Gabriel's horn
A horn you can fill with paint, but never paint.
Evangelista Torricelli, 1640s
The painter's nightmare
Imagine a horn that starts wide at one end and narrows forever, never quite closing. It is infinitely long, made by spinning the curve y = 1/x round the x-axis, starting at x = 1.
The painter Ravi is asked to paint the outside. He dips his brush and starts. However much paint he brings, he never finishes: the surface goes on forever, and its area is infinite.
Then his apprentice has a sly idea. "Why not just fill it?" she says. She stands the horn up and pours paint in. Amazingly, a finite amount of paint, just π cubic units, about 3.14, fills the whole infinitely long horn.
So the horn can be filled with paint, which would coat the inside surface, but its surface can never be painted. Ravi puts down his brush and asks for a pay rise.
The puzzle
A shape can have a finite volume but an infinite surface area.
See it for yourself
What's really going on
Calculus works out both. The volume adds up circular slices of area π(1/x)², which shrink so fast that their total converges: π(1 − 1/a) for a horn of length a, approaching π as a grows. The surface area adds up rings whose size shrinks only like 1/x, and that sum grows without limit, roughly like 2π times the natural log of the length.
The paint puzzle cheats a little: real paint has thickness, and the horn eventually becomes narrower than a molecule of paint, so you couldn't really fill it. In pure maths, though, the result stands. When Torricelli found it in the 1640s, many thinkers were shocked that an infinite object could have a finite size.
Think about it
Which grows faster as the horn gets longer, the volume or the surface? What does the animation's readout show?