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Thought Experiments in Mathematics: Gabriel's Horn
- Jun8 10y agoThis answer on Math SE illustrates the concept of finite volume-infinite area quite nicely (http://math.stackexchange.com/a/14632/14643 http://math.stackexchange.com/a/14632/14643). Note that the converse is proven only for surfaces of revolution and for differentiable functions. Finding a pathological counter example that violates these assumptions would be interesting.
- erostrate 10y agoMy intuition is that the converse is impossible in general. There is an axis along which an object with infinite volume is infinite, and taking cross sections along this axis you should be able to prove that the object has infinite area.
- Chinjut 10y agoThe converse is impossible in any context where the isoperimetric inequality is applicable.
- tgb 10y agoGood point. Though the Wikipedia article on it gives the most general form as requiring our region to have closure with finite Lebesgue measure. This is necessary of course by the example another corner brought up about swapping the inside and outside. But that means it exactly can't rule out the existence of regions with infinite volume (outside and inside) and finite surface area!
- Someone 10y agoSection 7.1.3 of the article has a proof that such a counterexample cannot exist. Of course, this being mathematics, changing the rules can produce a counterexample. A simple way of doing that is by declaring the inside of the shape the outside and vice versa. I expect one could construct a counterexample around a black hole, too.
- aoki 10y agoah, thank you. i did not really understand why this was really any more "surprising" than the 2D analogue (finite area under an infinite curve). there are at least some other people who think it's not :-)
- kps 10y agoThe link is an image of a print-formatted copy of the Wikipedia article https://en.wikipedia.org/wiki/Gabriel%27s_Horn https://en.wikipedia.org/wiki/Gabriel%27s_Horn which is more easily read directly on many devices.
- zeroer 10y agoThe most interesting thing about it is that at one point people thought it was a paradox.
- Someone 10y agoIt took lots of very intelligent mathematicians to realize that infinities are weird. Even today, it takes quite some schooling to come to terms with some of the _easy_ examples. For example, I expect that easily over 95% of university graduates disagree with the statement "There are as many rational numbers as there are prime numbers.". I even fear that is true for the easier to believe "0.9999999... = 1"
- deleted 10y ago[deleted]
- llamaz 10y agoThere probably is _some_ mathematical formalism out there where there are less prime numbers than natural numbers. For example, most mathematicians would agree that there are as many reals in [0, 1] as there are in the real line. But according to a different definition of size, namely the most famous measure from measure theory, the Lebesgue measure --- the length of [0, 1] is 1 while the real line is undefined (I think). The person on the street would probably prefer the Lebesgue measure as more intuitive. I guess my ultimate point is that imprecise statements are harder to prove wrong. There's a quote, I don't remember who said it, that you should mistrust precise statements rather than imprecise ones, because it's precisely precise statements that can be proven wrong. edit: found the quote: http://www.brainyquote.com/quotes/quotes/r/raymondsmu190329.html http://www.brainyquote.com/quotes/quotes/r/raymondsmu190329....
- tanderson92 10y agoBut the Lebesgue measure is not a counting measure so it cannot be in any way related to the "number of numbers"...?
- Someone 10y ago
- roywiggins 10y agoI'm a big fan of the Alexander horned sphere, which is even weirder. The interior is simply connected (like the interior of a sphere) but the exterior isn't. https://en.wikipedia.org/wiki/Alexander_horned_sphere https://en.wikipedia.org/wiki/Alexander_horned_sphere
- mkagenius 10y agoHow is this different from this: "a plain sheet of paper can have infinite surface area and finite (0) volume" ?
- danbruc 10y agoAn ideal sheet of paper, a plain, is a two dimensional object and has by definition no volume. In case of Gabriel's horn we really have a three dimensional object bounded by a two dimensional surface. The bounding surface itself has no volume just like a plain.
- ktta 10y agoBut an ideal sheet of paper can have finite surface area and infinite perimeter length. See the Stackoverflow link in a comment above.
- danbruc 10y agoOf course, in both cases it is interior - volume respectively area - and boundary - surface area respectively boundary length - but that is not the same thing as area and volume of a two dimensional object. The analog for Gabriel's horn would be volume and four dimensional hypervolume which is of course also zero because it is a three dimensional object. The paradox goes from interior to boundary, what mkagenius suggested is taking the interior one dimension up.
- ezequiel-garzon 10y agoI cannot resist: it's a plane, not a plain.
- danbruc 10y agoDamn it, I even looked it up because I was unsure. But the German word Ebene translates to plane and plain and I missed that.
- deleted 10y ago[deleted]
- blippityBlop 10y agoOkay, tra la la. But can we all take note that the volume is PI, and that PI's decimal places are also infinite. In this moment, I take issue with the idea that PI represents a "finite" volume. PI, is more accurately a "known-volume-other-than-infinity." Meanwhile, angels dance upon the head of a pin somewhere.
- tedunangst 10y agouhm, 2/7 has infinite decimals. it's not infinite.
- blippityBlop 10y agoIt's 22/7, and zeno's paradox aside, pi has an infinite component to it, if you look at it the wrong way. That the horn's tail (or mouth piece, as it were) extends into an infinitely small value approaching zero, is as fair as the never-ending irrational fraction pi represents. Otherwise, the horn's volume is undefined. Where does the tip of the horn cross zero? It doesn't??? Hmmm, that sounds like a non-finite value to me. Discussions like this often wax political more than analytical. There are four lights.
- drdeca 10y agoYou can just re-scale it to have volume 1 if you want. Also, uh, you realize that pi is less than 5, and more than 0, right? If you draw a circle with radius 1, the area would be pi just as much as the volume of this at some scaling. Would you say that such a circle has infinite area? That seems like it would be refusing to use a term as it is generally used. You can object by saying that circles do not exist, I suppose. That would be consistent. But saying that a unit circle has infinite area is either false, or using a term in a nonstandard way. Similarly, you could reject such a horn (perhaps because it isn't bounded), but calling the volume infinite is like calling the area of a unit circle infinite. Maybe you just object to abstract objects in general? If so, alright. People are still going to study them though.
- blippityBlop 10y ago
- keithnz 10y agoFor programmers it is helpful to visualize the horn such that the volume = features and the surface = possible bugs
- sixstringtheory 10y agoI remember finding this gem in my calculus book's exercises, and toiled over it and debated with my professor and lost sleep over it. Well, I didn't disprove it, but I learned a lot of good stuff along the way :)
- kevinwang 10y agoIs this similar to the Dirac Delta function, which bounds a finite area, with infinite length? Could one say they are different dimensional analogues of each other?
- thomasahle 10y agoThe Dirac Delta isn't really well defined from a calculus point of view. It's basically a half line in space, so its area should be 0.
- Chinjut 10y agoFor what it's worth, this is a 3d version (finite volume, infinite surface are) of a phenomenon which might be more easily grasped in 2d (finite area, infinite perimeter). And this idea of finite area described by y = f(x) [between this curve and the x-axis, say], with infinite perimeter, amounts to just the same thing some function f whose integral over an infinite range (e.g., from x = 1 to infinity) is finite. If you prefer to think discretely, this is essentially the same phenomenon as an infinite series with finite sum (splitting our curve into the block from x = 1 to 2, the block from x = 2 to 3, the block from x = 3 to 4, etc., the total area is the sum of the series of areas in these blocks, while the total perimeter is automatically infinite as at least length 1 on top and bottom is contributed in each of these blocks). So, consider, for example, y = 1/x^2, from x = 1 to infinity: infinite perimeter but finite area (as its antiderivative is -1/x + C, which only increases by a finite amount over this range; in the same way, differentiating any function with a finite asymptote yields examples, and every example comes in essentially this fashion). In moving to 3d and making a surface of revolution out of our starting curve, we are now integrating πf^2 instead of f itself, so we need f^2 to yield a convergent integral rather than f itself, but otherwise everything is just the same. Thus, we can get away with f(x) = 1/x, as in Gabriel's Horn, but also, just as well, with f(x) = x^{-p} for any p > 1/2. Not that these are the only such things; again, any convergent infinite integral or series yields examples. So, just as well, y = 1/2^x or such things would work.