Today, we’re going to talk about the Awesome Theorem.
I’ve mentioned this before, but that is literally the name of the theorem. Gauss, its discoverer, named it theorma egregium, which means the awesome theorem.^{1} And the theorem is awesome enough that the name stuck.
Before I tell you what the theorem is and why it’s so awesome, let me remind you what we talked about last time.
Curvature is, unsurprisingly, how much and how surfaces curve. When you first think of curvature, it seems like it should simply be due to how we are visualizing our surface. For instance, a sheet of paper can be bent into a cylinder, which has different curvature. However, if I were an ant living on the paper, unaware of a third dimension away from the paper, I could never tell whether or not the paper was bent.
This kind of curvature can be measured by looking at principal curvatures, which essentially measure how much a surface bends in two different directions. For a cylinder, this would be the curvature along the cylinder and the curvature across the cylinder.
Of course, these curvatures aren’t intrinsic to the surface. A flat sheet would have two zero principal curvatures, but it can be bent into a cylinder.
So is there some sort of curvature intrinsic to the surface? How would you measure it?
That is the subject of the awesome theorem.
Gauss’s theorem says this: The product of the principal curvatures is intrinsic to the surface.
Pretty impressive, huh?
Yeah, okay, at first glance it may not seem that impressive. But it really is. Let’s talk about why.
What does intrinsic mean?
Intrinsic means that this curvature, called the Gaussian curvature in his honor, can be measured by ants living on the surface. Or, in other words, it can be measured using only measurements on the surface, without referencing the third dimension we usually visualize it in. We can measure the curvature using just things like lengths, angles and areas, and, as long as those things don’t change, this curvature can’t either.
That may not seem very impressive, but think about it this way. How do we know the Earth isn’t flat? After all, some people still claim it is!
Most of the ways scientists have shown that the Earth isn’t flat use the fact that there is a third dimension to travel in. For instance, looking at the moon and its phases requires we can look up. Looking at the angles made by sunlight at noon on the same day in different locations requires both a sun, and the ability to either dig a well or build a tower. Watching ships appear over the horizon requires that a ship has height above the ocean’s surface.
But if we were truly two dimensional beings living on the surface of a sphere, we could still tell the Earth wasn’t flat.
You might point out that you could always just travel around the equator, and so the Earth must be round. But that only shows the Earth wraps around itself, not that it’s round. After all, a cylinder wraps around itself, but it’s not a sphere, even if we add ends to it.
What could we actually measure?
There are a few things you can measure. The first one is something we already mentioned: the angles of a triangle! Positive Gaussian curvature, like on a sphere,^{2} makes the angles of any triangle add up to more than 180 degrees. On a surface with zero Gaussian curvature, like a cylinder or a plane, the angles add up to exactly 180 degrees. And on a surface with negative Gaussian curvature (which is harder to draw, but looks like a saddle point if you zoom in), the angles add up to less than 180 degrees.
So, our two-dimensional scientists could tell they lived on a sphere by measuring angles of triangles!
Another way has to do with circles. As you likely know, the circumference of a circle is (where is the radius) and the area is .
But, like with triangles, that only works if the Gaussian curvature is zero.
Again, look at a sphere. If you take a small circle near the north pole, those formulas will work pretty well.^{3}
But if you increase the radius large enough, that circle will be the entire northern hemisphere!
If you recall formulas for spheres, this means that if your Earth has radius , then your circle has circumference and area . That doesn’t seem too bad till you realize that these formulas use the wrong radius.
Weird.
A bit more calculation shows that, if our circle is the northern hemisphere, its radius is , so the circumference is and the area is !^{4} If we compare coefficients, this shows that both the areas and circumferences of circles on a sphere are less than their counterparts on a flat surface.
This is a general theme for curvature. “Positive” curvature makes angles larger than expected, but makes surfaces close in on themselves (like the sphere), which makes areas and circumferences of circles smaller than expected. “Negative” curvature, it turns out, does the opposite: angles are smaller, but areas and circumferences are larger.
Let’s end with two practical applications of this incredible theorem.
The first has to do with maps.
As you are no doubt aware, there are a lot of different maps of the Earth. There’s the famous and standard Mercator:
Or, maybe you prefer a polar map, like the azimuthal equidistant projection:
Or, perhaps you’re an anarchist, and like the Hammer retroazimuthal projection:
Why are there so many different maps?
It comes down to the fact that none of them can be perfect.
And that’s because of the awesome theorem.
Maps are printed or displayed on flat surfaces, and flat surfaces have zero Gaussian curvature. Spheres, like the Earth, have positive Gaussian curvature. But remember that Gaussian curvature can be measured using only distances and angles and areas.
Since a map and a sphere have different curvatures, it means that something has to distort. If you want angles to be correct, your distances and areas will be wrong. If you want areas to be correct, your angles will be off. Gauss’s theorem shows that no flat map can represent the Earth perfectly.
You need a globe for that.
The last example is the most famous one: eating pizza.
Some pizza comes with a very thin, floppy crust. If you try to just pick it up, it’s going to flop over and make a mess of everything.
While you could just eat the plate with the pizza, another way to get that pizza-y goodness into your mouth is to bend the slice in half.
If you’ve ever done this, you’ll notice the pizza doesn’t flop around near as much.
Why is that?
The pizza, to start, is flat, with zero Gaussian curvature. And while it can flop around, the base of the crust doesn’t really stretch like a balloon would. That means that when you fold the slice in half, the Gaussian curvature stays zero. Then, in one direction, the principal curvature is zero (as it has to be since the product of curvatures is zero), while the other direction has some other principle curvature.
But the Gaussian curvature of the pizza is fixed! That means that the nonzero curvature can change a bit bigger or a bit smaller, but the zero curvature direction is fixed. If it bent up or down in that direction, the product of principal curvatures couldn’t be zero!
Thanks, Gauss, for saving my pizza.
Now let’s go enjoy some.
- Google translates it as “the excellent theorem,” while Wikipedia translates it as “the remarkable theorem.” Whatever adjective you use to describe it, it’s pretty awesome. ↩
- The Gaussian curvature is positive since the two principle curvatures are both or are both , depending on which normal vector to the surface you pick. In either case, that leaves you with Gaussian curvature . ↩
- It’s important to note that the radius on these surfaces must be measured along the surface, not in like you might be visualizing it. ↩
- As a warning, these are not general formulas for any circle on a sphere, just for this special hemispheric one. The general formula is more complicated. ↩