Math Mutation 55: The Painter's Paradox A while back, in episode 21, you may remember that we discussed an odd figure known as a fractal, that had an infinite perimeter but a finite area. But in a recent discussion a co-worker of mine, Wayne, mentioned a simple non-fractal 3-D figure known as Gabriel's Horn that has a similar property: it has an infinite surface area, but only a finite volume! This means, bizarrely, that you can fill up the figure by pouring in a finite amount of paint, but you can never paint its surface. Thus the definition of this figure is sometimes referred to as the "Painter's Paradox". It was first discovered way back in the 17th century by Evangelista Torricelli, the same Italian physicist who invented the barometer. How is Gabriel's Horn constructed? It's actually very simple. Take the graph of y = 1/x, drawn starting from x=1 and continuing on to infinity. Now rotate the graph in a circle around the x-axis, forming what looks like a giant horn, with the wide end open at x=1 and the narrow part trailing off forever to infinity. If you look at the circular cross-section at any given point, its radius is 1/x, so it's area is pi r squared, or pi times 1/(x squared). Now let's calculate the surface area and volume of this figure. You may recall from our discussion a few weeks ago that integrals, or infinite continuous summations of 1/(x squared) converge to a finite sum, since the values diminish very quickly and become negligible. So if we find the volume by taking the integral from one to infinity, we can easily determine that the total volume of this figure adds nicely to a value of pi. But to find the total surface area, we need to take the integral of the perimiter, which is 2 pi r at any given point, or 2/x. So this surface area is proportional to an integral of 1/x, which we know diverges to infinity! So how can this be? How can we have a finite volume, but an infinite surface area? Ultimately, in the particular case of Gabriel's Horn, this descends from our misfortune that sums of 1/x grow to infinity, while sums of 1/(x squared) converge-- so paradoxically, adding a dimension to a figure whose size diminshes as 1/x will turn an infinite value into a finite value. Just like our discussion a few weeks ago of energy transmission in two-dimensional worlds, where this put chamistry and physics into upheaval, here this causes strange properties for Gabriel's Horn. Incidentally, you can see from this analysis that the Horn is really just a representative of a huge family of similar figures, created by "stretching" an infinite two-dimensional graph into three dimensions using a converging function. For example, take the graph of y = sin x from x=1 to infinity, then just thicken it by stretching the graph in three dimensions by a factor of 1/(x squared) at each point, looking at the solid formed by the area between the graph's curve and the x-axis. You will similarly get a figure with infinite surface area and finite volume. The real problem here, of course, is that we are defining figures that can never be physically built-- you can't construct a horn that actually descends out to infinity. Any finite piece of Gabriel's Horn that can be built, can obviously be both painted and filled with paint. If we were describing a figure that could physically be constructed, I'd be a lot more worried about this paradox! But as it is, our description of the figure is really just an interesting mathematical game. You might find it fun to look up other converging and non-converging integral functions in a calculus text, and figure out more strange figures you can build by rotating, stretching, or otherwise extending the dimensions, creating out-of-sync area-related and volume-related properties. And this has been your math mutation for today. References:
  • Gabriel's Horn at Wikipedia