A Trumpet With Finite Volume You Could Never Paint

Take the curve y = 1/x for x ≥ 1 and spin it around the x-axis. You get a long, narrowing horn called Gabriel's Horn — sometimes Torricelli's Trumpet, after Evangelista Torricelli, who described it in a 1641 manuscript and called the result De solido hyperbolico acuto.

Its volume is the integral of π(1/x)² from 1 to infinity, which evaluates to π. A finite number — about 3.14159 cubic units. Its surface area is the integral of 2π(1/x)√(1 + 1/x⁴) from 1 to infinity, which diverges to infinity.

So you could fill the horn with π units of paint, but you could not coat its inner wall — the wall has infinite area. Torricelli's contemporaries took the result as evidence that infinity itself was suspect. Thomas Hobbes argued the construction was impossible. Galileo refused to write about it.

The paradox is only intuitive, not logical. The horn's wall thins faster than its area grows, so the same paint that fills the volume effectively does coat the wall — at every point, with thickness shrinking to zero. Saying the wall has infinite area is true; saying you would need infinite paint to coat it is not, once you allow the coating to thin.

It was one of the earliest worked examples that Cavalieri's geometry of indivisibles — the proto-calculus Torricelli helped develop — could give finite answers to questions about unbounded regions.