Stay in touch – I’ll post something funny again in about BusyBeaver(10) days.

]]>That’s such an extremely elegant way to explain why this works. I like it a lot 🙂

]]>http://www.thingiverse.com/thing:10472

Fun observation: the image of slices (eg. ℵ({1}×[0,1]) ) are cantor sets.

]]>>I am a geometric derivatives skeptic.

I know. I got quite a bit of amusement out of resolving this using them, just because of that.

> I think that everything that can be proven using them can be proven without them without complicating the proof much.

It’s a matter of elegance. Perhaps it will be more clear in my solution to

> Also, I don’t quite understand why is approximately .

Actually, I’m going to step back in the proof a bit because I can make things a bit neater.

To prove , let’s solve for in .

Well, consider . We expand to our first term in the geometric approximation: . Some simple algebraic manipulation results in .

All this is a formalization of the fact that a high geometric derivative of a function means that the difference between and is in some sense small. So if it becomes bigger and bigger, they go to the same point (in terms of the value of x that would be needed to achieve the same result…).

The reason I like this a lot more than your approach is that it is conceptually smaller. It’s a result of the basic idea of the geometric derivative,

]]>(Also note that I responded to you on the “Cubed” entry.)

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