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Topic: Gabriel's Horn (Read 10746 times) |
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towr
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Re: Gabriel's Horn
« Reply #25 on: Jan 17th, 2007, 1:52am » |
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on Jan 16th, 2007, 5:19pm, rmsgrey wrote:| A possible real world example of a finite area enclosing an infinite volume: a black hole |
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I think that in non-euclidean space it needn't be a problem to have an infinite space enclosed in a finite area. (Our space might be non-euclidean, although there's not really a way to tell. )
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Icarus
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Re: Gabriel's Horn
« Reply #26 on: Jan 17th, 2007, 7:15am » |
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on Jan 17th, 2007, 1:52am, towr wrote:
I think that in non-euclidean space it needn't be a problem to have an infinite space enclosed in a finite area. |
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Yes. In fact, rmsgrey gave a nice example early in this thread.
By, the way - I am trying to modify your greasemonkey script so that it works on the "Modify" page as well as the new post page, but nothing is working. Do you know how to accomplish this?
Also - what's everyone's favorite javascript resource? I've finally decided it's time I learn it. (I'd prefer something that doesn't try to spoonfeed you "hello world" - I learned to code 30 years ago, and still do it regularly as part of my job today.)
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SMQ
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Re: Gabriel's Horn
« Reply #27 on: Jan 17th, 2007, 7:29am » |
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on Jan 17th, 2007, 7:15am, Icarus wrote:| Also - what's everyone's favorite javascript resource? |
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Well, I'm still fairly new to Javascript, but I primarily use the language spec for abstract concepts together with quirksmode for practical details. On the HTML side I've found the Gecko DOM reference to be substantially more useful in real-world situations than the W3C DOM specification. But, of course, your mileage may vary.
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towr
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Re: Gabriel's Horn
« Reply #29 on: Jan 17th, 2007, 9:21am » |
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on Jan 17th, 2007, 7:15am, Icarus wrote:| By, the way - I am trying to modify your greasemonkey script so that it works on the "Modify" page as well as the new post page, but nothing is working. Do you know how to accomplish this? |
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Changing the "include" in the header seems crucial.. Took me ten minutes to figure out that that was why none of my attempts worked.
And the previewbutton has a different name on that page.
I've reuploaded the updated script. In case you don't want to have to tinker with it yourself.
Btw, a great tool when making scripts to change pages is the DOM inspector in firefox, ctrl-shift-i
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| « Last Edit: Jan 17th, 2007, 9:32am by towr » |
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rmsgrey
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Re: Gabriel's Horn
« Reply #30 on: Jan 17th, 2007, 9:58am » |
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on Jan 16th, 2007, 6:04pm, Ulkesh wrote:
I'm not so sure what you mean by infinite volume in your above description. If you mean an arbitrary volume of mass collapsing into a singularity, this is only supported by physical models not backed-up by experiment at this scale (obviously). If you mean the time dilation described by general relativity when entering the gravitational field of a black hole, it of course depends on your frame of reference, so I'm still a little unclear...  |
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Some models of the effect of a gravitational singularity involve the stretching of space nearby to the extent that the singularity is infinitely far from the event horizon - for a 2D space-time, Gabriel's Horn would be an example - if you put a circle with diameter 2 symmetrically on the horn, it "encloses" an infinite area.
The dubious physical reality of such a model is why I only described it as a possible example rather than an actual one.
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rmsgrey
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Re: Gabriel's Horn
« Reply #31 on: Jan 17th, 2007, 10:13am » |
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on Jan 17th, 2007, 7:33am, Ulkesh wrote:
Well, I'm not about to spend the time verifying the maths, but the conclusions are quite clear - that, under whatever assumptions they're making, their model cannot support an infinite volume enclosed by a finite area - either the area becomes infinite, or the concept of volume breaks down, or they derive impossible topological properties of the enclosed space.
So it looks like black holes aren't an example after all - unless the model is wrong.
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Icarus
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Re: Gabriel's Horn
« Reply #32 on: Jan 17th, 2007, 10:38am » |
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In this case, they are defining a concept of volume, so the model cannot be wrong (at least in this sense). Rather, you would need to use a different concept of volume to get infinite volume with finite horizon in a spacetime.
on Jan 17th, 2007, 9:21am, towr wrote:
Changing the "include" in the header seems crucial.. Took me ten minutes to figure out that that was why none of my attempts worked.
And the previewbutton has a different name on that page. |
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The change to the include was the first thing I did, but I couldn't get it to work. Of course, part of the problem was that I was unaware that if you edit installed scripts, the edited version is not used until it is uploaded again. (Is there a way to do this without restarting the browser?) Anyway, your version works fine. Thanks again. One change I did make was to change the regex to match single character names as well, just in case I feel the need to insert special variable symbols such as  or  .
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towr
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Re: Gabriel's Horn
« Reply #33 on: Jan 17th, 2007, 11:14am » |
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on Jan 17th, 2007, 10:38am, Icarus wrote:| One change I did make was to change the regex to match single character names as well, just in case I feel the need to insert special variable symbols such as or . |
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hmm, yes.. When I made the script I hadn't really decided yet whether to use [] or $$. But the latter avoids a lot of problems (like array indices and all the usual tags), so I guess that restriction in the regex I put in isn't needed.
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ThudnBlunder
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Re: Gabriel's Horn
« Reply #34 on: May 13th, 2008, 7:14pm » |
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on Apr 22nd, 2003, 4:11pm, Icarus wrote:
As for objects with infinite volume but finite surface area - I don't think this is possible. One well-known property of a sphere is that it maximizes volume for a given surface area. If infinite volume in finite surface area were possible, then you could improve (infinitely so!) on a sphere. |
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Yet in 1658, just 15 years after Torricelli's discovery of Gabriel's Horn, it seems that Huygens and de Sluze showed that if the upper half of the cissoid y2 = x3/(1 - x), which has a vertical asymptote at x = 1, is revolved around the x-axis we get an infinite volume (with a goblet-shaped base) and yet its surface area is finite.
I checked the volume, but the surface area looks a bit messy.
I get dy/dx = (3 - 2x)/2x(1 - x) if anyone wants to finish it off. 
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| « Last Edit: May 13th, 2008, 8:23pm by ThudnBlunder » |
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Eigenray
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Re: Gabriel's Horn
« Reply #35 on: May 13th, 2008, 7:56pm » |
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on May 13th, 2008, 7:14pm, ThudanBlunder wrote:| if the upper half of the cissoid y2 = x3/(1 - x), which has a vertical asymptote at x = 1, is revolved around the x-axis we get an infinite volume (with a goblet-shaped base) and yet its surface area is finite. |
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Are you sure about that?
1+y'2 = (4-3x)/[4(1-x)3],
so
2 y  {1+y'2} > C*(1-x)-2,
and the integral diverges.
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ThudnBlunder
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Re: Gabriel's Horn
« Reply #36 on: May 13th, 2008, 8:10pm » |
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on May 13th, 2008, 7:56pm, Eigenray wrote:
Are you sure about that?
1+y'2 = (4-3x)/[4(1-x)3],
so
2y {1+y'2} > C*(1-x)-2,
and the integral diverges. |
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No, I'm not. My source is Nonplussed! Mathematical Proof of Implausible Ideas by Havil.
I quote the author: "Huygens and de Sluze added to the mathematical unease of the time by reversing the conditions: their solid has finite surface area and infinite volume."
Yet on the next page he quotes a letter from de Sluze to Huygens, describing the solid as a drinking glass that had small weight, but that even the hardiest drinker could not empty.
So is de Sluze referring to a drinking glass made from a finite volume of material, not surface area?
I hope you didn't use my dy/dx. I think it was wrong. 
I now get (3 - 2x)/2x(1 - x)
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| « Last Edit: May 13th, 2008, 8:39pm by ThudnBlunder » |
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Eigenray
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Re: Gabriel's Horn
« Reply #37 on: May 13th, 2008, 9:52pm » |
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Well the volume is infinite. But actually it's clear that the surface area of any such shape must be infinite as well: projecting onto a plane can only shrink surface area, but it covers the plane x=1 completely.
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ThudnBlunder
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Re: Gabriel's Horn
« Reply #38 on: May 14th, 2008, 7:43am » |
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on May 13th, 2008, 9:52pm, Eigenray wrote:| Well the volume is infinite. But actually it's clear that the surface area of any such shape must be infinite as well: projecting onto a plane can only shrink surface area, but it covers the plane x=1 completely. |
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Yes, and it looks like those (1-x) factors aren't going anywhere.
So it is a curious claim for a mathematician like Havil to make.
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brotherbandit
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Re: Gabriel's Horn
« Reply #39 on: Aug 10th, 2013, 10:46am » |
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Hi, I'm new.
on Jan 16th, 2007, 6:04pm, Ulkesh wrote:
I'm not so sure what you mean by infinite volume in your above description. If you mean an arbitrary volume of mass collapsing into a singularity, this is only supported by physical models not backed-up by experiment at this scale (obviously). If you mean the time dilation described by general relativity when entering the gravitational field of a black hole, it of course depends on your frame of reference, so I'm still a little unclear... |
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okay, first of all, time dilation is part of Einstien's Special relativity theory.
The theory of General relativity states that space and time are linked in a single surface called "spacetime" and in this surface, all matter floats around, creating disturbances due to their mass. This disturbance, theoretically, creates gravity.
A black hole is an object so massive that it's disturbance is often depicted as a gabriel's trumpet, whose sides are composed of spacetime.
Thus, based on this paradox, a black hole has an infinite amount of surface (i.e. spacetime) but a finite amount of disturbance (i.e. gravity)
on Jan 17th, 2007, 9:58am, rmsgrey wrote:
if you put a circle with diameter 2 symmetrically on the horn, it "encloses" an infinite area. |
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That "circle" would be the black hole's event horizon.
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| « Last Edit: Aug 10th, 2013, 10:53am by brotherbandit » |
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towr
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Re: Gabriel's Horn
« Reply #40 on: Aug 10th, 2013, 12:54pm » |
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on Aug 10th, 2013, 10:46am, brotherbandit wrote:Welcome
Quote:| okay, first of all, time dilation is part of Einstein's Special relativity theory. |
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Notwithstanding, it's also part of general relativity as http://en.wikipedia.org/wiki/Gravitational_time_dilation
Special relativity tells us that the clocks of a a satellite run slower because of their speed (losing 7 s/d); general relativity tells us the clocks on earth run slower still because of the gravity well it sits in (losing us 45 s/d). With as end-result that the satellite's clock are actually running faster wrt us (by 38 s/d).
http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html
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