A Few of My Favorite Spaces: SO(3)

This article was published in Scientific American’s former blog network and reflects the views of the author, not necessarily those of Scientific American

Last week, I got excited when Toby Hendy followed me on Twitter. She’s a physics Ph.D. student with a YouTube channel mostly about physics with a few hair videos thrown in. I recognized her from her video about Dirac’s belt trick, which she illustrates with her very long braid.

I wrote about the trick, which I now think of as the Waitress pie trick, after watching the Tony Awards ceremony in 2016. Talking with Hendy online last week reminded me that I’d been wanting to write more about SO(3), the space lurking behind the hair and pies.

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SO(3) is short for the special orthogonal group* in three dimensions, and it’s also known as the three-dimensional rotation group. It's the collection of rotations of three-dimensional space that preserve one distinguished point called the origin. It might be easiest to imagine as all the ways you can rotate a ball—be it a basketball or the Earth itself—that leave the center of the ball fixed. 

In addition to being a set, SO(3) is a group, a mathematical term indicating that there is a little bit more structure to the set. A group requires both a set and a way to combine elements of the set to get other elements of the set. It’s pretty easy to convince yourself that performing one rotation that fixes the center point of a ball and then a second rotation will produce another rotation. A group also requires an identity rotation (doing nothing) and for each rotation to have an inverse that undoes it.

So far we've defined SO(3) as a group of rotations. Aside from some perhaps unfamiliar terminology, it’s pretty straightforward. Now we get meta.

We want to understand the structure not of the sphere we’re rotating but of the space of rotations itself. How many dimensions’ worth of information do we need to describe all the rotations? Can we figure out a way to visualize not just individual rotations but this entire space of rotations?

First, let’s think about how we can describe rotations. One way is to note that every rotation of the sphere that fixes the center goes around a line through the center. If you’re thinking about a globe, it’s probably the line connecting the North and South poles. In SO(3), we’re allowed to rotate around any such axis, and we can rotate by any amount. So one way of specifying a rotation is to specify an axis (or, equivalently, two opposite, or antipodal, points on the sphere) and an amount of rotation. One way to visualize this idea is by imagining a solid ball. For convenience, we’ll say it has radius 1/2. (When a mathematician chooses a number other than 0 or 1 for “convenience,” you know she’s got a trick up her sleeve!) Every point in the sphere other than the center is a certain distance away from the center along a certain axis. To understand the correspondence between rotations and points in the solid ball fully, we have to get into some more details.

It’s fun to play with these ideas on your own to figure out why each convention might have been chosen, but for now, you’ll have to take my word for it. Let’s imagine a point in the ball other than the center of the ball. It has a distance d, between 0 and 1/2, along some axis through the center. Imagine lining up your line of sight with this axis of rotation, orienting the ball so your eye is closer to the point you chose than to the center. Then rotate by whatever portion of a turn is specified by the distance d. So if d=1/4, rotate by a quarter turn. It might take a little pondering, but it turns out all rotations can be described using distance and axis, so we get a correspondence between points in the ball and rotations of the sphere.

It’s fun to try to imagine taking a walk through the solid ball and looking at what rotations of the sphere result. For example, if you walk along a ray coming from the center of the ball, you’re fixing an axis and rotating further and further around that axis. What happens when you get to the edge of the ball? At that point, you’ve rotated by half a turn, or 180 degrees, in one direction. When you rotate further, it’s equivalent to rotating the other direction by less than 180 degrees, which corresponds to a point on the same ray but on the opposite side of the center, or lining up your eye with the same axis, but having the opposite side closer.

This means that in order to describe how rotations are related to each other, we want to be able to walk along one ray to the edge of the ball and then pop over to the opposite side of the ray and keep walking the same direction. So basically we want a solid ball with antipodal points glued together. Regular readers of A Few of My Favorite Spaces might find this a little familiar. Gluing antipodal points of a round object together is just what we did to get a projective plane! And indeed, SO(3) is topologically the same as real projective 3-space, or RP³.

While I was researching this post, I came across a new paper and series of videos by David Pengelley and Daniel Ramras about the relationship between SO(3) and the belt or pie trick. The paper assumes a bit of mathematical background, but the videos don't require as much, and both of them are helpful for thinking about what exactly your hair or arm are doing while they're rotating around and around and eventually getting back to the starting point.

*Strangely, understanding the words in the name of the space is not really necessary to understand this post, but it seems weird for you to leave this post without at least a brief explanation of the name. SO(3), the 3-dimensional special orthogonal group, is a collection of matrices. In mathematics, a matrix is a rectangular array of numbers, which seems to spectacularly undersell its utility. The reason we care about matrices here is that they provide an easy way to represent transformations called linear functions.

In the case of SO(3), the matrices we’re interested are square matrices, so they have the same number of rows and columns (3 of each). The word orthogonal means that the columns of the matrix have to be orthogonal (perpendicular, but fancier) to one another, and the word special means the matrices have to have determinant 1; basically, the transformations they represent can't make things bigger or smaller.

Read about more of my favorite spaces: The Cantor Set Fat Cantor Sets The Topologist’s Sine Curve Cantor's Leaky Tent The Infinite Earring The Line with Two Origins The House with Two Rooms The Fano Plane The Torus The Three-Torus The Möbius Strip The Long Line Space-Filling Curves The Wallis Sieve Two Tori Glued along a Slit The Empty Set The Menger Sponge The Connected Sum of Four Hopf Links Borromean Rings The Sierpinski Triangle Lexicographic Ordering on the Unit Square The SNCF Metric The Mandelbrot Set Fatou's Pancake The Pseudosphere The Douady Rabbit The Poincaré Homology Sphere The Kovalevskaya Top A 6-Holed Torus The Real Projective Plane The 1-Dimensional Sphere The Loch Ness Monster The Koch Snowflake The Bicylinder The Catenoid