Gabriel's Horn, Inside vs. Outside Surface area

[bob012345]

Consider Gabriel's Horn, the mathematical object formed by a surface of revolution of the curve x= 1/x from x=1 to infinity. It is known that one can fill the horn with a volume of Pi cubic units of paint but it would take an infinite amount to paint the surface. I think they usually mean the outer surface. But what of the inner surface which must be by definition already *covered in paint? And for a purely mathematical object how can the inner surface differ from the outer surface? Consider a sphere or box with a 1D plane defining the surface. Aren't the inner and outer areas the same? I'm concerned with only the mathematics of the situation and not practical aspects such as the size of atoms. Thanks.

* Of course, this may be a mistaken assumption on my part! Contemplating the infinite is tricky.

 

[fresh_42]

The inner and outer surfaces do not differ. Here's what Wikipedia has to say about this apparent paradox:

"Since Gabriel's horn has a finite volume, it can be filled with a finite amount of color. However, covering an infinitely large area requires an infinite amount of "paint. Looking at the inside of the horn, covering it seems, on the one hand, to require an infinite amount of color because of the infinitely large area. On the other hand, the inside is completely covered in the filling of the horn, for which only a finite volume is needed.

This apparent paradox does not take into account that in a real covering with color, the color layer has a certain thickness. When this finite large thickness becomes larger than the radius of the horn, the color fills the entire cross section of the horn. Then the required amount of color is no longer determined by the surface, but by the volume. Thus, the required amount of color can not be determined by multiplying the infinite area by a finite thickness of the color layer. On the other hand, if one assumes an infinite thin layer of paint without a volume property, one can not compare its nonexistent volume with the volume of the body."

 

[bob012345]

The inner and outer surfaces do not differ. Here's what Wikipedia has to say about this apparent paradox:

"Since Gabriel's horn has a finite volume, it can be filled with a finite amount of color. However, covering an infinitely large area requires an infinite amount of "paint. Looking at the inside of the horn, covering it seems, on the one hand, to require an infinite amount of color because of the infinitely large area. On the other hand, the inside is completely covered in the filling of the horn, for which only a finite volume is needed.

This apparent paradox does not take into account that in a real covering with color, the color layer has a certain thickness. When this finite large thickness becomes larger than the radius of the horn, the color fills the entire cross section of the horn. Then the required amount of color is no longer determined by the surface, but by the volume. Thus, the required amount of color can not be determined by multiplying the infinite area by a finite thickness of the color layer. On the other hand, if one assumes an infinite thin layer of paint without a volume property, one can not compare its nonexistent volume with the volume of the body."

Thanks. I though the inner and outer surfaces should be equal. Ok, I can see how people have resolved the paradox nicely. Put into simple words, the rate the paint layer gets thin is faster than the expansion of the surface area so a finite amount of paint can cover an infinite surface area. From the Wikipedia page "Gabriel's Horn" (and I'm not sure exactly what page you quoted from since I don't see that quote here);

"Since the horn has finite volume but infinite surface area, there is an apparent paradox that the horn could be filled with a finite quantity of paint, and yet that paint would not be sufficient to coat its inner surface. The "paradox" is resolved by realizing that a finite amount of paint can in fact coat an infinite surface area — it simply needs to get thinner at a fast enough rate. (Much like the series 1/2N gets smaller fast enough that its sum is finite.) In the case where the horn is filled with paint, this thinning is accomplished by the increasing reduction in diameter of the throat of the horn."

They added one statement "The converse phenomenon of Gabriel's horn – a surface of revolution that has a finite surface area but an infinite volume – cannot occur:". Being an optimist, I'd say that doesn't completely rule out the possibility of someday making a Tardis.



Thanks again!

 

[fresh_42]

Thanks. I though the inner and outer surfaces should be equal. Ok, I can see how people have resolved the paradox nicely. Put into simple words, the rate the paint layer gets thin is faster than the expansion of the surface area so a finite amount of paint can cover an infinite surface area. From the Wikipedia page "Gabriel's Horn" (and I'm not sure exactly what page you quoted from since I don't see that quote here);

It's more that the paint atoms fill the available circular section at some point so that the infinite surface beyond this point doesn't come into account: at some point the paint cannot flow any deeper.

I changed the language version of Wikipedia and let Google translate it. The different language versions of Wikipedia entries are not identical, so one can look for the one which fits best.

 

[bob012345]

It's more that the paint atoms fill the available circular section at some point so that the infinite surface beyond this point doesn't come into account: at some point the paint cannot flow any deeper.

I changed the language version of Wikipedia and let Google translate it. The different language versions of Wikipedia entries are not identical, so one can look for the one which fits best.

Interesting. Remember, I was only thinking about mathematical paint, not limited by any practical considerations of atoms. :)

 

[fresh_42]

Interesting. Remember, I was only thinking about mathematical paint, not limited by any practical considerations of atoms. :)

Yes, but if it is infinitely thin, then it won't get you a volume at all.

 

[LCKurtz]

I used to give my classes the following idea when discussing the problem whether it takes an infinite amount of paint to paint an infinite surface. So let's consider the xy plane laying flat and think about painting it. The paint we will use is high quality non molecular which spreads so well that it has this property:
If an area is painted with any thickness $t>0$ the paint can spread to a larger area.

Definition: The plane can be painted with a volume $Q$ of paint if for any $A>0$, the interior of a circle with area $A$ about the origin can painted with that amount of paint.

Under these assumptions a drop of paint (the good stuff) is sufficient to paint the plane.

 

[lavinia]

It might be interesting to look at the volume of a small collar neighborhood of small radius, ε, around the surface. One would think superficially that its volume would be well approximated by the surface area times ε.

BTW: Along the same idea, the loxodrome or rhumb line is the path that makes a constant angle with the meridian lines on the earth. It circles infinitely many times around each pole but has finite length.

 

[Rada Demorn]

Yes, but if it is infinitely thin, then it won't get you a volume at all.

I am afraid @fresh_42 that Wikipedia's got it wrong.

You see starting with a thin layer of paint of height h say at x=1 and letting this height of layer of paint progressively diminish as brush moves alongside x-axis ( say, h=1/x ) you can paint the outside surface with a limited amount of paint too!

The situation is completely the same as with the inside volume case.

 

[fresh_42]

I am afraid @fresh_42 that Wikipedia's got it wrong.

You see starting with a thin layer of paint of height h say at x=1 and letting this height of layer of paint progressively diminish as brush moves alongside x-axis ( say, h=1/x ) you can paint the outside surface with a limited amount of paint too!

The situation is completely the same as with the inside volume case.

Thanks for pointing this out. But if you paint the interior, you wouldn't have to continue up to infinity and thus a finite area is sufficient, due to the thickness of paint. In your example with thickness 1, we would have stopped at once. So in this case with the given restrictions the Wikipedia entry is correct. However, you might object that it doesn't explain the paradoxon.

Therefore you asked another question: Why is the volume smaller (finite) than the volume of paint we would need for the exterior (infinite area times constant thickness)? Or which curve do we need to make use of the fact that $V \sim \int f(x)^2\, dx$ and $A \sim \int f(x)\,dL_f$ such that $V < \infty$ and $A = \infty$. In this case it all comes down to the harmonic series and the paradoxa there, or the tiny difference between
$$
\sum_{n\in \mathbb{N} }\dfrac{1}{n} \text{ and } \sum_{n\in \mathbb{N} }\dfrac{1}{n^{1+\varepsilon}}\; , \;\varepsilon > 0
$$