Three Dollar Quill

All right. As I get a better grasp on what this article is saying, I'm further from sure that I can directly use it. Basically, it defines two infinite families of algebraic numbers that are suitable as bases, but these families are not exhaustive. For instance, if I'm understanding the families correctly, neither of them contain the plastic ratio, which is suitable. However, messing around with the families did reveal something interesting.

\(x^2 - 3x + 1 = 0\) has greater solution \(\frac{3 + \sqrt{5}}{2}\), which is between 2 and 3, so it uses ternary digits. The carrying looks a little confusing, until you see that a value of three or above in a place carries both up and down at the same time. Unless I'm missing something, there's no maximal version of this base, because there's no way to construct the equivalent of two consecutive ones in phinary. My intuition is that the base isn't doesn't have a maximal version if the negative coefficients are all consecutive, and all of them are have magnitude equal to the highest digit in the base, except for the last one, which has magnitude equal to one more.

Anyway, I don't have further ideas of what to do with all of this just yet. While I was typing this up, I tried to implement complex integers. There's a little work to do to get the display of them nice, but they should be ready to go for purposes of doing arithmetic to them, which means I can think about moving IntegerBase code into a base class and extending it to work with complex bases. I really should see about writing proper validators and invariant checkers, though...

Oh well, none of that is happening right now, because it's late and I'm going to get ready for bed now.

Good night.