Most of us learn some amount of Euclidean geometry in school. We learn that the interior angles of triangles add up to 180°. We learn how to prove that lines are parallel, or shapes are congruent or similar. Even though we live on a planet that is not flat, our everyday intuition is on a scale that makes us feel like Euclidean geometry is the natural way to think about shapes, lengths, and angles.
I think it’s a real shame that more students are not exposed to non-Euclidean geometry early in their educations, but that’s a column for another time. Suffice it to say that if one is lucky enough to encounter geometry beyond that which takes place on a perfectly flat plane, one will learn that there is much more to geometry than two-column proofs and the Pythagorean theorem.
The intuition we develop in Euclidean geometry does not prepare us well for non-Euclidean geometry. One of the delights I found when I first started studying hyperbolic geometry (one of the flavors of non-Euclidean geometry) was that many things that seem so obvious as not to require any kind of justification are flat-out wrong when we leave the flat Euclidean plane.
For example, in non-Euclidean geometry, there is no such thing as a pair of triangles that are similar but not congruent. The geometry of a sphere is another flavor of non-Euclidean geometry, so we can think about it on a globe. On the Earth, there is a triangle that connects the North Pole with Quito (the capital of Ecuador) and Libreville (the capital of Gabon). This triangle is close to being a triangle with three right angles, or 270° of internal angle. (The Atlantic Ocean is growing, so in a few billion years, it will be even closer.) On the other hand, the "golden triangle" in Texas (a name only Denton seems to use for the region spanned by Dallas, Fort Worth, and Denton), has a total internal angle of 180.002°, just a smidge more than a flat triangle. (Thanks to David Radcliffe for designing the tool I used to figure that out.)
If you play with more triangles on a globe (or a grapefruit), you will find that the total area of the triangle increases proportionally to the total angle of the triangle. You can't find a golden triangle-shaped triangle on the earth with cities that are thousands of miles apart. Spherical geometry won't let you do that. In hyperbolic geometry the relationship goes the opposite way: the larger the triangle, the less total angle it has.
In Euclidean geometry, there is no natural measuring stick. Over the course of history, humans have chosen many different ways to measure flat distances. All of them are somewhat arbitrary. And in fact, I think it’s in part our experience with flat geometry that makes us think it's normal for units of measurement to be arbitrary.
But it doesn’t have to be that way. In spherical and hyperbolic geometry, the formulas that relate the size of a shape with its internal angle always have these pesky coefficients that pop out. The formula for the area of a spherical triangle is K2(a+b+c−180), where a, b, and c are the measurements (in degrees) of the three angles of the triangle, and the area of a hyperbolic triangle is K2(180−a−b−c). The K2 factor comes from the intrinsic curvature of the space, or how the space is bending at every point. On a sphere, that curvature is related to the radius of the sphere. A smaller sphere has higher curvature, and a bigger sphere has lower. (A baseball is more curved than a basketball.)
To most mathematicians, that K2 is a bit distracting. Two spheres of different sizes have more in common than not, and why should you have to keep track of these coefficients? It’s a recipe for arithmetic errors. Wouldn’t it be better if K2 were just 1 no matter what? Good news: This paradise can be yours! Just set your unit of measurement to be the radius of the sphere you’re looking at. Instead of meters, feet, furlongs, or fathoms, the correct base unit of length for people who live on the earth is the radius of the earth itself.
History of metrology buffs may point out that the meter was originally defined in terms of the size of the earth. Specifically, a meter was defined to be a 10 millionth of the distance between the Equator and North Pole along a line of longitude. (The meter has since been redefined in terms of the speed of light in a vacuum.) But come on, to relate it to a quarter of the circumference and not the radius? That’s a real misstep, if you ask me. So I think we need to scrap the meter and starting over with the Radius (abbreviated R).
The radius of the earth is about 3,963 miles, so we will need to get really comfortable with scientific notation to start expressing everyday measurements in terms of Radii. I’m 2.589×10−7 R tall. I live about 0.25 R away from where I grew up. I’m working on a sewing project that requires me to make a lot of 3.983×10−9 R bias tape. I’m sure you’ll agree that the extra work is minor compared to the satisfaction of knowing you are using the only correct unit of length the planet has seen fit to provide us with.
“But Evelyn,” you say. “The earth is not a sphere.”
I take a deep breath. “Frankly, there’s no way all the governments on the planet could cooperate effectively enough to maintain a massive conspiracy about the shape of the earth for mi—”
“No, no,” you laugh. “I mean, it’s an oblate spheroid. It’s not a perfect sphere.”
Fair enough. It’s true, the Earth is not completely spherical. It bulges out slightly at the equator (more than a billiard ball, despite what you may have heard), making it an oblate spheroid rather than a sphere. We’ve got a couple of options here. First, we could not worry about it. The difference between the largest and smallest distances between the center of the earth and a point on the surface is 13 miles out of almost 4,000, about 0.3%. Nothing we’re measuring could be so important that we need to be off by 1 part in 300, could it? The other option is to just pick one value and go with it. The International Union of Geodesy and Geophysics has a few options for radius of the earth, including the equatorial radius (3963 miles) and three different average radii, all of which are pretty close to 3959 miles.
No matter what the final decision is on the precise definition, we can rest assured that the earth has provided us with a beautiful absolute unit and we’ll never have to argue about how to measure anything ever again.