The last couple of days, I've been doing quite a bit of thinking about hairy balls.

No! Wait! Don't leave! It's not what it sounds like!

I've been trying to get a picture of what 'bundles' are like. My best effort so far goes like this. Manifolds are too abstract to deal with intuitively for me. So let's just use a simple accesible manifold, just a sphere in 3D (aka a ball), as a stand in for all manifolds. Now, to get a tangent bundle corresponding to the manifold, we have to take the set of all the tangent spaces to all the points on the manifold, more or less. That's too abstract. But the tangent spaces are vector spaces, and vectors are easy to visualize - they're just arrows! More, tangent bundles are apparently just important instances of 'fiber bundles'. That is, bundles are made up of abstract things called fibers, which end up corresponding to the individual tangent vector spaces to each point on the manifold. So. A nice picture for all this that our balls sprout hair*. All kinds of hair - neat, frizzled, dreaded, etc. The hair as a whole is now a picture corresponding to a 'tangent bundle' to the manifold, with the hairs being the 'fibers' making up the bundle. Sensible, right? So, we've got tangent bundles pictured as hairy balls now. Why the hell did we want to do that? Well, I don't really know, because I'm not yet sure how one *uses* tangent bundles. But, I've read that if you want to deal with differentiational thingies with manifolds, you end up dealing with bundles. That covers stuff like 'velocities' and also far more bizarre things.

That actually fits nicely with our image of hairy balls. Say you take such a hairy ball, and give it a good shake. Run about the room waving it in the air. If the hairy balls are sufficiently hairy, you can even outfit a cheerleading squad with them, and have them do some cheers. Now, instead of watching the cheerleaders, try watching the hair on the balls. I know this is really hard, but just try it. What's the hair doing? Well, it's swishing all over the place - and if the balls are sufficiently hairy, you can only see the hair - not the balls. And the swishing is *not* random - it depends on just what the balls are doing - how they're accelerating, or whether they are deforming under there, or whatever. So by looking at the hair of the balls, we can get lots of good information about the balls. What's more, the hair makes life a lot easier - it's a lot easier to notice a hairy ball accelerate and jitter about than a naked one. If we imagine ourselves in a huge empty room, with our viewpoint moving with the ball, it's going to be hard to see how the ball is moving if it's bald. But if it's hairy, we're in like Flynt - we just watch the hair.

So that's why tangent bundles are useful. I think.

Hey! Don't look at me like that. I didn't write all that just because it was an excuse to talk about hairy balls and cheerleaders in the same sentence. No siree Bob! And it's not my fault your mind is in the gutter! Kids these days...

* - As seen on TV! Yours for just 19.95 plus shipping and handling, but only if you call now! "I'm not just the company president - I'm a customer!"

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