Voder-Vocoder

I was reading some old notes from some math research I did in 2004. I was looking at the Soddy Equation—

(a + b + c + d)^2 = 2(a^2 + b^2 + c^2 + d^2)

—which relates the curvatures of four kissing circles (see Descartes’ theorem). Example of four kissing circles.

Given three kissing circles, one can find two possible fourth circles (it’s a quadratic, right?). This makes a cluster algebra-like thing and graphically makes a Apollonian gasket.

So far no big suprise. This has all been known since Rene Descartes.

* * *

We had an interesting idea. Draw a Apolonian Gasket. For each of the infinite number of circles in the gasket, perform a circle inversion of all the OTHER circles with respect to that circle. Then plot that out. Repeat the process on all the new circles, ad infinitum. You now have an infinite number of circles, topologically dense in R^2, with the property that no two circles intersect in anyway but tangentially.

I never came up with a decent way to program a computer to display such a set.

I called this the Dionysian Gasket.

Why do I find trigonometry so beautiful?

(source)

By definition, versin(A)=1-cos(A), and exsec(A)=sec(A)-1.

Correct me if I’m wrong, but if an algorithm is patented, and you develop an application that uses it, you can be sued for infringement, even if you never sell that application to anyone else.

Story:

Some units of time (18 months? Four years? Does it matter?) ago I was in Virginia visiting relatives. I got to hang with my cousin’s three kids. They were playing the card game bullshit–which they called something else, because they were very proper children. (I still yelled “BULLSHIT!” when appropriate. They were shocked.)

After they switched to war—the worst card game ever—I refused to play. Instead, I taught them to play hearts. The youngest (maybe seven at the time) never developed any strategy for the game, and the middle child’s meds wore off halfway through the game, but I enjoyed playing a real game.

Why am I thinking about card games recently?

Last month, I wandered through the Westside GenericMegaBookstore and I saw a book, The Pocket Idiot’s Guide to Texas Hold’em (Should be The Idiot’s Pocket Guide to …). I bought it and enjoyed it a lot. It stressed the mathematics behind the game of poker. (Do not ever play a betting game if you are unfamiliar with the concept of expectation value.)

Last semester, we played Set in my methods class.

My boss at the pizza place quit smoking recently. He immediately took the money he saved and bought lotto tickets. I almost explained the notion of expectation value to him. Instead, I just told him that if I am going to gamble, I prefer to play games where skill can make a difference.

This all got me thinking about invented games, like William Shroyer’s Clubs. (All games are invented. I just was friends with Wes in High school.) Webb reminded me of his game Sparts. We didn’t like Sparts because it was to baroque.

So I’ve been thinking about game theory, game design, gambling, and mathematics.

From an interview with Gödel’s biographer, Rebecca Goldstein. (via MeFi)

He had meant his incompleteness theorems to prove the philosophical position to which he was, heart and soul, committed: mathematical Platonism, which is, in short, the belief that there is a human-independent mathematical reality that grounds our mathematical truths; mathematicians are in the business of discovering, rather than inventing, mathematics. His incompleteness theorems concerned the incompleteness of our man-made formal systems, not of mathematical truth, or our knowledge of it. He believed that mathematical reality and our knowledge of mathematical reality exceed the formal rules of formal systems. So unlike the view that says there is no truth apart from the truths we create for ourselves, so that the entire concept of truth disintegrates into a plurality of points of view, Gödel believed that truth - most paradigmatically, mathematical truth - subsists independently of any human point of view. If ever there was a man committed to the objectivity of truth, and to objective standards of rationality, it was Gödel. And so the usurpation of his theorems by postmodernists is ironic. Jean Cocteau wrote in 1926 that “The worst tragedy for a poet is to be admired through being misunderstood.” For a logician, especially one with Gödel’s delicate psychology, the tragedy is perhaps even greater.

Somewhere in here is a proof of the law of sines.

I am enjoying Mathematical Writing by Donald E. Knuth, Tracy Larrabee, and Paul M. Roberts.

“I look forward to the day when a Pulitzer Prize will be given for the best computer program of the year.” —Knuth

Here you go…

Instead of doing math the other day, I was playing minesweeper compulsivly. I had an idea: minesweeper on a hexagonal grid:

UPDATE: I found XBomb for unix and Hexatron Hex Mine. Duh.

“All accessible microstates are equally probable.”

In other words, don’t be surprised at your cards. It’s as likely as anything else that could have happened.

Mark said some things the other day that I didn’t respond to because I was drunk. Here we go:

General Platonism is the belief that there exist transcendent truths. That’s it. Nothing more here. One of these truths is “(ZFC)=>(1+2=3)”. I won’t even comment on whether or not ZFC is true.

So, to conclude, General Platonism is a belief, yes, but it cannot be considered a religion.

Been doing some Java programing for math. I’m getting paid to do this too! I’ve written two applets that deal with a rotor-based sandpile model.