paradox · counting
Gabriel's horn
Finite paint for the inside. No end of paint for the surface.
A shape can have a finite volume but an infinite surface area. You could fill it with a finite amount of paint, yet never have enough paint to coat its outside.
Rotate the curve. Compare the volume you can pour in with the surface you would need to paint.
The horn is formed by rotating the curve y = 1/x, from 1 onward, around the x-axis.
The volume of the resulting shape is finite. It approaches π, about 3.14.
The surface area is infinite. On the slider it keeps growing as the horn gets longer.
A finite amount of paint could fill the horn, yet no finite amount of paint could cover its inner surface.
The paradox comes from the difference between volume and surface area in an infinitely long, narrowing shape.