Gabriel's Horn is formed simply by graphing out y=1/x, for x>1 (because you can't divide by zero, and this removes some asymptotic nonsense). The idea is that you rotate the curve about the x axis to make a shape looking like one of those old-timey horns, with the mouthpiece being at the end of infinity. Observe!

What's interesting about this particular shape, though, is that the volume of the shape as you go out along the x axis is always decreasing but never gets to zero, and the sum of all the volume converges on π. In other words, the cumulative volume of the horn as you approach the mouthpiece is always getting closer to π ("π units of volume," to be precise), but never quite reaches it. If you added it up all the way to infinity, it would reach π units of volume (pro tip: this is impossible).

OK, so what's so interesting about this? Well, if you checked out the Wikipedia page linked above, then you might have seen the painter's paradox. Because the curve goes on to infinity but the volume it expresses is convergent on π, you could fill such an object with π volume of paint - gallons, liters, whatever system of measure you're working with (I don't feel like doing math right now, just talking about it). However, if you were to try to paint the outside of the horn, it would take an infinite amount of paint. But... wait a minute: it has finite volume, but infinite surface area? Yep, that's the paradox.

Of course, such an object could never actually be constructed, let alone painted. Actual horns are made up of matter, which has mass and takes up space, so once you get to the point where the horn needs to be less than a hydrogen atom wide, you're kind of boned because there's nothing for you to build it with. Even if you found Euclidean horn-stuff, which I just invented (it isn't made up of discrete components and is infinitely divisible), the constraints of real-world paint present you with a distracting pseudo-solution to the paradox. Paint is made of molecules, and no matter how infinitesimally small the horn gets, you'll always need a minimum number of molecules to "cover" the horn. Because the horn goes on forever, the cumulative volume of paint required to cover the outside to a length x diverges to infinity as x approaches infinity. Filling the inside with real paint would still require a finite amount because eventually, the horn would get so narrow that not even one molecule of the paint could squeeze in (even if we ignore surface tension), and you could never actually fill it all the way to infinity. But this is irrelevant to the point that the volume of the horn is a convergent sum while its surface area diverges - even with Euclidean paint, the inside of the horn would still be filled by a finite 3D volume, though the 2D surface area is still infinite.

Ultimately, the paradox arises because it is counterintuitive to most people that something could be finite in terms of volume, but infinite in terms of surface area. By ignoring the limitations of the real world, we can show with math that this is actually possible (possible to express without contradiction, that is), but it still grates against our intuitions and causes lots of people to scratch their heads and go, "What?" Neat trick, isn't it?

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