Why is North North? Part 2, The Hairy Ball Theorem

A few days ago I wrote too many words about how north, south, east, and west are defined.

The purpose then was to answer the question of why you can't walk north forever, but you can walk east forever. The answer was that the definition of north/south has a "coordinate singularity", a point where the definition breaks down and can't uniquely name different directions, and also that walking north by definition leads you to the singularity.

We also saw that the east/west scheme has a coordinate singularity at the poles, but that east/west was conveniently designed to never lead you to the singularity like north and south were.

Then at the end I raised a question: can we do better? Is there a directional scheme we can choose for the earth so that there are no coordinate singularities anywhere? If so, what would that scheme look like?

Going abstract

To be able to answer this question we first need to think clearly about what constitutes a direction scheme. This concept may not be obvious, but it's pretty simple when you think about it. A direction scheme is a rule that points out a direction at every point on a sphere. Imagine arrows painted on every point of the sphere. When you stand at a point, wherever the arrow points is the direction associated with that scheme.

The direction scheme called "north." At every point there is an arrow pointing north. If you walk along the arrows you are heading north.

A hypothetical direction scheme called "crazynorth." Wherever you are on the sphere, if you walk along the red line there then you are heading crazynorth. We'd never actually use this scheme, but we want to consider all possibilities in general.

We'll also require that a direction scheme be smooth, that is it doesn't change abruptly at any point, because it would be confusing for your directions to suddenly switch just because you moved a millimeter.

A rule that assigns an arrow like this to every point on a surface is called a "vector field" by mathematicians.

One way to picture vector fields is to imagine that the surface in question has hair growing out of it that you then comb. Every hair has a corresponding arrow that represents whose size and direction represents how parallel to the surface the hair is. If the hair is completely flat against the surface then we draw an arrow with length 1. If the hair is sticking straight out of the surface we draw no arrow.

Hairs on a surface and how they correspond to arrows (or vectors) on that surface.

Singularities (again)

What does a coordinate singularity mean in the context of vector fields? In the last post we said a coordinate singularity is a point where directions are not defined. When we interpret arrows from a vector field as directions, we can see that the directions are undefined when the arrows have zero length (so you can't tell which way they point), or they have infinite length (because that's just not defined), or when the surrounding vector field isn't smooth and changes abruptly at a point (because at the point of change there are multiple definitions for the direction).

Two kinds of singularity. In (a) there is a point with a zero length arrow. In (b) there is a point of discontinuity, where the vector field changes abruptly. In both cases we can't use the vector fields to uniquely distinguish directions. (a) is the kind of singularity east/west has, while north/south has the type in (b)

Thus, the our question about finding singularity-free direction schemes becomes "can we find a smoothly varying vector field on a sphere that such that all the arrows are finite and not zero at every point?"

 In terms of the hairy surface, a coordinate singularity then corresponds to a place where the hair sticks straight up.

Golly!

The Hairy Ball Theorem

(Try not to laugh too hard.) Now the best part. It turns out there is a definite answer to the question I just asked. The answer is no, it is not possible. Every smooth vector field on a sphere has at least one point where the the arrow has zero length. In terms of hairy balls, every time you arrange the hair on a sphere there will be a point where the hair stands straight up. 

An attempted vector field with singularities that look like hairy cowlicks at the poles.

This means that any direction scheme we try to devise on the surface of a sphere will have a point where the direction is not defined! We will always have the same problem we had with the north/south, east/west direction schemes. There will always be points where our direction scheme isn't defined. The usual north,south,east,west scheme is the best system we can get.

I'm not going to explain how you prove this theorem, but it is very general. The Hairy Ball Theorem applies to spheres, but also any shape that is sphere-like (in the sense that you can squeeze it into a sphere shape without tearing any holes). So you also can't find a singularity-free direction scheme for an egg, or a banana, or any simple 3D object. It doesn't apply to other shapes though: it's easy to find good direction schemes on a donut, or on an infinite plane.

And the Hairy Ball Theorem doesn't just apply to maps and globes. It implies that it is impossible to design a radio antenna that doesn't have a blind spot. It implies that there is always at least one wind cyclone on the earth (because there must be a point with no wind).

The center of a cyclone is a zero singularity in a surface wind pattern

Electrical radiation acts like vectors on a sphere, so every antenna has a point with zero signal. Image courtesy https://de.wikipedia.org/wiki/Datei:Felder_um_Dipol.jpg

Math is cool, man.