Hairy circle of spheres

Well hairy isn’t too accurate unless there are hairs that are closed circles, but these are the concessions one makes for a dumb play on the Hairy Sphere Theorem. This post doesn’t really have much to do with that theorem.

What’s below are some pictures suggesting Seifert fibrations of S^1 x S^2, the circle of spheres. View this 3-manifold as an interval of concentric spheres where you have to imagine gluing the inner sphere to the outer sphere.

A Seifert fibration of a 3-manifold is a filling of the 3-manifold with circles so that around each point a teeny tiny enough chunk looks like it’s filled with parallel lines. Here these circles get chopped into intervals. The interval going through the north pole connects up to give one circle, and so does the interval through the south pole. We call these circles the singular fibers.

All the other intervals connect up with a fixed number of others to form circles. These are the regular fibers. Near each point on a singular fiber, a regular fiber passes by some fixed number of times, the order of the singular fiber. In the picture above this number is 5 for both singular fibers.

Here they have order 2.

Here they have order 3.

Here they have order 1 and so they aren’t that special. A homeomorphism would make all the fibers appear as radial arcs, the S^1‘s of the S^1 x S^2.

In the three examples above, red fibers are shown at regularly spaced latitudes (along fixed longitudes) going from the north pole to the south pole. The yellow fibers are copies rotated to other longitudes.
Let’s look at just the red ones as the orders of the singular fibers increase.

Order 1

Order 2

Order 3

Order 4

Order 5

Order 6

Order 7

Order 8

Order 16

Order 25

Neat, I guess.

While we’re here. Have some eye candy. There’s more in this set.

I think all these below come from the order 1.

(Oh, and these pictures are a result of experimenting with Grasshopper for Rhino and some non-photorealistic rendering.)