Am I right side up, or upside down?
And is this real, or am I dreaming?
— The Dave Matthews Band,
Last Thursday (November 17) marked the birthday of August Ferdinand Möbius (1790-1868). Most of us know him best for the Möbius strip, but he was a serious mathematician who contributed to several different areas of science. My Ph.D. thesis involved Möbius transformations, which are very useful in describing rotations and other operations; in fact you can use quaternions to perform Möbius transformations, and we know how powerful quaternions are. His professorship at Leipzig was actually in astronomy, though that’s certainly not what he’s best-known for today.
I thought about several things to write concerning Möbius, but frankly, I think the famous Möbius strip is a great subject in itself, even though precedence for the strip’s discovery goes to Johann Listing. If you’re inclined, get some plain paper (typing paper will work just fine) and follow along. In fact, I’m going to give you homework in this post, in the spirit of all the classes I’m not teaching this term.
Möbius may not have been the first to think of a Möbius strip, but he independently discovered it while working on the mathematical study known as topology. I wrote briefly about topology (and the deep relationship donuts have with coffee cups) before in explaining the fascinating quantum phenomenon known as the Aharanov-Bohm effect. In brief, topology is the study of the properties of objects that don’t have to do with their specific shape: you can stretch and squeeze and deform things in topology, as long as you don’t rip holes or heal existing holes. In that way, a donut is like a coffee cup: they both contain one hole, the middle of the donut or inside the handle of the mug.
Using topology, some imagination, and some trickery, we can create two-dimensional surfaces of many kinds using an ordinary piece of paper. We’re all familiar with the usual x-y coordinate system, known as Cartesian coordinates (for mathematician-philosopher Rene Descartes). It’s useful for so many things, but there are times it breaks down, even for something that’s two-dimensional. For example, even though Earth is (roughly) spherical, the directions you travel can be broken down into two directions: North-South (longitude lines) and East-West (latitude lines). However, there are two places where the latitude-longitude coordinate system breaks down: the North and South Poles, where a single latitude corresponds to all the longitudes simultaneously. In fact, no coordinate system can describe the surface of a sphere without a breakdown at at least one point! It’s not for lack of trying, it’s just a mathematical fact. (There are coordinate systems that break down at only one point, but they’re less useful for cartography.)
Here are some rules for the mathematical toys to follow:
- The things we are creating are two-dimensional in a mathematical sense, even though we live in three spatial dimensions. A piece of paper has thickness and two sides, but we’re going to pretend neither of those things are true.
- Similarly, we’ll ignore the edges of the paper. More on that soon.
- I’ll define a regular x-y set of coordinates, but of course that’s arbitrary. For example, unlike on Earth where the North and South poles are defined because of rotation, on an ideal mathematical sphere, every point is created equal. We must distinguish between “coordinate singularities”, where our coordinate system breaks down, and stuff like creases or holes (which I won’t discuss in this particular post).
A normal piece of paper can stand in for an infinite plane, in a topological sense. Just as the paper doesn’t have to be perfectly flat (let’s ignore creasing it for now), topology doesn’t tell us anything about how curved the plane is—that gets back into geometry. However, if you curl the paper back entirely on itself and tape the sides together so that it makes a cylinder, that’s obviously a different kind of shape than a regular piece of paper. If you imagine a universe built that way, traveling perpendicular to the direction the paper curls (the y-direction), you’d never reach an end, but if you traveled in the direction of the curling (the x-direction), you’d eventually end up where you started. Your Cartesian coordinate system holds up everywhere: x never turns into y, and there is no “north pole”.
Now take a piece of paper (a thin strip works better than a full sheet) and connect one edge to the other, but give it a twist—creating a Möbius strip. (Actually, there are two types of Möbius strip, depending on if you give the paper a clockwise or counter-clockwise twist; these different strips are chiral versions of each other.) It still looks kind of like a cylinder, but there’s a big difference: if you’re traveling in the x-direction, by the time you’ve reached your starting x-coordinate, “North” and “South” will have switched! (Remember Rule 1: even though our paper has two sides, the mathematical Möbius strip has no “sides”. You can help see this effect by coloring along the top edge of your paper before you constructing the strip.) You need to traverse the entire length one more time before everything is back to how it started.
Homework problem #1: make both types of Möbius strip, and show that you can’t turn one into the other without cutting or breaking the tape—just like your right and left hands aren’t equivalent. There is no place where the coordinate system breaks down, yet a single coordinate system doesn’t cover the entire Möbius strip. This kind of shape is called non-orientable: there’s no consistent way to define a “North” direction that works for the whole strip. If you’re mathematically sophisticated, you can start nattering on about fiber bundle spaces and manifolds, but I’ll spare you that (unless you really want me to explain it).
What about our friend the sphere? It’s not easy to make one out of a sheet of paper. Using the same kinds of constructions as our cylinder and Möbius strips, you have to compress the entire top edge into a single point and the entire bottom edge into a single point before you tape together the right and left edges. The top edge, in other words, becomes the North pole and the bottom becomes the South pole. Generally, if you’re making a sphere out of paper, you don’t try to do that, but we’re playing topologist today and can pretend our paper is both stretchy and compressible.
Homework #2: Naively, you might have thought that you could make a sphere by just taping the top and bottom of the paper together. Here’s the square representing that situation, which also is the universe of the classic video game Asteroids (and its innumerable imitators). Figure out what shape this actually makes and leave your answer in the comments.
Homework #3: Now for a really fun one, which you won’t be able to do with paper, but perhaps can figure out anyway. Imagine taping the left and right sides together like a cylinder, but taping the top and bottom together like a Möbius strip. Draw the result (or locate a picture online that illustrates it), and leave your answer in the comments.
We don’t live in a two-dimensional universe, but the shapes we’ve been playing with have three- and four- and so-on-dimensional versions. You can see why this stuff might be important in cosmology: if our universe is like a sphere (for example), then it won’t go on forever. Light traveling out from a galaxy will eventually return to that galaxy, though far in the future. So far, our observations are consistent with the universe going on forever in every direction, but all that means is that the distance to wrap around could be significantly larger than the universe we observe. No sane person suggests the universe has an edge, but it could have a shape, and part of the challenge of observation is to detect the signatures of that shape. In a future installment of the Universe in a Box series (here’s part 1, part 2, and part 3), I’ll come back to that question, but for now, have fun with paper and glue and glitter.