On 08/05/2012 04:43 PM, Kevin Wampler wrote:

> "Question: Why can't you extend the complex numbers to 3D space?
>
> Answer: Because the hairy ball theorem says so."
>
> But that seems a pretty weak explanation of why such an extension isn't
> possible if you're insisting that the extension be a field. After all
> such an extension isn't possible to 4d, even though the hairy ball
> theorem *doesn't* apply in that case. If you're going to take this view
> why bother with the hairy ball theorem at all? You can prove more by
> other means.

OK, point taken.

It's just that the "other means" are usually incomprehensible, whereas 
"you can't comb a sphere flat" is a pretty intuitive description of why 
you can't do something.

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