Paint paradox
Consider the curve y = 1/x, over the region x = 1 to infinity. The volume of revolution about the x-axis is Pi, but the surface area is infinite.
So you can fill the cone shape with paint, but you can't paint it. Explain!
So you can fill the cone shape with paint, but you can't paint it. Explain!
Labels: puzzle
posted by Chris at
3:41 PM
8 Comments:
This is the well known paradox of Gabriel's Horn.
A consequence of this is that you can effectively paint the inside by filling it with paint, but you can't paint the outside!
Since the "walls" of the horn are of infinitesimal thickness, I think that this makes it even more of a paradox!
Hi Wiz. I hadn't heard of Gabriels's Horn before. I'll check that out. Being an occasional pedant, I'll mention that the walls are not infinitesimally thin - they have zero thickness.
I could paint the outside though!
No explanation is required as far as I can see. It just is what it is.
When we think about infinity, funny things happen.
My suggestions:
-If you're painting the inside charge by surface area, not paint used.
-If you're looking for steady work, paint the outside, and charge by the hour
Cam
Cam, shame on you, that's a cop-out response :)
The inside surface area is the same as the outside surface area. So there is a problem. (Of course, I know how to resolve it).
Oh, all right. One could say that if the paint is able to fill a volume it must have a discrete thickness. If the paint has a discrete thickness it can't actually fill the volume as the horn becomes narrower than the thickness of the paint, thus our infinite surface area remains unpainted despite having a sufficient supply of paint and a finite volume.
Cam
That's more like it. That pretty well disposes of the paradox.
You could also say the paint has "infinite thinness" instead of discrete thickness. Then it could fill the horn but still be able to spread over the infinite surface.
I think this explains the paradox better!
Hi Wiz. You mean "infinitesimal" thickness (or thinness).
One proposition is that you paint with thickness inversely proportional to x.
I'm not sure if one can go the whole hog and paint with zero thickness paint.
But I think the paradox arises because one thinks in terms of real paint.
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