
Gabriels Horn: Infinite Surface, Finite Volume - iliis
http://blog.plover.com/math/gabriels-horn.html
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lutusp
A quote: "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_." [Emphasis added]

It would have been much clearer to have said, "The volume of the horn can be
filled with a finite volume of paint." There's no reason to use language
seemingly designed to create a aura of mystery.

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leephillips
When I was an undergraduate at Hampshire College my advisor, David Kelly, used
a draft version of the linked text in an experiment to teach introductory
calculus using nonstandard analysis rather than limits. This was probably in
1978, and I was a teaching assistant for the class. I think the experiment
worked pretty well.

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graeham
The snake example is good, but for me it is more intuitive to think of the z-y
plane as having very small volume but infinite surface area as the plane
extends infinately in both z and y.

