Wizard
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November 9th, 2024 at 5:39:31 PM permalink
I love a good paradox and think they are worthy of their own thread, as opposed to passing mention in the Easy Math Puzzles thread.

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:

  1. What is the volume of the horn?
  2. 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.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
ThatDonGuy
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November 9th, 2024 at 5:52:55 PM permalink

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.

Dieter
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November 9th, 2024 at 6:35:49 PM permalink
I seem to remember that this shape takes an infinite amount of paint to cover the outside, but can only hold a finite amount of paint.
After a certain point, the inside diameter is smaller than a paint molecule, and you have to leave a tiny infinite gap.
May the cards fall in your favor.
Wizard
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November 9th, 2024 at 7:22:24 PM permalink
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.


link to original post



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.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
Wizard
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November 9th, 2024 at 7:49:49 PM permalink
Here is my calculus to show the volume and surface area (PDF).
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
ThatDonGuy
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November 10th, 2024 at 8:41:46 AM permalink
Quote: Wizard

The big question is how can it be that the volume is finite, but the surface area is infinite? Thus, the paradox.
link to original post


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.
UsernameRemorse
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November 10th, 2024 at 8:54:44 AM permalink
Enclosing 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.
ThatDonGuy
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November 10th, 2024 at 9:53:41 AM permalink
Quote: UsernameRemorse

Enclosing 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.
link to original post


No, it doesn't - in fact, it causes another paradox.
unJon
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November 10th, 2024 at 12:37:32 PM permalink
Is it a paradox? Seems to me you would first need to ask how much surface area PI volume of paint can cover.

And the answer seems to be:
An infinite surface area given some idealized paint of infinitesimal thickness.
The race is not always to the swift, nor the battle to the strong; but that is the way to bet.
Wizard
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November 10th, 2024 at 6:27:20 PM permalink
I'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.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
AutomaticMonkey
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November 10th, 2024 at 6:58:43 PM permalink
Quote: Wizard

I'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.
unJon
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November 10th, 2024 at 8:16:17 PM permalink
Quote: Wizard

I'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



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.
The race is not always to the swift, nor the battle to the strong; but that is the way to bet.
Wizard
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November 11th, 2024 at 1:39:33 AM permalink
Quote: AutomaticMonkey

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



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.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
ThomasK
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November 11th, 2024 at 2:22:19 AM permalink
Quote: unJon

Quote: Wizard

I'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



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.
link to original post


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?
"When it comes to probability and statistics, intuition is a bad advisor. Don't speculate. Calculate." - a math textbook author (name not recalled)
Dieter
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November 12th, 2024 at 1:57:54 AM permalink
Quote: AutomaticMonkey

Quote: Wizard

I'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.
May the cards fall in your favor.
ThomasK
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November 12th, 2024 at 8:25:12 AM permalink
Quote: Wizard

Quote: AutomaticMonkey

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



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.
link to original post


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.
"When it comes to probability and statistics, intuition is a bad advisor. Don't speculate. Calculate." - a math textbook author (name not recalled)
Wizard
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November 12th, 2024 at 9:13:57 AM permalink
Quote: ThomasK

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.
link to original post



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.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
ThomasK
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November 13th, 2024 at 6:56:37 AM permalink
Quote: Wizard

Quote: ThomasK

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.
link to original post



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.
link to original post


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.)
"When it comes to probability and statistics, intuition is a bad advisor. Don't speculate. Calculate." - a math textbook author (name not recalled)
Wizard
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November 14th, 2024 at 8:06:52 AM permalink
I added to my solution to show that an infinite series of cans, each with height 1 and radius 1/i for i = 1 to infinity had a finite volume and infinite surface area. It made use of two famous infinite series in mathematics. Click the link if interested.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
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