Monday, February 14, 2011

On rotations of spheres

Consider a sphere living in some number of dimensions. The sphere we are used to is the 2-sphere, denoted by S2. Mathematicians call it this because even though it lives in 3-dimensions, it is itself a 2-dimensional object (in the sense that small sections of it look like the 2-dimensional plane). One can similarly talk about the n-sphere, which lives in n+1 dimensions. Thus for example, the 1-sphere, S1, can be thought of all all points on a plane that are of distance one from the origin.

When one has a geometric object one of the most obvious things to do is to ask what rigid movements of the object will take it to itself. Thus, for example, for a sphere, a rotation about some axis through the sphere's center rigidly moves the sphere to itself.

Rotations seem simple, but they can be surprisingly tricky. For the 1-sphere if one does a rotation and then another rotation one is left with a rotation. This is an important and helpful property. It says essentially that rotations form what mathematicians call a subgroup of the group of all rigid motions. It turns out that this is still true for the 2-sphere, S2 even when one uses different axises for the two rotation Now, you might find surprising the fact that for S3, the sphere of three dimensions living in four dimensions, this breaks down. The composition of rotations is not necessarily a rotation. In some sense the not obvious fact is not why this breaks down for three dimensions, but why it still holds true in two dimensions.

The above is well known to mathematicians or to many people who have played around a bit with geometric objects. But, I recently learned a related fact that I found startling. Call a rotation "periodic" if we eventually get every point back to where we started. So for example, if one repeat a 90 degree rotation (π/2 for those using radians) four times we will have every point back where we started. Now, it turns out that for rotations of S2 even though the composition of rotations is a rotation, the composition of periodic rotations is not necessarily periodic. Once one knows that this is true it is easy construct examples. Consider two rotations around perpendicular axises each of 30 degrees (π/6 in radians). It isn't difficult to show that their composition although still a rotation is not periodic. This is a good example of how even basic geometry can surprise us even when we think we understand it.


Anonymous said...

Rotations not closed? What do we mean by rotation? Do rotations have to be rotations about some n-subspace by some angle? I always just thought of all the orientation preserving isometries as "rotations", but I guess that's not how the term's used...

Joshua said...

Yes, in this context, rotation about some angle is what is meant. The relevant point then is that you can have two things that are both rotations and they compose to a non-rotation. Of course, this is still an orientation preserving isometry, but there's no way of writing it as a rotation about some axis.