Well, it’s Gabriel, Gabriel playin’!
Gabriel, Gabriel sayin’
“Will you be ready to go
When I blow my horn?”
– Cole Porter, Anything Goes
The commenters in last week’s Sunday Function proposed an excellent idea for this week. As we did then, we’ll start simple and work up to it. Graph the curve f(x) = 1/x.
Now take the horizontal axis and think of it as an axle, one that can rotate smoothly. Hook a motor onto it and spin it up to a nice fast clip. The graph of the function will trace out a surface in three-dimensional space. In the business, we call it a surface of revolution. It looks like this:
Now I’ve truncated the plot at x = 4 for reasons of space. But in reality you can imagine that in fact the surface keeps going on forever in that direction in the shape of an ever-narrowing trumpet. That shape is called Gabriel’s Horn, and though I’m leaving off the actual explicit specification for the expression describing it, it’s still a function (of two parametric parameters or of y and z) nonetheless.
There’s two things we might want to calculate. First, the volume contained in the trumpet. Imagine you have a penny, and you drop it into the trumpet such that it acts as a plug or a manhole cover. It’s perfectly perpendicular to the x-axis. The penny will have a radius of f(c) = 1/c, because at the particular point c that it sticks it clearly must have the same radius as our original function f(x) evaluated at c. What’s the volume of that penny? It’s the area, pi*r^2, times the thickness of the penny. Knowing that r = 1/x for whatever generic x the penny sticks at and calling the thickness dx, we have a volume (call it dV) for the penny of dV = pi/x^2 dx. So if we fill the entire trumpet with custom pennies of appropriately varying radii and add up their volumes, we’ll have the volume of the trumpet. Looks like a job for calculus!
So despite being infinitely long, the trumpet has a finite volume. It’s a little counterintuitive, but it’s perfectly true nonetheless.
What about the surface area? It’s a little harder to find the expression for surface area of a solid of revolution, but you can take my word that it’s this, with the third expression having our particular function plugged in:
That integral looks fairly tricky to evaluate directly. I think it could probably be done, but we won’t have to. The term under the square root is bigger than 1, and so the 1/x term is a lower limit for the behavior of the integrand. But 1/x diverges, so the integral is infinite. Our infinitely long horn has infinite surface area.
Infinite surface area and finite volume. That’s weird. As people like to say about this shape, you can fill it with paint, but you can’t paint it. It’s true, but it’s not so implausible as it sounds because physical paint isn’t just a pure 2d surface. Real paint has thickness. If you poured real paint into a real Gabriel’s Trumpet, eventually it would fail to keep coating the interior once it narrowed down to a thinner diameter than a paint molecule.
Even Gabriel might think that’s pretty cool.