Gabriel's Horn

We all love a paradox, and here's one that really makes my mind boggle. Imagine a three dimensional shape with finite volume, but with infinite area, what would it look like? Are you imagining complicated surfaces that turn in on themselves in all weird and bizarre manners? Take a look at this:

This structure is called Gabriel's Horn, after the horn the Archangel Gabriel blows to announce judgement day. It is also called Torricelli's Trumpet, as its properties were first studied by Italian mathematician and physicist Evangelista Torricelli (inventor of the barometer) in the 17th century.
It is formed by taking the curve of y=1/x in the domain of x greater than or equal to 1, and rotating it around 360 degrees. Although first discovered before the invention of calculus, today, by using integration, we can show that it has a volume of PI, and an infinite surface area.

Without looking too deep into the mathematics here, the paradox can be considered from a painter's point of view. Imagine the Gabriel's Horn was a real object, we could fill it with a finite amount of paint, but we could never really fill it, as not one drop of paint would ever reach the bottom. Also, you could never paint the entire surface of the horn, although we have already filled it with a finite amount of paint. Strange.