Horn O' Everything
April 15, 2004 7:35 AM Subscribe
Absolutely, The Universe Could Be Funnel-Shaped At an extreme enough point, you would be able to see the back of your own head. It would be an interesting place to explore - but we are probably too far from the narrow end of the horn to examine it with telescopes.
Frank Steiner’s Quantum Chaos group
It's fascinating to speculate - but does it matter? I'm not being rhetorical; I can't understand the value of knowing it's shaped like a doghnut, horn or giant spacekitten.
posted by Pericles at 8:19 AM on April 15, 2004
posted by Pericles at 8:19 AM on April 15, 2004
This is interesting. However, questions are always brought up in my mind whenever the large scale structure of the universe is being considered. Perhaps this is naive, but would knowing the topology perhaps explain the initial conditions (i.e. at the big bang)? I have always thought of the initial universe being like a single point- how then does something with a weird topology come out of that? Also, they talk about curvature of the universe in their paper on this; I get the impression that it is always assumed that the curvature, once it is measured, is constant for all the universe (or having some uniform property- for instance, no one ever suggests a spiral shape). Why would this necessarily be so?
(Ok, this non-cosmologist will stop asking naive questions.) Good stuff, mcgraw, thanks.
posted by jjray at 8:36 AM on April 15, 2004
(Ok, this non-cosmologist will stop asking naive questions.) Good stuff, mcgraw, thanks.
posted by jjray at 8:36 AM on April 15, 2004
Oddly, googling "picard topology" returns no results. I wonder if it's simply an analogue of Gabriel's Horn, which would explain the "finite volume" comment. My calculus students always start gibbering incomprehensibly when I tell them that something can have finite volume but infinite surface area. (If that doesn't confuse you immediately, think about painting the interior: you couldn't ever coat the inside surface with a brush, but you could pour in a 4-gallon can and fill the whole thing.)
posted by gleuschk at 9:08 AM on April 15, 2004
posted by gleuschk at 9:08 AM on April 15, 2004
From the paper by Steiner's group, they reference for the Picard topology another paper that talks about Picard groups. However, I cannot find a good explanation of the connection anywhere.
One issue about these apparent paradoxes is their utilization of the properties of continuity to exist. Gabriel's Horn assumes that everything is continuous. However, if we were to attempt to realize it physically, the paint we would use to fill it would not be continuous at a small enough scale, so even if we had a cone like that with "infinite" volume, enough paint would eventually fill it up because somewhere the local volume is more narrow than the paint. Another of these sorts of paradoxes (that I would love to see applied to a nugget of gold) is the Banach-Tarski paradox.
posted by jjray at 10:38 AM on April 15, 2004
One issue about these apparent paradoxes is their utilization of the properties of continuity to exist. Gabriel's Horn assumes that everything is continuous. However, if we were to attempt to realize it physically, the paint we would use to fill it would not be continuous at a small enough scale, so even if we had a cone like that with "infinite" volume, enough paint would eventually fill it up because somewhere the local volume is more narrow than the paint. Another of these sorts of paradoxes (that I would love to see applied to a nugget of gold) is the Banach-Tarski paradox.
posted by jjray at 10:38 AM on April 15, 2004
This is fascinating, but as an amateur astronomy/physics enthusiast, I'm a bit amused by how our ideas about the universe at large are continually changing as our ability to observe it improves. Every few years, someone comes out with a new theory based on the latest data that doesn't match up with the old theories...I look forward to future hypothesies stating that the universe is shaped like a star, a silly straw, and a turtle.
Of course, that's science for you...always changing when the "facts" do. ; )
posted by The Card Cheat at 10:59 AM on April 15, 2004
Of course, that's science for you...always changing when the "facts" do. ; )
posted by The Card Cheat at 10:59 AM on April 15, 2004
That's a different Picard Group than the one they're apparently talking about, jjray. Then's paper, referenced in the Steiner group's paper, describes what they mean by the Picard Group. It's just a three-dimensional analogue of the classical so-called "modular group", better known as PSL(2,Z). Then's paper is fairly reasonable, provided you're down with some complex analysis and are willing to ignore some buzzwords like "orbifold".
The Picard Group described in your MathWorld link is an invariant, rather than a uniquely defined group. It's a group attached to any order in an algebraic number field (or, actually, attached to any commutative ring -- MathWorld is rarely as complete in their descriptions as one might like). So the Picard group for the integers is zero, while the Picard group for the ring k[x,y,z]/(xy-z^n) is isomorphic to Z/nZ..
Also, Gabriel's Horn isn't really a paradox, just nonintuitive. You can get way more bizarre behavior from fractals (like Sierpinski's gasket), but Gabriel's Horn just points out that "common sense" doesn't work so well when you start talking about infinite quanties/objects. Banach-Tarski is similar, though much more sophisticated, since it calls for the Axiom of Choice (which might be regarded as the grandaddy of all ways in which infinity is slippery stuff).
posted by gleuschk at 11:21 AM on April 15, 2004
The Picard Group described in your MathWorld link is an invariant, rather than a uniquely defined group. It's a group attached to any order in an algebraic number field (or, actually, attached to any commutative ring -- MathWorld is rarely as complete in their descriptions as one might like). So the Picard group for the integers is zero, while the Picard group for the ring k[x,y,z]/(xy-z^n) is isomorphic to Z/nZ..
Also, Gabriel's Horn isn't really a paradox, just nonintuitive. You can get way more bizarre behavior from fractals (like Sierpinski's gasket), but Gabriel's Horn just points out that "common sense" doesn't work so well when you start talking about infinite quanties/objects. Banach-Tarski is similar, though much more sophisticated, since it calls for the Axiom of Choice (which might be regarded as the grandaddy of all ways in which infinity is slippery stuff).
posted by gleuschk at 11:21 AM on April 15, 2004
Uh. Oh, I was looking at Then's paper, and I didn't look closely at the MathWorld link. Sorry, I should have been reading more closely instead of assuming "same name means same thing". I should have been more careful, because I have found other shortcomings in the MathWorld descriptions in the past as well. Thanks for the clarification on that.
Also, and I am not trying to quibble, Gabriel's Horn is a paradox in the sense that it is seemingly contradictory, though is actually true. In the sense of essentially self-contradictory, however, I agree that it is not.
posted by jjray at 11:52 AM on April 15, 2004
Also, and I am not trying to quibble, Gabriel's Horn is a paradox in the sense that it is seemingly contradictory, though is actually true. In the sense of essentially self-contradictory, however, I agree that it is not.
posted by jjray at 11:52 AM on April 15, 2004
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posted by mcgraw at 7:37 AM on April 15, 2004