December

On the fun side, I wrote an essay, Cat's Cradle, that isn't about math! That's the first since December 2021. Someday there will be more.

On the serious side, I'm still working on those essays about continued fractions and the pattern. I have many things to report.

First, I finally did something that I should have done long ago. When I started writing the first group of essays, I thought there were only going to be three of them, so I just filed them away under the old stand-alone essay Continued Fractions. Then, later, when I started writing the second group of essays (the epilogue), well, I thought there were only going to be three of them, so I just tacked them onto the end. As a result, the tree of essays was a mess.

So, I added one essay and rearranged everything.

Here are links to the three happy little root nodes.

Continued Fractions
The Markov Project
The Markov Project (Epilogue)

How's the actual work coming along? Well, in May I had plans for five more essays. Since then, I finished one essay and thought of three more essays that I want to write. So, now I have plans for seven more essays.

On the positive side, …

• I think many of the remaining essays should be relatively short and sweet. But then I always think that.
• The essay that I did finish, More About Symmetry, revealed to me so many amazing new things that I had to split it into five parts.
• I did a bunch of work on Details About Operations. It isn't quite ready yet, but when it is, it will be full of nice details.
• How lucky I am that I found the perfect math problem! There's plenty of math that's beyond me, but this problem is right at my level. It has some depth. And, it involves integers, modular arithmetic, matrices, and group theory, some of my favorite things.

Finally, I'd like to pass along some advice from another writer. Unfortunately I don't remember the name of the writer or the exact wording of the advice, but the general idea was something like this.

When you get stuck, delete the part you like.

When you've written something that you particularly like—a word, a phrase, a sentence, a paragraph—the fact that you like it can blind you to the fact that it doesn't really fit into the structure you're trying to create.

Of course, the reason I mention the advice is that I had to use it recently. I liked the following paragraph a lot, so I'm just going to save it right here.

The third symmetry, remember, is a statement about the original expansion E(u). A true statement? Probably. I've verified it by experiment, and I imagine I'll find a proof in [Aigner]. In any case, it's the first statement here.

• Exactly two rotations have reflection symmetry.
• Any one rotation has reflection symmetry.
• Any one reflection is a rotation.
• Every reflection is a rotation.

The first statement is the strongest. Each implies the next, and the last two are equivalent. For some pertinent examples, see An Interesting Detail.