Mon, 10 Nov 2008
Gabriel's Horn is not so puzzling
Elementary calculations, with calculus, allow one to show that although the Horn has finite volume, it has an infinite surface area. This is considered paradoxical, because it says that although an infinite amount of paint is required to cover the interior surface of the horn, the entire interior can be filled up with a finite amount of paint.
The calculations themselves do not lend much insight into what is going on here. But I recently read a crystal-clear explanation that I think should be more widely known.
Take out some Play-Doh and roll out a snake. The surface area of the snake (neglecting the two ends, which are small) is the product of the length and the circumference; the circumference is proportional to the diameter. The volume is the product of the length and the cross-sectional area, which is proportional to the square of the diameter.
Now roll the snake with your hands so that it becomes half as thick as it was before. Its diameter decreases by half, so its cross-sectional area decreases to one-fourth. Since the volume must remain the same, the snake is now four times as long as it was before. And the surface area, which is the product of the length and the diameter, has doubled.
As you continue to roll the snake thinner and thinner, the volume stays the same, but the surface area goes to infinity.
Gabriel's Horn does exactly the same thing, except without the rolling, because the parts of the Horn that are far from the origin look exactly the same as very long snakes.
There's nothing going on in the Gabriel's Horn example that isn't also happening in the snake example, except that in the explanation of Gabriel's Horn, the situation is obfuscated by calculus.
I read this explanation in H. Jerome Keisler's caclulus textbook. Keisler's book is an ordinary undergraduate calculus text, except that instead of basing everything on limits and on limiting processes, it is based on nonstandard analysis and explicit infinitesimal quantities. Check it out; it is available online for free. (The discussion of Gabriel's Horn is in chapter 6, page 356.)
[ Addendum 20081110: A bit more about this. ]