wu :: forums - Print Page


    
      
        wu :: forums
        (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi)
      

        riddles >> medium >> Gabriel's Horn
        
(Message started by: THUDandBLUNDER on Apr 21^st, 2003, 1:17pm)

Title: Gabriel's Horn
Post by THUDandBLUNDER on Apr 21^st, 2003, 1:17pm

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?

Title: Re: Gabriel's Horn
Post by THUDandBLUNDER on Apr 21^st, 2003, 1:32pm

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! :o

Title: Re: Gabriel's Horn
Post by aero_guy on Apr 21^st, 2003, 4:03pm

yup... math can be funny sometimes.

Title: Re: Gabriel's Horn
Post by James Fingas on Apr 22^nd, 2003, 8:23am

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.

Title: Re: Gabriel's Horn
Post by THUDandBLUNDER on Apr 22^nd, 2003, 8:52am

Quote:

so the whole integral is larger than the integral from zero to one of 1/sqrt(x), which is infinite.

But [smiley=smallint.gif]x^-1/2.dx = 2x^1/2 + c
= 2 in this case

Title: Re: Gabriel's Horn
Post by James Fingas on Apr 22^nd, 2003, 1:20pm

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.

Title: Re: Gabriel's Horn
Post by Icarus on Apr 22^nd, 2003, 4:11pm

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.

Title: Re: Gabriel's Horn
Post by redPEPPER on Apr 23^rd, 2003, 6:04am

Makes me think of the Menger sponge.

http://ems.gphys.unc.edu/nonlinear/fractals/images/sponge.gif

As the number of iterations --> oo, the volume --> 0 and the surface --> oo.

Title: Re: Gabriel's Horn
Post by THUDandBLUNDER on Apr 25^th, 2003, 12:08pm

An excellent explanation, Icarus.

Title: Re: Gabriel's Horn
Post by rmsgrey on Jan 14^th, 2007, 8:04am

on 04/22/03 at 16:11:24, 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.

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.

Title: Re: Gabriel's Horn
Post by Locke64 on Jan 14^th, 2007, 11:12am

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.

Title: Re: Gabriel's Horn
Post by THUDandBLUNDER on Jan 14^th, 2007, 11:23am

on 01/14/07 at 11:12:45, 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.

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. ;)

Title: Re: Gabriel's Horn
Post by SMQ on Jan 15^th, 2007, 6:55am

on 01/14/07 at 11:12:45, Locke64 wrote:

Do those codes work for everyone else?

They presently only work for people using the Firefox (http://www.mozilla.org/firefox) web browser with the Greasemonkey (https://addons.mozilla.org/firefox/748/) extension and a user script (http://dwarfrune.com/~smq/wuforumssymboldisplay.user.js) I wrote to display those codes (mostly for viewing old posts where they were used).

--SMQ

Title: Re: Gabriel's Horn
Post by Icarus on Jan 15^th, 2007, 7:21pm

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.

Title: Re: Gabriel's Horn
Post by towr on Jan 16^th, 2007, 12:48am

I made myself a script that turns tags like $int$ $surd$ into proper images in the message text
like: http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/int.gif http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif

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 $sqrt$ :P

Title: Re: Gabriel's Horn
Post by hiyathere on Jan 16^th, 2007, 10:49am

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.

Title: Re: Gabriel's Horn
Post by towr on Jan 16^th, 2007, 1:14pm

on 01/16/07 at 10:49:02, 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.

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.

Title: Re: Gabriel's Horn
Post by Grimbal on Jan 16^th, 2007, 2:41pm

on 04/21/03 at 13:32:41, 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! :o

I cannot reproduce your results. What do you take in the interval [0,1]?

And why don't soap bubbles take this shape?

Title: Re: Gabriel's Horn
Post by THUDandBLUNDER on Jan 16^th, 2007, 4:46pm

on 01/16/07 at 14:41:14, 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?

Bit late to object now, Grimmy - my idea was refuted by James Fingas three years ago. ;)

Title: Re: Gabriel's Horn
Post by rmsgrey on Jan 16^th, 2007, 5:19pm

A possible real world example of a finite area enclosing an infinite volume: a black hole

Title: Re: Gabriel's Horn
Post by Ulkesh on Jan 16^th, 2007, 6:04pm

on 01/16/07 at 17:19:31, rmsgrey wrote:

A possible real world example of a finite area enclosing an infinite volume: a black hole

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... :-/

Title: Re: Gabriel's Horn
Post by Icarus on Jan 16^th, 2007, 6:25pm

on 01/16/07 at 00:48:41, towr wrote:

Nice! That makes creating readable math posts a lot easier.

SMQ's script is still required for reading the old posts, though.

Title: Re: Gabriel's Horn
Post by THUDandBLUNDER on Jan 16^th, 2007, 7:18pm

on 01/16/07 at 18:25:27, Icarus wrote:

SMQ's script is still required for reading the old posts, though.

Yes, thanks a lot for those scripts, SMQ and towr.

Title: Re: Gabriel's Horn
Post by THUDandBLUNDER on Jan 16^th, 2007, 8:43pm

on 01/16/07 at 18:04:51, Ulkesh wrote:

...so I'm still a little unclear... :-/

Possibly wormholes. Welcome.....to the real world. ::)

Title: Re: Gabriel's Horn
Post by Grimbal on Jan 17^th, 2007, 1:00am

on 01/16/07 at 16:46:22, THUDandBLUNDER wrote:

Bit late to object now, Grimmy - my idea was refuted by James Fingas three years ago. ;)

:-[ <- "embarassed"

Title: Re: Gabriel's Horn
Post by towr on Jan 17^th, 2007, 1:52am

on 01/16/07 at 17:19:31, rmsgrey wrote:

A possible real world example of a finite area enclosing an infinite volume: a black hole

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. )

Title: Re: Gabriel's Horn
Post by Icarus on Jan 17^th, 2007, 7:15am

on 01/17/07 at 01:52:26, 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.

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.)

Title: Re: Gabriel's Horn
Post by SMQ on Jan 17^th, 2007, 7:29am

on 01/17/07 at 07:15:47, Icarus wrote:

Also - what's everyone's favorite javascript resource?

Well, I'm still fairly new to Javascript, but I primarily use the language spec (http://www.ecma-international.org/publications/standards/Ecma-262.htm) for abstract concepts together with quirksmode (http://www.quirksmode.org/js/contents.html) for practical details. On the HTML side I've found the Gecko DOM reference (http://developer.mozilla.org/en/docs/Gecko_DOM_Reference) to be substantially more useful in real-world situations than the W3C DOM specification (http://www.w3.org/DOM/DOMTR). But, of course, your mileage may vary.

--SMQ

Title: Re: Gabriel's Horn
Post by Ulkesh on Jan 17^th, 2007, 7:33am

If anyone out there is well-versed in general relativity (not me!), this might be of interest.

http://arxiv.org/PS_cache/hep-th/pdf/0508/0508108.pdf

Title: Re: Gabriel's Horn
Post by towr on Jan 17^th, 2007, 9:21am

on 01/17/07 at 07:15:47, 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?

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 (http://tcw2.ai.rug.nl/~towr/wwusym.user.js)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

Title: Re: Gabriel's Horn
Post by rmsgrey on Jan 17^th, 2007, 9:58am

on 01/16/07 at 18:04:51, Ulkesh wrote:

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.

Title: Re: Gabriel's Horn
Post by rmsgrey on Jan 17^th, 2007, 10:13am

on 01/17/07 at 07:33:23, Ulkesh wrote:

If anyone out there is well-versed in general relativity (not me!), this might be of interest.

http://arxiv.org/PS_cache/hep-th/pdf/0508/0508108.pdf

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.

Title: Re: Gabriel's Horn
Post by Icarus on Jan 17^th, 2007, 10:38am

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 01/17/07 at 09:21:26, 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.

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 http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/x.gif or http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/z.gif.

Title: Re: Gabriel's Horn
Post by towr on Jan 17^th, 2007, 11:14am

on 01/17/07 at 10:38:16, 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 http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/x.gif or http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/z.gif.

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.

Title: Re: Gabriel's Horn
Post by ThudanBlunder on May 13^th, 2008, 7:14pm

on 04/22/03 at 16:11:24, Icarus wrote:

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 (http://curvebank.calstatela.edu/diocles/diocles.htm) 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. :)

Title: Re: Gabriel's Horn
Post by Eigenray on May 13^th, 2008, 7:56pm

on 05/13/08 at 19:14:32, ThudanBlunder wrote:

if the upper half of the cissoid (http://curvebank.calstatela.edu/diocles/diocles.htm) 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.

Are you sure about that?
1+y'² = (4-3x)/[4(1-x)³],
so
2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gify http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif{1+y'²} > C*(1-x)^-2,
and the integral diverges.

Title: Re: Gabriel's Horn
Post by ThudanBlunder on May 13^th, 2008, 8:10pm

on 05/13/08 at 19:56:25, Eigenray wrote:

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)

Title: Re: Gabriel's Horn
Post by Eigenray on May 13^th, 2008, 9:52pm

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.

Title: Re: Gabriel's Horn
Post by ThudanBlunder on May 14^th, 2008, 7:43am

on 05/13/08 at 21:52:13, Eigenray wrote:

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.

Title: Re: Gabriel's Horn
Post by brotherbandit on Aug 10^th, 2013, 10:46am

Hi, I'm new.

on 01/16/07 at 18:04:51, Ulkesh wrote:

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 01/17/07 at 09:58:32, rmsgrey wrote:

if you put a circle with diameter 2 symmetrically on the horn, it "encloses" an infinite area.

That "circle" would be the black hole's event horizon.

Title: Re: Gabriel's Horn
Post by towr on Aug 10^th, 2013, 12:54pm

on 08/10/13 at 10:46:27, brotherbandit wrote:

Hi, I'm new.

Welcome

Quote:

okay, first of all, time dilation is part of Einstein's Special relativity theory.

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 7http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/mu.gifs/d); general relativity tells us the clocks on earth run slower still because of the gravity well it sits in (losing us 45 http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/mu.gifs/d). With as end-result that the satellite's clock are actually running faster wrt us (by 38 http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/mu.gifs/d).
http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html