Posted: Mon Nov 26, 2007 1:10 pm
We don't actually know that space is infinite; we're pretty sure that it doesn't have a boundary, the same way that the Earth doesn't have an edge that you can fall off of, but as for being infinite in volume we can't really tell. As Dragon pointed out, most of what we know is strictly local, thanks to the finiteness of the speed of light and whatnot.
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Posted: Fri Nov 30, 2007 1:41 pm
Speaking of the finiteness of the speed of light, we wouldn't really be able to "see" the edge (boundary) of the universe anyway. If the "edge" existed, and there was a photon that started at the edge at the beginning of the universe, it would still have only traveled a finite distance between then and now. If the boundary is farther away from us than (speed of light)x(age of universe), neglecting curvature and expansion and etc, we wouldn't even be able to know it was there.
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