Gabriel's Horn, Inside vs. Outside Surface area

In summary, Gabriel's Horn is a mathematical object with a finite volume but infinite surface area. It is known that a finite amount of paint can fill the horn, but it would take an infinite amount of paint to cover the surface. This apparent paradox is resolved by realizing that a finite amount of paint can cover an infinite surface area if it gets thinner at a fast enough rate. This is similar to the concept of a decreasing series, where the sum is finite even though the individual terms approach zero. The inner and outer surfaces of the horn do not differ and the paradox is further explained on the Wikipedia page for Gabriel's Horn. It is also mentioned that the converse phenomenon of a surface of revolution with a finite surface area but infinite volume cannot occur
  • #1
bob012345
Gold Member
2,068
899
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.
 
Last edited:
Mathematics news on Phys.org
  • #2
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."
 
  • Like
Likes bob012345
  • #3
fresh_42 said:
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!
 
  • #4
bob012345 said:
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.
 
  • Like
Likes bob012345
  • #5
fresh_42 said:
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. :)
 
  • #6
bob012345 said:
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.
 
  • Like
Likes bob012345
  • #7
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.
 
  • #8
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.
 
  • Like
Likes jim mcnamara
  • #9
fresh_42 said:
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.
 
  • Like
Likes fresh_42
  • #10
Rada Demorn said:
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
$$
\sum_{n\in \mathbb{N} }\dfrac{1}{n} \text{ and } \sum_{n\in \mathbb{N} }\dfrac{1}{n^{1+\varepsilon}}\; , \;\varepsilon > 0
$$
 

Related to Gabriel's Horn, Inside vs. Outside Surface area

1. What is Gabriel's Horn?

Gabriel's Horn, also known as Torricelli's Trumpet, is a mathematical shape that resembles a trumpet with infinitely long, curved sides and a finite volume. It is named after the archangel Gabriel in the Bible.

2. What is the difference between the inside and outside surface area of Gabriel's Horn?

The inside surface area of Gabriel's Horn is infinite, while the outside surface area is finite. This means that the inside of the shape can hold an infinite amount of liquid, but the outside can only be covered by a finite amount of paint, for example.

3. How is it possible for the inside surface area to be infinite?

The inside surface area of Gabriel's Horn is infinite because the shape has an asymptote, which is a line that the curve approaches but never touches. This means that the surface area of the shape continues to increase without ever reaching a limit.

4. Does this mean that Gabriel's Horn can hold an infinite amount of liquid?

No, the fact that the inside surface area is infinite does not necessarily mean that the shape can hold an infinite amount of liquid. In reality, the shape would never be able to hold an infinite amount of liquid due to physical constraints such as gravity and the size of molecules.

5. Why is Gabriel's Horn considered a paradox?

Gabriel's Horn is considered a paradox because it challenges our intuitions and common sense. It has a finite volume but an infinite inside surface area, which goes against our understanding of basic geometry and shapes. This paradox has sparked many debates and discussions among mathematicians and philosophers.

Similar threads

Replies
10
Views
4K
Replies
2
Views
951
  • Precalculus Mathematics Homework Help
Replies
4
Views
11K
  • Mechanics
Replies
4
Views
3K
  • General Math
Replies
16
Views
4K
  • Calculus
Replies
4
Views
5K
  • Sci-Fi Writing and World Building
Replies
21
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
3K
  • Advanced Physics Homework Help
Replies
1
Views
2K
Back
Top