Gabriel's Horn Paradox
For this one, consider the graph of y=1/x, for values of x from 1 to infinity. It will look like this for the values of x from 1 to about 17.
Then, rotate that graph around the x axis. It will make an infinitely long horn. The end of it will look like one of these horns you sometimes see at sporting events.
The questions are:
- What is the volume of the horn?
- What is the surface area of the horn?
I'll stop there for now to give the other math wizards on the forum some time to have fun answering the questions above.
The cross-section of the horn at x has area PI / x^2 and circumference 2 PI / x
If +INF represents positive infinity:
The volume = INTEGRAL(1, +INF) (PI / x^2) dx = PI * INTEGRAL(1, +INF) (1 / x^2) dx = PI * (-1/(+INF) + 1/1) = PI
The surface area = INTEGRAL (1, +INF) (2 PI / x) dx = 2 PI * INTEGRAL(1, +INF) (1 / x) dx = PI * (ln (+INF) - ln 1) = positive infinity
Something else to note:
Despite the fact that its surface area is positive infinity, it can be painted with a finite amount of paint, as you can create a larger horn with the cap at x = 1/2 and the curve y = 2 / x; the surface area of the smaller horn is contained within the volume of the larger horn, which has finite volume.
After a certain point, the inside diameter is smaller than a paint molecule, and you have to leave a tiny infinite gap.
Quote: ThatDonGuy
The cross-section of the horn at x has area PI / x^2 and circumference 2 PI / x
If +INF represents positive infinity:
The volume = INTEGRAL(1, +INF) (PI / x^2) dx = PI * INTEGRAL(1, +INF) (1 / x^2) dx = PI * (-1/(+INF) + 1/1) = PI
The surface area = INTEGRAL (1, +INF) (2 PI / x) dx = 2 PI * INTEGRAL(1, +INF) (1 / x) dx = PI * (ln (+INF) - ln 1) = positive infinity
Something else to note:
Despite the fact that its surface area is positive infinity, it can be painted with a finite amount of paint, as you can create a larger horn with the cap at x = 1/2 and the curve y = 2 / x; the surface area of the smaller horn is contained within the volume of the larger horn, which has finite volume.
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I agree!
I'll have to chew on your note.
The big question is how can it be that the volume is finite, but the surface area is infinite? Thus, the paradox.
Quote: WizardThe big question is how can it be that the volume is finite, but the surface area is infinite? Thus, the paradox.
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Another way of thinking about it:
The "interior surface area" of the horn equals the "exterior surface area", and you can paint the interior by filling it, yet supposedly you cannot paint the exterior.
Quote: UsernameRemorseEnclosing the horn within another finite volume does not resolve the paradox of the infinite surface area requiring an infinite amount of paint. The infinite surface area remains a property of the horn, and covering it entirely would, in theory, require an infinite amount of paint, assuming the paint layer has zero thickness.
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No, it doesn't - in fact, it causes another paradox.
And the answer seems to be:
Area = pi
Circumference = 2pi.
What if we filled the circle with paint? We can't say that part of the paint went to painting the circumference. I'm not sure why we can't say that, other than this example shows the circumference is greater than the area. In other words, there isn't enough paint in the middle to paint the circumference. This is true everywhere on Gabriel's Horn.
Just rambling here.
Quote: WizardI'm not sure how to put this in words, but I think it's bad math to think of the volume as including the surface area. For example, consider a circle of radius 1.
Area = pi
Circumference = 2pi.
What if we filled the circle with paint? We can't say that part of the paint went to painting the circumference. I'm not sure why we can't say that, other than this example shows the circumference is greater than the area. In other words, there isn't enough paint in the middle to paint the circumference. This is true everywhere on Gabriel's Horn.
Just rambling here.
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How accurate is it to say it even has volume, being it's not closed? If this were a physical structure and you expanded it in a tank of water it would displace no water.
Maybe if the right terms were added to close it off on each end and they were continuous curves with the rest of the horn the volume would then go to infinity as the surface area does.
Quote: WizardI'm not sure how to put this in words, but I think it's bad math to think of the volume as including the surface area. For example, consider a circle of radius 1.
Area = pi
Circumference = 2pi.
What if we filled the circle with paint? We can't say that part of the paint went to painting the circumference. I'm not sure why we can't say that, other than this example shows the circumference is greater than the area. In other words, there isn't enough paint in the middle to paint the circumference. This is true everywhere on Gabriel's Horn.
Just rambling here.
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Volume is 3D. Surface area is 2D. In theory a minuscule amount of a volume and cover an immense amount of a surface area. Not a stretch to say a finite amount of 3D material can cover an infinite amount of 2D area.
Imagine a 1 by 1 cube. You can slice such a cube into an infinite amount of squares. If you lay those infinite amount of squares next to each other you could cover an infinite plane.
No real paradox here.
Quote: AutomaticMonkeyHow accurate is it to say it even has volume, being it's not closed? If this were a physical structure and you expanded it in a tank of water it would displace no water.
Maybe if the right terms were added to close it off on each end and they were continuous curves with the rest of the horn the volume would then go to infinity as the surface area does.
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Well, nobody says we can't take an integral unless it is a closed shape because the water would pour out otherwise. The pi calculation is just simple integration. I don't think there is any error there.
Quote: unJonQuote: WizardI'm not sure how to put this in words, but I think it's bad math to think of the volume as including the surface area. For example, consider a circle of radius 1.
Area = pi
Circumference = 2pi.
What if we filled the circle with paint? We can't say that part of the paint went to painting the circumference. I'm not sure why we can't say that, other than this example shows the circumference is greater than the area. In other words, there isn't enough paint in the middle to paint the circumference. This is true everywhere on Gabriel's Horn.
Just rambling here.
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Volume is 3D. Surface area is 2D. In theory a minuscule amount of a volume and cover an immense amount of a surface area. Not a stretch to say a finite amount of 3D material can cover an infinite amount of 2D area.
Imagine a 1 by 1 cube. You can slice such a cube into an infinite amount of squares. If you lay those infinite amount of squares next to each other you could cover an infinite plane.
No real paradox here.
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I wonder whether "dimension" actually is created by infinity alone, as by stacking infinitely many squares, in order to create a cube:
The infinite plane is also defined by infinitely many, infinitely long parallel lines which are 1D.
Would infinitely many lines, arranged like a fan, i.e. intersecting in a single point, still define an infinite plane, when fanned out all around a full circle?
Or would the lines "pile up" in that point of intersection, creating the additional dimension for 2D?
Would this pile of lines have a finite height, as in the case of the cube made from infinitely many squares, and how high would it be?
Quote: AutomaticMonkeyQuote: WizardI'm not sure how to put this in words, but I think it's bad math to think of the volume as including the surface area. For example, consider a circle of radius 1.
Area = pi
Circumference = 2pi.
What if we filled the circle with paint? We can't say that part of the paint went to painting the circumference. I'm not sure why we can't say that, other than this example shows the circumference is greater than the area. In other words, there isn't enough paint in the middle to paint the circumference. This is true everywhere on Gabriel's Horn.
Just rambling here.
link to original post
How accurate is it to say it even has volume, being it's not closed? If this were a physical structure and you expanded it in a tank of water it would displace no water.
Maybe if the right terms were added to close it off on each end and they were continuous curves with the rest of the horn the volume would then go to infinity as the surface area does.
link to original post
Thanks.
I normally think about the x>1 side, but this just showed me the x<1 side is almost as interesting.
Quote: WizardQuote: AutomaticMonkeyHow accurate is it to say it even has volume, being it's not closed? If this were a physical structure and you expanded it in a tank of water it would displace no water.
Maybe if the right terms were added to close it off on each end and they were continuous curves with the rest of the horn the volume would then go to infinity as the surface area does.
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Well, nobody says we can't take an integral unless it is a closed shape because the water would pour out otherwise. The pi calculation is just simple integration. I don't think there is any error there.
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Maybe the thing is that the value of pi may not be interpreted as a finite volume in the case of the infinitely long Gabriel's Horn. It rather is the upper limit: Any finite horn will always have a volume less than pi. Longer horns will have a bigger volume but no horn exists which would exceed the volume of pi.
The alledged paradox would therefore originate in comparing a finite volume like that of a sphere of radius "cubic root of 3/4", which has the finite volume of pi, to an infinite object which approaches this value but never reaches (nor exceeds) this volume.
This principle of approching a limit in infinity is quite familiar: A fair coin is said to come up heads with a probability of 0.5, as do tails with a probability of 0.5, "after infinitely many tosses". Any finite sequence however, especially short sequences, will in general deviate from the 0.5 vs. 0.5 probabilities. Only after more and more tosses the observed frequencies will converge towards these values.
Quote: ThomasKMaybe the thing is that the value of pi may not be interpreted as a finite volume in the case of the infinitely long Gabriel's Horn. It rather is the upper limit: Any finite horn will always have a volume less than pi. Longer horns will have a bigger volume but no horn exists which would exceed the volume of pi.
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I think we will have to disagree, although it may just be semantic in nature. In my opinion, the alleged paradox is not the result of an infinitely long horn being impossible. In calculus we deal with limits to infinity all the time. As others have said, I think the problem is in comparing two dimensions with three dimensions. It just isn't kosher mathematically.
Quote: WizardQuote: ThomasKMaybe the thing is that the value of pi may not be interpreted as a finite volume in the case of the infinitely long Gabriel's Horn. It rather is the upper limit: Any finite horn will always have a volume less than pi. Longer horns will have a bigger volume but no horn exists which would exceed the volume of pi.
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I think we will have to disagree, although it may just be semantic in nature. In my opinion, the alleged paradox is not the result of an infinitely long horn being impossible. In calculus we deal with limits to infinity all the time. As others have said, I think the problem is in comparing two dimensions with three dimensions. It just isn't kosher mathematically.
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I had this vague recollection that math distinguishes between integration methods and found information about the "improper integral". INTEGRAL(1, +INF)(1/x^2) seems to be the most prominent example for that. So maybe it is a question of preconditions of well-defined, yet restricted, mathematical procedures which, when compared, produce the impression of a paradox.
(I also found keywords Riemann, Darboux and Lebesgue with respect to integrals.)
