Talk:Gabriel's horn

Not paradox

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 piece of metal can also have those two quantities... 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 completely different paradox. btw, I am changing nothing as this is my own personal opinion... --Llamatron 06:54, 18 Aug 2004 (UTC)

c1c2c3c4c- it is not a paradox, just flawed logic. as y approaches x it never equals o it is always .00000001, .00000000001 etc thus volume is created infinitely. even after fixing that mathematical mistake, infinity is a physical impossibility the entire idea of this being a paradox even assuming they hadn't flubbed the math is stupid. — Preceding unsigned comment added by 2601:245:CF01:7D2A:5C09:A240:3837:28B8 (talk) 12:39, 13 June 2015 (UTC)

Paint without thickness

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)

The paint thing is taken off, but you CANNOT fill it with infinitely thin paint. (as if you could even see it). It would take FOREVER for you to pour paint into it, thus you could pour 20 times the amount it should hold, and it STILL wouldn't fill. Assuming the end of the horn itself generates gravity. (plus, in a more humorous note, Gabriel was therefore blowing the horn before he even existed.) --74.134.8.244 23:26, 10 May 2007 (UTC)

I am a novice. However, one thing strikes me. PI being transcedental we can never find out how much paint is needed to fill it(otherwords we do not know the volume but we have a name for what we do not know: PI). Therefore infinite area might suggest absurdity and not unpaintable infinate surface. We may go on and conjuncture that 'either of the infinite area-volume pairs must have irrational value'. ~rAGU (talk)

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)

Of course you will have trouble evaluating that expression in brackets at 1. --Spoon! 22:30, 12 September 2007 (UTC)

Why? It looks to me like [ln(1+sqrt(2)) - sqrt(2)] or [-0.5328...] 00:35, 24 January 2008 (UTC) —Preceding unsigned comment added by 81.159.28.152 (talk)

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. ${1 \over 1}+{1 \over 1}+{1 \over 1}+{1 \over 1}+{1 \over 1}+\cdots$ is obviously infinite (you are adding 1 each time) and so a divergent series
2. ${1 \over 1}+{1 \over 2}+{1 \over 4}+{1 \over 8}+{1 \over 16}+\cdots$ is obviously finite as you never get above 2 and so a convergent series
3. ${1 \over 1}+{1 \over 2}+{1 \over 3}+{1 \over 4}+{1 \over 5}+\cdots$ (the harmonic series) is in fact infinite
4. ${1 \over 1}+{1 \over 4}+{1 \over 9}+{1 \over 16}+{1 \over 25}+\cdots$ 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 suppose I'm simply going to have to accept that this has been figured out by better brains than mine. It is counterintuitive that a shape of infinite length could have less than infinite volume.

*Septegram*Talk*Contributions* 20:35, 27 February 2007 (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)

Mistake corrected

A limit was evaluated as being equal to infinity which is nonsensical; I rephrased it for correctness. Jake 06:52, 27 November 2006 (UTC)

History

The article could be improved by adding a bit of history to it. Perhaps a whole section with it. When and where did Evangelista Torricelli first describe Grabriel's Horn? And the like. --Bilgrau 08:28, 28 March 2007 (UTC)

Minor change

At the end of Mathematical Definition it says "...it was considered paradoxical as, by rotating an infinite _area_ about the x-axis, an object of finite volume is obtained.". I changed it to read 'curve' instead of 'area', because building a solid of revolution is rotating a curve, not an area, the area is a mere consequence of the rotating curve. If anyone disagrees I'd like to hear their opinion, please. Christophe Lasserre 16:23, 16 November 2007 (UTC)

Infinite Sums of Infinity Paradox

I have removed the following as it looks wrong --Rumping (talk) 22:56, 16 May 2010 (UTC)

Another interesting paradox of Gabriel's Horn is the "Infinite Sums of Infinity" paradox. Since Gabriel's horn is the function of $1 \over x$ rotated about the x-axis, its volume can be estimated by dividing the rotation into slices and adding the sum of their volumes. By making the slices infinitely thin, their volume approaches the definite integral of $1 \over x$ from 1 to a.

$V=\int _{1}^{a}{1 \over x}\mathrm {d} x=\ln a$

Taking the limit as a approaches infinity yields infinity itself. Therefore, the paradox forms that an infinite sum of infinities equals a finite number - in this case, $\pi$.

Yes, it is wrong. The slice radius is $r={\frac {1}{x}}$, so its side area is $\pi \cdot r^{2}={\frac {\pi }{x^{2}}}$, not ${\frac {1}{x}}$. Consequently, as the thickness is $dx$, the slice volume is $dV={\frac {\pi }{x^{2}}}dx$, not ${\frac {1}{x}}{dx}$, and eventually

$V(a)=\int \limits _{x=1}^{x=a}dV=\int \limits _{1}^{a}{\frac {\pi }{x^{2}}}{dx}=\pi \left[-{\frac {1}{x}}\right]_{1}^{a}=\pi \left[\left(-{\frac {1}{a}}\right)-\left(-{\frac {1}{1}}\right)\right]=\pi \left(1-{\frac {1}{a}}\right)$

CiaPan (talk) 11:40, 6 August 2015 (UTC)

Diagrams

They would be better if they ranged x>=1 rather than x>0. I don't have the tools to fix these. -- SGBailey (talk) 22:28, 11 July 2010 (UTC)

Paint

Gabriel's horn has an infinite surface area, so in theory you could not paint one. But since Gabriel's Horn has a volume of π cubic units, one could supposedly fill it with π cubic units of paint. If you did this, and then emptied the horn, would it be painted? —Preceding unsigned comment added by Pandamonia (talk • contribs) 20:59, 7 November 2010 (UTC)

I wondered the same thing the first time reading this article. Back then it actually had the paint analogy contained in the article. Thinking strickly in theory on your question and it can really have you in circles!

More realistically though, I'm gonna say "no" because any substance sufficiently thin to have a chance of "painting" the horn (it would need to be infinitely thin) would cease to have any properties one would associate with a liquid or even matter, let alone "paint" and therefore you couldn't even begin to fill anything with it.

And of course not only couldn't you paint it in theory, but also not in practice, regardless if the horn had a truely infinite area or not. Any substance that one could qualify as being "paint", would be far to thick to reach the tiny recess of the narrow part of the horn. Indeed, the molecules themselves would be too large. Conversely, consider filling the finite volume and shape of the horn. It would be very easy to fill such a shape with paint even if it did have an infinite volume! The paint would dam itself at some narrow point and then fill from that point up.

The paint analogy was once in the article and while its a cute way of illustrating the paradox, it simply offered too many problems and was ultimately edited out. See above discussion at top of this talk page on the subject. But to answer your question, "If you filled the horn with paint and then emptied it, would it be painted?" In theory? Well, both yes and no I suppose because thats the paradox after all. Racerx11 (talk) 01:16, 28 December 2010 (UTC)

I don't think we should be discussing the paint using concepts like molecules and friction, etc. Those come from the real world, not mathematics. What did Euclid know about molecules? The idea that for any quantity of water "x" there is a quantity of water that is "x/2" is a perfectly reasonable assumption for humans to have made until experimental evidence contradicted it. No revision of mathematics was necessary. We merely accept that mathematical things don't work like real things, and keep going. We fill up Euclidean planes (which have zero thickness) with lines that have zero thickness, which contain points of zero anything. A sphere's surface has zero thickness. Mathematics is FULL of things that can't exist in the physical world. Why should "physical impossibility" be permitted as a discussion-point for this horn when it's not a discussion-point for zero-thickness things that are all over the place in mathematics? Either the area under the curve is finite or it isn't. If it's finite, then it's perfectly reasonable to talk about painting this horn's interior using a finite amount of paint.

I have tried to find a source to document one method for painting the EXTERIOR of this horn. Simply make a horn that was built spinning the graph "y=2/x". Every point on this curve is twice as far from the x-axis as the same point on the "y=1/x" curve having the same x-value. Fill the larger horn with the finite amount of paint that we know fills it. Insert the smaller horn all the way into the interior of the larger horn. (It's even possible to calculate the finite volume of paint that will be forced to overflow the flared rim of the larger horn by the displacement.) Let it sit for awhile. Remove the smaller horn from within the larger horn. It will come out having been painted by having been dipped. This method is obvious but I'm just not finding it anywhere. Please do the research and add it in.2600:1700:6759:B000:1C64:8308:33BC:E2D6 (talk) 08:54, 7 November 2023 (UTC)Christopher Lawrence Simpson

Cissoid of Diocles

I'm fairly certain the bit about the cissoid of Diocles is wrong. If the cup has infinite height, it must have infinite area, just like Gabriel's Horn. The book referenced is on Google Books, although the images have been removed which doesn't help. This Stack Exchange discussion also throws doubt on the claim of finite area. 109.155.174.230 (talk) 21:01, 15 October 2011 (UTC)

A German translation of the book, Julian Havil: Verblufft, pages 85-87 shows an image of a cissoid $y^{2}={\frac {x^{3}}{1-x}}$ rotated around the y axis and claims the same thing (infinite volume enclosed by a finite-area surface) without giving a proof. The author cites a portion of a letter from Sluse to Huygens, dated 12 April 1658. This letter can be found in Oeuvres complètes de Christiaan Huygens (1888), page 167-168. Translating their writing from Latin, Huygens and Sluse discuss the space between the asymptote and the cissoid, which they find to be finite (see Hans Niels Jahnke: A history of analysis, pages 60-61), so rotation around the y axis is a wrong interpretation. I will delete the section from the article, because the (mis)information presented therein appears to be the result of a misunderstanding. Olli Niemitalo (talk) 14:44, 6 November 2011 (UTC)

As far as I can tell the point with the Huygens/de Sluse analysis of the cissoid of Diocles is that the area between the cissoid and the line $x=1$ is finite. If this area for non-negative $y$ is rotated about the line $x=0$, then the resulting volume is also finite and looks like a glass tumbler or jar infinitely extended in one direction. Meanwhile the volume inside the rotated cissoid is clearly infinite, and it is the comparison of these two volumes which is interesting. The area here is not a surface area as in Gabriel's Horn, but in effect a cross-sectional area.--Rumping (talk) 01:41, 26 January 2012 (UTC)

and now there is an image of this container Rumping (talk) 09:08, 9 November 2021 (UTC)

New Further reading section

The material comes from elsewhere. Paradoctor (talk) 23:43, 14 November 2013 (UTC)

Capitalization of "horn"

The title and most of the external links spell "Horn" capitalized. The article lowercases it as "horn." I think we should settle on one spelling. What should it be? EpsilonCarinae (talk) 18:50, 13 June 2020 (UTC)

Title

As far as I can determine, only authors of recreational mathematics books, and the odd textbook, use the name "Gabriel's horn". Even that is a late 20th and 21st century affectation; there being nothing before the 1980s that I can find.

It isn't known as that in most places; in the 17th to 19th century everyone just called it the "solidum hyperbolicum acutum" ("solido hyperbolico acuto" being the ablative case, of course) or translations of the same into Italian/French/English/German/whatever. We've got a published source that has researched the names into the 18th century, and I can confirm from my own findings that mathematics dictionaries into the 19th century continued likewise. Barlow's 1814 New Mathematical and Philosophical Dictionary has "Hyperbolicum Acutum", for example. "acute hyperbolic solid" is even given in Julian Havil's controversial-to-this-talk-page book, on page 83. (I hope that this conotroversy is addressed now, by the way, by the mathematics professor that not only made the same error as Havil, but documented making that error.) The original Latin "solidum hyperbolicum acutum" can be found even in this century in non-recreational mathematics books; Steven G. Krantz and Brian E. Blank's 2006 book on single variable calculus uses that name (on page 620), for example.

It is somewhat saddening that the two names first introduced for this are not the thing's proper name, promoting a false notion of what its primary name is, and a supposed connection to Christianity that isn't even right. (The people who back in the 17th century tried to connect this with religion almost certainly knew their own religion well enough to know that Gabriel isn't stated to have a horn. They never called it that, or a trumpet.)

Jonathan de Boyne Pollard (talk) 12:08, 27 September 2021 (UTC)

Section "Apparent paradox" needs rewriting

The section Apparent paradox is not well written and would greatly benefit from rewriting. 2601:200:C000:1A0:B513:B27B:407D:C1BB (talk) 18:27, 5 April 2022 (UTC)