Gabriel’s Horn and the Painter’s Paradox

By Anupum Pant

If you take the plot of y=1/x and plot it from 1 to infinity, you’ll see that the plot seems to never meet the x axis. Now, take this plot and spin it fast with the x axis as the axis of rotation. You’ll then have a horn shaped solid object which is endlessly long. A mathematical object also known as the Gabriel;s horn or Torricelli’s trumpet.

Mathematically, this object is interesting because it can contain a finite amount of volume, but it’s surface area is infinite. That is to say, you can fill the horn/trumpet with a finite amount of paint, yet the whole paint it contains would not be enough to paint the inside surface of the object – known as the painter’s paradox. However, there’s a catch about painting the inner part of this horn. As WIkipedia puts it.

In fact, in a theoretical mathematical sense, a finite amount of paint can coat an infinite area, provided the thickness of the coat becomes vanishingly small “quickly enough” to compensate for the ever-expanding area, which in this case is forced to happen to an inner-surface coat as the horn narrows. However, to coat the outer surface of the horn with a constant thickness of paint, no matter how thin, would require an infinite amount of paint.