Talk:Gabriel's Horn

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Well, it seems as though in the context of the horn, the paradox mentioned defeats itself. A finite volume of paint could have an infinite surface area, if a peice of metal can also have those two quantites... Either I am missing something, or the original creators of the paradox were missing something (I shall not say what). Does anyone have any external information about the "paradox"? The time argument seems to be of a completly different paradox. btw, I am changing nothing as this is my own personal opinion...

--Llamatron 06:54, 18 Aug 2004 (UTC)

The final sentence reads "If the paint is considered without thickness, it would further take infinitely long time for the paint to run all the way down to the "end" of the horn." To me it seems irrelevant how much time it would take to paint the horn (since I'm not paying anyone to do it ;). The relevant point is that if the paint is considered to be without thickness, then any volume of paint can cover any surface area. Matt 15:53, 24 Dec 2004 (UTC)

Unfortunately the paint conceptualisation, while clever, misses the essence of the paradox. The problem rests on the apparent incompatibility of the formulas of area and volume where infinite quantities are concerned, at least in some circumstances. Coming up with an infinitely thinnable paint is obviously a bit of a cheat. However, if you're going to run with the paint thing, then the infinite time needed to complete the coverage does become important, because it's actually the paradox reasserting itself in a restated way - the reason the time required is infinite is simply that the length (thus area) of the horn is infinite, and there's no getting around that. - toh 01:23, 2005 Mar 10 (UTC)

I do not understand the infinite time or "length (hence area)" points. We know that "length infinite, so volume infinite" would be wrong. We could come up with a another horn based on y=1/x^2 where both the area and the volume would be finite, but the length and time would still be infinite; somehow I find that less surprising. --Henrygb 16:49, 17 Mar 2005 (UTC)

It seems an odd method of explanation to me also. Since we seem to be in agreement, I have removed it and added a resolution to the paradox that hopefully explains things in a more clear-cut way (I think that the best way to illustrate this would be with some diagrams; hopefully someone will come along and give it a go). - hitman012 15:00, 10 September 2006 (UTC)

[edit] Exact integral

It is not needed for the proof as we have

$A = 2\pi \int_1^a \frac{\sqrt{1 + \frac{1}{x^4}}}{x}\mathrm{d}x > 2\pi \int_1^a \frac{\sqrt{1}}{x}\ \mathrm{d}x = 2\pi \ln a$

but some people might be interested to note that

$A = 2\pi \int_1^a \frac{\sqrt{1 + \frac{1}{x^4}}}{x}\mathrm{d}x = 2\pi \ln a + \pi \left[\ln \left( 1 + \sqrt{1 + \frac{1}{x^4}} \right) - \sqrt{1 + \frac{1}{x^4}} \right]_1^a$

--Henrygb 01:43, 8 May 2006 (UTC)

[edit] Readability

I'm sure this is a fascinating subject. Do you suppose this could be rewritten so the average layperson (say, with a US high school education) could have the foggiest idea of what it means? I'm trying to understand how an object with finite surface area could have infinite volume, and unfortunately the information in the article isn't helping. Septegram 19:01, 7 September 2006 (UTC)

What in particular is the problem? The horn extends an infinite length to the right. So measuring from the left to a particular point, its surface area and volume both increase as the point moves to the right. Because of its particular shape, the surface area increases without limit, while the volume does not exceed a particular number. You can show this by doing the calculations. If you find this helpful, put it in the article. --Henrygb 21:11, 7 September 2006 (UTC)

No offense intended, but you've got to be kidding. "What in particular is the problem?"? Your response indicates to me that you are probably either a professional or enthusiastic amateur in a field that uses advanced mathematics on a regular basis. Unfortunately, it's all too easy when one is an expert to forget that there are plenty of people who aren't able to follow the steps that are obvious to you.

First, your summary is helpful, but no such summary exists on the main page.

Second, "you can show this by doing the calculations" is not useful for someone for whom runes would be clearer than the calculations on the main page. Please note that I did specify I was looking for an article that the average layperson (say, with a US high school education) could follow. I don't believe most high school students graduate with an understanding of calculus sufficient to follow the math on the main page (whether that's a crying shame or not is a separate subject). I know I didn't, and my challenge is compounded by the fact that I graduated college over 25 years ago.

Third, and here I'm doubtless going to display my ignorance, this appears to my untrained eye to be one of those cases of subsets of infinities. The diameter of the "horn" decreases but since the horn extends to infinity then it would seem that the volume it holds is also infinite.

So what I'm looking for is a plain-English explanation, I guess.

Septegram 22:22, 7 September 2006 (UTC)

It all depends how plain you want it. As you move right, you add a smaller amount each time. Consider the following series:

${1 \over 1}+{1 \over 1}+{1 \over 1}+{1 \over 1}+{1 \over 1}+\cdots$ is obviously infinite (you are adding 1 each time) and so a divergent series
${1 \over 1}+{1 \over 2}+{1 \over 4}+{1 \over 8}+{1 \over 16}+\cdots$ is obviously finite as you never get above 2 and so a convergent series
${1 \over 1}+{1 \over 2}+{1 \over 3}+{1 \over 4}+{1 \over 5}+\cdots$ (the harmonic series) is in fact infinite
${1 \over 1}+{1 \over 4}+{1 \over 9}+{1 \over 16}+{1 \over 25}+\cdots$ is in fact finite; it never exceeds $\pi^2 \over 6$

and the surface area is a bit like the third series while the volume is like the fourth. --Henrygb 22:52, 7 September 2006 (UTC)

I have added what I hope is a reasonable explanation. It could do with some work, but I made an effort to try and explain it without much reference to the mathematics involved; it's quite a difficult thing to understand without it, however. The diagrams on this page might help you understand the principle of a solid of revolution and grasp the explanation more easily. - hitman012 15:07, 10 September 2006 (UTC)

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Talk:Gabriel's Horn

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[edit] Exact integral

[edit] Readability

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