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Topic: Gabriel's Horn (Read 10745 times) |
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ThudnBlunder
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Gabriel's Horn
« on: Apr 21st, 2003, 1:17pm » |
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Gabriel's Horn is the surface of revolution produced by rotating the curve y = 1/x around the x-axis for x >= 1.
oo
The volume = [smiley=smallint.gif][pi]y2.dx = [pi]
1
oo
The surface area = [smiley=smallint.gif]2[pi]y[smiley=surd.gif][1 + (y')2)].dx = oo
1
This leads to the paradoxical conclusion that, while Gabriel's Horn can be filled up with just [pi] cubic units of paint, an infinite amount of paint is needed to paint its surface!
What do you make of that?
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| « Last Edit: Jan 16th, 2004, 9:52am by ThudnBlunder » |
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ThudnBlunder
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Re: Gabriel's Horn
« Reply #1 on: Apr 21st, 2003, 1:32pm » |
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And if we rotate the curve y = x-1/2 around the x-axis in the interval [0,1]
I believe we get an infinite volume contained within a finite surface area! 
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| « Last Edit: Apr 21st, 2003, 2:01pm by ThudnBlunder » |
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aero_guy
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Re: Gabriel's Horn
« Reply #2 on: Apr 21st, 2003, 4:03pm » |
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yup... math can be funny sometimes.
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James Fingas
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Re: Gabriel's Horn
« Reply #3 on: Apr 22nd, 2003, 8:23am » |
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I don't think the surface area of y=x-1/2 rotated around the x-axis on x in [0,1] is finite. I get this integral:
x=1
S.A = INT pi*1/sqrt(x)*sqrt(1 + 1/(4x3))dx
x=0
But the second square root is larger than 1, so the whole integral is larger than the integral from zero to one of 1/sqrt(x), which is infinite.
Physically, I'm saying that the surface area is larger than that of the YZ plane with a radius-1 hole cut out of it.
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| « Last Edit: Apr 22nd, 2003, 8:23am by James Fingas » |
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ThudnBlunder
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Re: Gabriel's Horn
« Reply #4 on: Apr 22nd, 2003, 8:52am » |
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Quote:so the whole integral is larger than the integral from zero to one of 1/sqrt(x), which is infinite.
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But [smiley=smallint.gif]x-1/2.dx = 2x1/2 + c
= 2 in this case
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| « Last Edit: Jan 16th, 2004, 9:23am by ThudnBlunder » |
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James Fingas
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Re: Gabriel's Horn
« Reply #5 on: Apr 22nd, 2003, 1:20pm » |
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Thud,
Right. I guess I got screwed up there. What I should have said was:
sqrt( 1 + 1/(4*x3)) is larger than 1/sqrt(4*x3), so the integral is larger than the integral of 1/(2*sqrt(x)*sqrt(x3)) = 1/(2*x2).
That integral is -1/(2*x) + c, which is infinite in this case.
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Icarus
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Re: Gabriel's Horn
« Reply #6 on: Apr 22nd, 2003, 4:11pm » |
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Concerning Gabriel's Horn, and why it can be filled with a finite amount of paint, but not painted...
When you paint something, you apply a coat of (roughly) equal thickness everywhere. If you were to create a second boundary around the Horn, displaced perpendicularly from the original by a tiny amount dx, you would find that the volume between the two boundaries was infinite (since the volume ~= Surface Area * dx)
So how can the volume of the Horn be finite? Because it's thickness is not constant. For any value of dx, there is a point along the horn after which the paint on the horn becomes much thicker than the horn itself. (You can't put a constant thickness of paint on the inside of the horn. There isn't room.)
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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redPEPPER
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Re: Gabriel's Horn
« Reply #7 on: Apr 23rd, 2003, 6:04am » |
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Makes me think of the Menger sponge.
As the number of iterations --> oo, the volume --> 0 and the surface --> oo.
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ThudnBlunder
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Re: Gabriel's Horn
« Reply #8 on: Apr 25th, 2003, 12:08pm » |
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An excellent explanation, Icarus.
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| « Last Edit: Jan 16th, 2004, 9:05am by ThudnBlunder » |
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rmsgrey
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Re: Gabriel's Horn
« Reply #9 on: Jan 14th, 2007, 8:04am » |
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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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I guess it depends partly on your definition of an enclosed volume - if you define it as either of two disjoint regions separated by the surface provided, then the "outside" of any finite volume is an example. If you restrict it to the finite region of the two provided, then it's trivially impossible to ever enclose an infinite volume. If you specify it being the smaller of the two regions, then you run into problems of comparing the two infinities.
In any case, if you're in a non-Euclidean space, then it is sometimes possible to divide space into two infinite regions with a finite boundary - for example, the infinite surface of a cylinder with finite radius and infinite length can be divided into two infinite regions with a finite closed curve. Whether either counts as being enclosed by the finite curve is another matter.
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Locke64
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Re: Gabriel's Horn
« Reply #10 on: Jan 14th, 2007, 11:12am » |
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Do those codes (smiley=smallint.gif and smiley=surd.gif) work for everyone else? They don't work for me, and I don't see any reason to use them if they don't work.
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ThudnBlunder
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Re: Gabriel's Horn
« Reply #11 on: Jan 14th, 2007, 11:23am » |
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on Jan 14th, 2007, 11:12am, Locke64 wrote:| Do those codes (smiley=smallint.gif and smiley=surd.gif) work for everyone else? They don't work for me, and I don't see any reason to use them if they don't work. |
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No, they are not functional at the moment.
But don't worry, they should be working again when Willie Wu (the admin) finishes his PhD at Stanford. ;)
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SMQ
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Re: Gabriel's Horn
« Reply #12 on: Jan 15th, 2007, 6:55am » |
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on Jan 14th, 2007, 11:12am, Locke64 wrote:| Do those codes work for everyone else? |
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They presently only work for people using the Firefox web browser with the Greasemonkey extension and a user script I wrote to display those codes (mostly for viewing old posts where they were used).
--SMQ
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Icarus
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Re: Gabriel's Horn
« Reply #13 on: Jan 15th, 2007, 7:21pm » |
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Indeed - the thing by far that I miss most about the older version of YaBB was the math symbolry we've had (as all long time readers know, as I complain about it regularly  ). But SMQ's greasemonkey script works great. I strongly suggest setting yourself up with it, then visit the start of the 0.999... thread at the top of the medium forum. Then you will see what we used to be able to do.
The only problem with it is that occasional "fake codes" such as [fake] get translated to a non-existent picture, instead of text. But on these rare occasions, I can turn off the greasemonkey and refresh the page to see what it should look like.
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towr
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Re: Gabriel's Horn
« Reply #14 on: Jan 16th, 2007, 12:48am » |
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I made myself a script that turns tags like $int$ $surd$ into proper images in the message text
like:  
It adds an extra button next to "post" "preview" "reset" to convert the tags in the message. The advantage of course is that other users don't need a script to see the symbols.
http://tcw2.ai.rug.nl/~towr/wwusym.user.js
Maybe if I ever find the time and inspiration, I'll make it a bit better. Perhaps up to the point it can translate pseudo-latex formulas.. But more immediately that it recognizes $sqrt$ 
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| « Last Edit: Jan 16th, 2007, 1:03am by towr » |
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Ghost Sniper
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Re: Gabriel's Horn
« Reply #15 on: Jan 16th, 2007, 10:49am » |
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You know, doing math on this forum will be a lot easier if common symbols can be used, so it would be easier to keep track.
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towr
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Re: Gabriel's Horn
« Reply #16 on: Jan 16th, 2007, 1:14pm » |
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on Jan 16th, 2007, 10:49am, hiyathere wrote:| You know, doing math on this forum will be a lot easier if common symbols can be used, so it would be easier to keep track. |
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Well, there are common symbols. They're in http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/
It's just a problem of making them accesible. The board used to have that feature, but it disappeared with the upgrade to a new version.
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Grimbal
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Re: Gabriel's Horn
« Reply #17 on: Jan 16th, 2007, 2:41pm » |
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on Apr 21st, 2003, 1:32pm, THUDandBLUNDER wrote:And if we rotate the curve y = x-1/2 around the x-axis in the interval [0,1]
I believe we get an infinite volume contained within a finite surface area! |
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I cannot reproduce your results. What do you take in the interval [0,1]?
And why don't soap bubbles take this shape?
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ThudnBlunder
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Re: Gabriel's Horn
« Reply #18 on: Jan 16th, 2007, 4:46pm » |
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on Jan 16th, 2007, 2:41pm, Grimbal wrote:
I cannot reproduce your results. What do you take in the interval [0,1]?
And why don't soap bubbles take this shape? |
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Bit late to object now, Grimmy - my idea was refuted by James Fingas three years ago. 
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rmsgrey
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Re: Gabriel's Horn
« Reply #19 on: Jan 16th, 2007, 5:19pm » |
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A possible real world example of a finite area enclosing an infinite volume: a black hole
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Ulkesh
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Re: Gabriel's Horn
« Reply #20 on: Jan 16th, 2007, 6:04pm » |
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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'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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Icarus
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Re: Gabriel's Horn
« Reply #21 on: Jan 16th, 2007, 6:25pm » |
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on Jan 16th, 2007, 12:48am, towr wrote:I made myself a script that turns tags like $int$ $surd$ into proper images in the message text
like:
It adds an extra button next to "post" "preview" "reset" to convert the tags in the message. The advantage of course is that other users don't need a script to see the symbols.
http://tcw2.ai.rug.nl/~towr/wwusym.user.js |
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Nice! That makes creating readable math posts a lot easier.
SMQ's script is still required for reading the old posts, though.
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| « Last Edit: Jan 16th, 2007, 6:26pm by Icarus » |
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ThudnBlunder
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Re: Gabriel's Horn
« Reply #22 on: Jan 16th, 2007, 7:18pm » |
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on Jan 16th, 2007, 6:25pm, Icarus wrote:| SMQ's script is still required for reading the old posts, though. |
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Yes, thanks a lot for those scripts, SMQ and towr.
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| « Last Edit: Jan 16th, 2007, 8:40pm by ThudnBlunder » |
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ThudnBlunder
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Re: Gabriel's Horn
« Reply #23 on: Jan 16th, 2007, 8:43pm » |
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on Jan 16th, 2007, 6:04pm, Ulkesh wrote:| ...so I'm still a little unclear... |
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Possibly wormholes. Welcome.....to the real world.
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Grimbal
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Re: Gabriel's Horn
« Reply #24 on: Jan 17th, 2007, 1:00am » |
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on Jan 16th, 2007, 4:46pm, THUDandBLUNDER wrote:| Bit late to object now, Grimmy - my idea was refuted by James Fingas three years ago. |
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 <- "embarassed"
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| « Last Edit: Jan 17th, 2007, 1:01am by Grimbal » |
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