Spherical geometry – a brainteaser or math funfact

This is a three part post. First there is a tiny bit of background, which you can skip if you don’t care why I wrote this post. Then there is an interesting mathematical challenge, that requires no calculations what so ever. It is however not a challenge at all if you already know how to prove the basics of spherical geometry. You can skip this as well, if you don’t want to think right now. And finally, after some spoiler-spacing, there’s the “Funfact”/solution to the challenge.

Tiny bit of background

So this morning I start thinking about spherical geometry in bed. Thinking about math at random times is not unusual for me, but not super frequent either, but on Thursday I had a four hour math test as a step towards getting a Massachusetts teacher’s license and it got me in math thinking mode.

It was a pretty good test, I think. The questions are varied and challenging, especially if you haven’t that particular kind of math for almost 20 years, or ever. For several of the more difficult questions I had to reconstruct bits of method that I couldn’t recall from first principles that I could. For instance I haven’t done much integration for the last 20 years and couldn’t remember the substitution rule, but I could figure it out from what I do remember about integration. Mind you, it was multiple choice, so I got some hints.

Something that I had never really done though, to my recollection, and which was on the practice test (I signed an NDA for the main test), was spherical geometry. And this morning I started thinking about that in bed.

Challenge

I’m going to start this challenge with some background as well, so scroll down past the picture if you just want “Math problem now!”

We live on a sphere, or close enough to one to make us appreciate problems like the ones I’m going to put to you. In most aspects of our lives (if we don’t work planning international air traffic corridors) we don’t appreciate the spherical nature of our world. In fact, our immediate surroundings are dominated by local bumps in the terrain and optical aberrations in the atmosphere to the extent that otherwise intelligent people can be flat Earthers. But spherical geometry becomes necessary if you’re flying across the Atlantic, or if you’re surveying all of North America and attempt to fit thousands of perfect squares onto it.

The jagged eastern border of Saskatchewan, Canada

The problem we face is first and foremost related to our indoctrination in flat geometry and our reliance on flat representations of the spherical Earth, the things we call maps. If you want the shortest route between two points on most flat maps, it is actually not what you get if you connect the two with a ruler (except on the equator, or straight North-South). It’s what is called a “great circle”. And the challenge to you is simply this:

• Think of two points on a sphere and how to find the shortest path between them. Use this to explain what a great circle is and how you justify that it’s the shortest path.
• If we define great circles as the straight lines of spherical geometry (which mathematicians do, I just looked it up), what can we say about parallel lines on a sphere?

That’s it. That’s what I was thinking about in bed this morning, and which I will explain after some spoiler space.

Spoilers ahead!

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Solution

Imagine two points on a flat surface, like a black(or white)board. Now draw spherical arch segments between the two, with different radii. Which arch is the shortest route? Second shortest? Third shortest?

We see that the bigger the radius, the shorter the arch segment between the two points, and we’re ready to move onto a sphere. Imagine the two points are somewhere along the 45th parallel (i.e at latitude 45° N). For instance two points on opposite coasts of the US, and (very roughly) 5000 km straight north from the equator. (Also imagine the Earth is a perfect sphere, and not slightly squeezed from the poles.) Now on most maps the straight line between these two will be along the parallel, but that is not the largest circle we can draw in 3D-space through those points. The largest circle is the one that has it’s center in the center of the Earth (and not just on the axis, like the 45th parallel) and therefore the other side dips down to 45° S on the points exactly opposite our two locations. And between the two points it goes slightly north. It’s a longer line on the flat map, but since it has the largest possible radius, it is the shortest path on the sphere.

Another property the great circle has, when we think of the sphere as perfect, is that it divides the Earth in two equal halves, like the equator, or a line of longitude. And that can be used to tells us something about parallel straight lines on a sphere. Namely that they don’t exist, and that “the 45th parallel” is either not parallel, or not straight. Surprise! It’s the latter.

If you take any straight line on a sphere it will be a great circle and it will divide the world into equal hemispheres. Let’s call them A and B. Now take any point on the sphere not on this first line and draw a straight line through that. Is it possible to have this second straight line be parallel to the first one, i.e. for them never to cross? No. Because this second line also divides the sphere into equal hemispheres.

Let’s do a proper proof by contradiction based on this.

Assume that you can have a parallel straight line. This creates equal hemispheres C and D, and since they are half the size of the sphere, they are equal in size to A and B. Since the lines don’t cross, either C or D has to be completely contained in one of the original hemispheres, which means it’s smaller, but we just said they were the same size, so we can’t have a parallel straight line on a sphere.

And that’s what I was thinking in bed this morning.