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Gabriel's Horn

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Re: Gabriel's Horn

#11

Post by parallel »

jveld wrote: post #10

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.

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

Re: Gabriel's Horn

#12

Post by mgob »

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.

Re: Gabriel's Horn

#13

Post by reagency »

parallel wrote: post #11
jveld wrote: post #10

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.

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

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

Re: Gabriel's Horn

#14

Post by wz1000 »

parallel wrote: post #11
jveld wrote: post #10

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.

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

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.

Re: Gabriel's Horn

#15

Post by thaumasiotes »

parallel wrote: post #11
jveld wrote: post #10

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.

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

Re: Gabriel's Horn

#16

Post by davidrusu »

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.

Re: Gabriel's Horn

#17

Post by crimsonalucard »

jveld wrote: post #10

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.

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.

Re: Gabriel's Horn

#18

Post by raldi »

parallel wrote: post #11
jveld wrote: post #10

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.

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

This sum's finiteness seems pretty intuitive:

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

Re: Gabriel's Horn

#19

Post by UncombedCoconut »

jveld wrote: post #10

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 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...)

Re: Gabriel's Horn

#20

Post by om2 »

crimsonalucard wrote: post #17
jveld wrote: post #10

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.

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 pa…

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.

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