[amelius]

But how will the number of neurons N grow with the number of turns in the spiral?

If N levels off, then the network has grasped the concept of a spiral and can generalize to arbitrary size.

If N doesn't level off, then the network isn't really learning the general case.

[therein]

I know this is going to sound cheesy but that's an amazing way to put it. It blew my mind.