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Gabriel's Horn (wikipedia.org)
98 points by wz1000 on June 16, 2015 | hide | past | web | favorite | 29 comments

Also check out the Menger sponge, which has infinite surface area and zero volume: http://en.m.wikipedia.org/wiki/Menger_sponge

You can do the first few levels with business cards:


As you go to higher levels, that "infinite surface area" thing starts to become difficult to manage. ;-)

Or equivalently any object with a fractal dimension between 2 and 3.

I think Sierpinski's triangle may also be of interest here, which, contrastly, approaches zero surface area: http://en.wikipedia.org/wiki/Sierpinski_triangle

not constrast; it's analogous, in lower dimension.

Sponge has infinite area and 0 volume. Sierpinski's triangle has infinite perimiter and 0 area.

I've thought it was paradoxical that infinitely long curves could have finite integrals ever since I first took calculus. For example, the integral of 1/x is |ln x|.

I wonder why it takes three dimensions before people start getting upset.

I'm not sure how you intended the example, but it's probably a bad one: If you take the integral of 1/x, say from x=1 to infty, you don't get a finite amount (because ln x goes to infinity).

On the other hand, if you take 1/x^2 as the integrand, you get the desired effect: its antiderivative is -1/x, so the integral of 1/x^2 from x=1 to infty is finite (actually, it's 1).

By the way: For good fun with convergent series, take a look at the problem of escaping a lion in a circular arena at equal maximum speeds. Here's an example link to both question and answer: http://puzzling.stackexchange.com/questions/8140/escaping-a-...

IIRC, there is an even more fascinating story associated to the history of this problem, as the first published solution was actually incorrect.

Thanks for the correction! It's been a couple years since I've actually done any calculus. I couldn't remember the exact function, so I had to google - I guess I read too quickly.

The paradox is not the convergence of infinite long curves..

Read the paradox description. To fill the horn with paint you need a finite amount. To paint the outside of the horn you need an infinite amount.

That is the paradox. The surface area of the inside is equal to the outside, yet one side requires a finite amount of paint while the other side requires an infinite amount.

At least that's how I understand the paradox.

That's not really the paradox. The amount of paint required depends on the thickness of the coat, and the inside is forced to get a thinner and thinner coat (since the space available gets thinner) while the outside is assumed to be painted with an even thickness of paint. If you painted the outside with a layer of paint whose thickness is proportional to the thickness of the curve, you could do it with a finite amount. (This is all assuming idea "paint" that is continuous and arbitrarily subdivisible, real paint is made of molecules.)

The intended "paradox" is that the surface area is infinite but the volume is finite.

From the article:

>Since the Horn has finite volume but infinite surface area, it seems that it could be filled with a finite quantity of paint, and yet that paint would not be sufficient to coat its INNER surface

It's the same thing. Imagine that you need one blob of paint for every tick-mark on the real axis. Infinite blobs of paint. (The resolution, in both scenarios, is that the size of a blob is not constant, and goes to infinitesimal)

I think you're misunderstanding the example.

An anti-derivative of 1/x is ln |x| (very different from |ln x|) when the domain excludes x=0. By the fundamental theorem of calculus, you can use it to integrate over intervals that exclude x=0. (On positive-real intervals F(x)=ln(x) works, and on negative-real intervals F(x)=ln(-x) works. In either case, F(x)=ln|x| is an equivalent formula. If you try a domain that includes x=0, you risk not just a problem with technicalities but also the practical problem that F(x)=ln|x| and G(x)=ln|x|+sgn(x) look no different...)

Or even just a convergent infinite series. Infinitely many positive numbers that sum to a finite number.

People don't tend to find that unintuitive. The Greeks had a nice geometric example in a square of side length 1:

    ├┘│3│   │       │
    ├─┘-│   │       │
    │ 64│   │       │
    ├───┘   │       │
    │       │       │
    │ 3/16  │       │
    │       │       │
    ├───────┘       │
    │               │
    │               │
    │               │
    │       3/4     │
    │               │
    │               │
    │               │
3/4 + 3/16 + 3/64 + 3/256 + ... is easy to visualize as successively filling in three quarters of an ever-smaller residual square. Intuitively, no matter how finely you detail it, you're never going to stop fitting inside the original area-1 square.

edit: better text art

Now squeeze the top-left piece so it's half as wide but twice as tall (i.e. the same height as the full rectangle), and do this recursively. Same area, right? Then stack the L shapes on top of each other. Then you have one of these horns.

Huh? The horn is a three-dimensional object; the square exists in 2-space. You can't make the horn from pieces of the square even if you allow deformation.

I don't think that's paradoxical. We deal with convergent infinite series all the time in everyday life.

10/3(3.333...) is 3 + 3/10 + 3/100/ + 3/1000 + 3/10000...

π(3.1415...) is 3 + 1/10 + 4/100 + 1/1000 + 5/10000...

Its quite easy to see that both of these series will be finite, and many numbers, lets say 4 or 3 + 5/10 will be greater than either of them.

Or in general, any infinite size n-dimensional structure can be embedded in some finite size n+1 dimensional structure.

This sum's finiteness seems pretty intuitive:

    9 + 0.9 + 0.09 + 0.09 + 0.009...

Good example. Minor nitpick, you listed 0.09 twice.

I always describe this to "non-mathy" people when they ask what could possibly be fascinating/beautiful/etc about math. I'd like to think I've changed at least a mind or two.

I wonder what sound it would make. Can we model such a horn assuming a from, say, 1cm to some value p and simulate it?

Depends on how many of the laws of physics you want to ignore. The horn has infinite length and sound waves have a finite speed (assuming sound is defined as the usual compression wave through a medium). So a sound wave starting at the narrow end would take an infinite amount of time to reach the other end. But the sound it self shouldn't be anything special.

One of the reasons that these sort of paradoxes can exist is because real numbers can get arbitrarily small. In principle, you could build a horn that very closely matches Gabriel's horn, but it would necessarily be an approximation of the mathematical object because you do not have available particles of arbitrary size with which to build. This last notion hints at an important property of the real numbers: they can be divided into pieces smaller than any size you can specify. They are useful abstractions upon which to build extremely successful models of reality [1], because at those scales, real numbers work very well. However, at extremely small scales, the correspondence between real numbers and reality breaks down. What does it mean to model a horn made of atoms with numbers that are arbitrarily small fractions of the width of an atom? This is not a 'real numbers / axiom of choice bad' rant, but rather I'd like to point out that real numbers exist in our minds, and using our physical intuition about to reason about them at very small scales is bound to lead to paradoxes.

[1] Electromagnetism, classical mechanics, etc

This reminds me of a Putnam problem from a few years back, was something along the lines of:

Construct a set of discs in R^2 s.t. no infinite straight line can be drawn without intersecting at least one disc, and the sum of the areas of all the discs is finite.

Take a set of discs with radii 1/n. Their total area is finite but the sum of radii is infinite, so we can just use them to cover the X and Y axes. Yeah, this is pretty similar to Gabriel's horn.

(approximate) logarithmic spiral of discs whose radii are the harmonic sequence?

(Awarded 1 point of 10 for being unable to specify the precise piecewise-linear spiral function)

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