I don’t know if everyone has a favorite math thing. Given the kind of person who subscribes to my newsletter there’s a greater-than-average chance you do. I certainly have one, and because it’s my newsletter I get to tell you about it!

When I was in high school I took an intro to calculus class. One thing we learned was how to use it to determine the surface area and volume of different kinds of geometric shapes. I won’t go into the details, but suffice to say you can, from just the equation of a shape, determine those properties. For example, if you have a sphere, you can figure out, from first principles, what the equations are for its area and volume (the familiar 4 pi r² and 4/3 pi r³ ).

Another thing we learned about is what’s called a surface of revolution. It’s where you take some function and revolve it around the one of the axes in a coordinate system. Wikipedia has a somewhat high-level description of how this works, as does Wolfram (with some examples shown).

As an example, with malice aforethought, take the simple function y = 1 / x. When you graph it you get a fun little curve:

This is decently straightforward. At x = 1, y also equals 1. But as x gets bigger (to the right), y gets smaller, and the graphed curve gets closer and closer to the x-axis. But it never quite gets to y = 0; even at very large x, y is very small but not 0. If x = a billion, y = 1/billion, which is small but not 0.

The same is true going the other way: at small x (toward the left) y gets bigger. At x = 1/billion, y = a billion. 

Here’s the (first) fun bit. Imagine that graphed curve is a wire. Now imagine grabbing the right hand side of that graph with your right hand, and the top part with your left. Keeping your right hand still, pull the top of the curve down toward you, spinning it around the x-axis. Keep rotating it all the way around until it’s back up to the top. What do you get?

You get this:

It looks like a trombone bell! Or a trumpet bell, I guess, but I played trombone, and again it’s my newsletter so trombone it is. This shape is called Gabriel’s Horn, in fact, after the archangel Gabriel who blows a horn at the End of Days in the Bible (though to my knowledge it’s never specified who blows the horn).

Here’s the second fun bit: I cut off the graph at pretty small values of x and y. What if we let it go on forever?

Then we get this shape:

Cooool. There are two very important things to note here. First, as x gets bigger (to the lower right), z and y get smaller, so the graph gets narrower and narrower. If we extend this out more and more, it gets closer and closer to a one-dimensional line.

Second, as x gets smaller, the graph flattens out because y and z get bigger. At very small values of x, y and z get huge, and the shape approaches being a plane. It never quite gets there, but it gets very flat.

This shape is very, very strange. You can use calculus to calculate its surface area and volume, and what you get will destroy your mind.

Look first at the narrow part. That gets close to being a line, which has no surface area and no volume, so as x gets bigger it contributes to those two properties less and less.

But at small x things get weird. The surface area grows without bound, as we say, meaning it gets infinitely big. It’s essentially an infinite plane, so its area is infinite as well. OK, fine, that’s obvious enough from looking at the graph.

But what about the volume? As X gets small, the horn itself gets closer and closer the y-z plane. At very large x, y and z are small, so there’s less and less room between the horn itself and that plane. That means there’s less and less volume the smaller x gets.

That means very small values of x also don’t contribute to the volume. Instead of the volume blowing up to infinity, it actually approaches a finite value. If you do the math, you find that volume is equal to… pi! Yup, our favorite number, 3.1415 etc. etc., in whatever units the axes are in.

Do you see the bizarreness now? Imagine this is an actual three-dimensional object, extending infinitely in all directions, and the axes are measured in, say, centimeters. It would take an infinite amount of paint to cover it, but you could fill its interior with just over three cubic centimeters!

I will give you a moment to collect the pieces of your brain that have exploded all over the room.

This is not a trick; there’s no weird gimmick. It’s math: Gabriel’s Horn has infinite area but finite volume.

This is not how we see things behave in the real world. If the area goes up the volume has to as well, usually even faster! But not in this case. Because the shape gets closer to all the axes in every direction, the volume is constrained.

But that seems paradoxical! You’d think the inside of the surface and the outside would be the same, so if you wanted to paint the inside of the surface it would take an infinite amount of paint, but if that’s so then how could it be filled with only a small amount? That paint still has to touch every part of the surface, so you need an infinite amount, yet pi cubic units fills it. 

Around and around we go. It’s definitely brain-twisty.

Now, in the real world there are other issues with this as well. If you try to paint the inside, eventually the part spread out over the y-z plane gets so narrow that a paint molecule won’t even fit, and the part along the x-axis gets so small you get the same problem. You can probably argue about quantum mechanics limiting things as well when you get small enough values, but whatever, it doesn’t matter, because we’re talking pure math here, and it doesn’t always exactly mimic the real world. It’s an idealization, where you can have infinite values and perfect lines and planes and all that.

In fact this apparent paradox caused a lot of consternation among mathematicians in the 17^th century. The nature of infinity is weird and not always easy to grasp. In this case, it really messes with our notion of area and volume, which we evolved through things we can grasp with our hands. Making the leap to pure math can really be hard when the concepts seem to contradict our more physical experience.

No wonder Gabriel (maybe) used it to announce Judgment Day. I can’t blame him.

Anyway, when we went through this math in high school my mind was well and truly blown (at least as much as the Archangel’s trombone). I was both delighted by the math and horrified by the implications, and to be honest it still freaks me out to this day. It’s a reminder to me that math describes reality, but it isn’t necessarily reality. This too has very deep and difficult philosophical implications, which I will leave to bigger brains than mine to probe, even as I enjoy the argument.

And finally, to paraphrase myself:

Math! I love this stuff.

I created the graphs at the Desmos site, where you can play with various functions and graphing. It’s pretty fun! All credit to them.

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