stoutfellow

I think I'm going to dig out my copy of _Winning Ways for Your Mathematical Plays_, the two-volume opus on combinatorial game theory that Conway wrote along with Richard Guy and Elwyn Berlekamp, and try to finally finish it. Volume One begins so breezily, and the concepts are so simple, that I float along with it until, about halfway through the volume, I realize I have no idea what they're talking about.

This time, it's for real. I'll *study* the damn thing if I have to.

A couple of mathematical researchers have discovered an unexpected pattern in the distribution of prime numbers: Who's next?

It's only a conjecture at this point, but there's a lot of experimental evidence to support it.

Weird stuff.

(H/t to nancylebov)

Today I returned the second (and last) midterm to my Linear Algebra class. I wasn't pleased with the scores, and I guess quite a few students were even more displeased. I hate when this happens - I feel as though I've failed them somehow.

At any rate, I went over the problems for them. One of the problems had several different lines of attack, and I went over two of the most direct methods. One student - one who'd done well - raised his hand: "Could you do :this:?" It was a good idea, and I quickly described what he'd find if he took that route.

Finishing the review, I continued with the current material, involving coordinates with respect to a basis. I showed them a few examples, pointing out the geometric content and showing how this allowed a shift from abstract vector spaces to things that look very much like R^n. I got just to the verge of discussing the mechanics of change-of-basis when time ran out.

The student with the good idea came up to me then, and said, "I see what I did wrong. I stopped too soon; I should have gone and :done this and this: Then I'd wind up with :this:" I agreed, and pointed out that he was doing precisely what I'd been talking about moments before, and that the result was precisely change-of-basis - the subject of the next class session. "Yes, you're on the right track!"

As he walked away, he was laughing softly, in tones of joy.

I may not have connected with quite a few students, but this kid's got it.

The Dean of the College of Arts and Sciences is a mathematician. With his administrative duties, he doesn't get as much time as he'd like to work on and talk about mathematics, but he does grab opportunities when he can. Yesterday he gave a departmental colloquium talk, on the algebraic analysis of Markov chains. That's off in the borderland between combinatorics and probability theory, far distant from my work on the classification of polygons.

Listening to him, though, was surprising. We're using the same, or very similar, tools in our different areas. We're both using group algebras (he's going me one better, looking also at semigroup algebras), and the stochastic matrices he's working with, viewed from the right angle, bear a strong resemblance to my central weighted transforms (although mine are a bit more broadly defined).

I mentioned this to him after the talk, and he wants to see some of it written up - but most of that comes from my work on G-gons, which I haven't written up, and can't, for the time being. But maybe we'll sit down sometime and chat about it.

This is, in large measure, what mathematics is about: identifying commonalities and studying their implications, showing connections between disparate fields. I'm wondering if what he's doing might have some application to my work. I don't see how as yet, but you never know...

Many years ago, I was messing around with different ways of counting on my fingers. Taking "up" and "down" as bits, it's possible to count to 1023, but certain numbers - e.g., 4 - become, um, unacceptable. I tried working with base six for a while, which allows counting up to 35, but eventually settled on a modified base-five system - kind of a Roman numerals thing, with the fingers of one hand representing 1, the thumb of that hand 5, the fingers of the other hand 10, and the other thumb 50. This lets me count up to 99, which is useful. (If I need to get over 100, I enlist my big toes.)

I just now discovered that I had reinvented chisanbop (although without the press-your-fingers-on-the-table bit). I swear, I didn't know.

There is a small piece of the classification problem that has been niggling at me almost from the beginning. It has to do with "magnitude forms", which are fairly simple first-order forms. Consider, for example, hexagons. The sum of the squares of the sides is a magnitude form. The sum of the squares of the short diagonals - if the hexagon is ABCDEF, I mean AC, BD, CE, DF, EA, FB - is another, and the sum of the squares of the long diagonals AD, BE, CF is yet another.

Unfortunately, these forms are not "invariant", which is a highly desirable property. The invariant magnitude forms for hexagons are... harder to describe, but there's an easy way to write each of the side/diagonal forms, above, in terms of the three invariant forms m1, m2, m3. Reversing that - writing the invariant forms in terms of the side/diagonal forms - has been a nagging problem. I've been able to work the reversal for everything up to hexagons, but a general reversal has eluded me.

Today, while contemplating magnitude forms on G-gons, I figured out the reversal in general for one set of possibilities, for another, for yet another... and it hit me that there was a definite pattern, and an easy one.

I now have a general technique for writing the invariant magnitude forms in terms of the side/diagonal forms, and it's almost the mirror image of the technique that runs the other way. I shoulda suspected...

[The song below is apropos, for I have slain a giant!]

I've decided that I'm trying to do too much, too soon in Taxonomy III. I've decided to detach a small piece of it, and move the rest all the way down to Taxonomy VI. The old IV, V, VI will each bump up one.

The problem is that, to do the Varignon stuff the way I wanted to, I had to build up a great deal of machinery before getting to the payoff, and much of that machinery is useful, separately, for other things. So, I'll take a piece of the machinery and exploit it in the new Taxonomy III, discussing first-order pure classes. Another chunk will go into the new IV and V, covering various kinds of mixed classes. Then, when I get to VI, the remaining machinery will be of a digestible size.

I don't regret the work I put into III this summer; I've learned a bunch of new things which will illuminate the other papers. Still, it's a bit frustrating.

I'm groping my way towards an understanding of first-order classes of G-gons. I already have a pretty good picture of those for traditional polygons, and most of the techniques I developed there carry over. Some things need significant revamping, and I have to keep reminding myself of the changes. At one point I had to dredge up memories from a seminar I took more than forty years ago...

Patterns are emerging, complex enough to be entertaining but simple enough to be graspable. This is the best stage of research, when "Here Be Dragons" starts to be erased and rivers, mountains, and other geographic features start getting names - descriptive ones, not "P River" or "J Mountain" but "cocentroidal class", "doubly centroidal orthoclass", and the like, coded to remind the user of their meaning.

This is fun.

I still have to finish writing Taxonomy III.

I'm somewhat annoyed with myself.

The "something unexpected" in my research which I spoke of was an illusion; I had neglected one significant fact, and I didn't need software support to figure it out. It didn't happen, and it doesn't happen; I was able to reduce the general case to the Zn case, where I'd already worked out what happens.

What galls me is that I made the *exact same mistake* back when I began investigating polygons. Admittedly, it only took a couple of days to figure it out this time, as opposed to the two or three weeks it took last time; that improves my mood some, but not a lot.

On the other hand, this week's misadventure does suggest some possible linkages between... well, no, I won't try to explain that. I may be able to salvage something useful, is all.

I said before that one of the purposes of abstraction was avoiding the need to do the same thing over and over. There's another purpose, though, as I am re-learning in my study of G-gons.

Consider the concept of area. I mentioned before that a mathematician's notion of area isn't quite consonant with the layman's - that a polygon can have negative area, for example. There's more, though.

There's no particular ambiguity about the area of a triangle or quadrilateral. The area of a pentagon, however, is the sum of two completely independent quantities. Any pentagon can be written as the sum of two pentagons, one of which is an "affine-regular pentagon" and the other an "affine-regular pentagram". One way of determining whether a pentagon ABCDE is affine-regular is this: each of the sides - say CD - is parallel to a diagonal - in this case, BE - and the diagonal is precisely phi times as long as the side, where phi is the golden ratio. There's a similar test for affine-regular pentagrams. Anyway, the area of the original pentagon is the sum of the areas of the two affine regular polygons, which are completely unrelated. The same sort of thing applies to polygons with n sides; if you let m be (n-1)/2, rounded down, the area of an n-gon is the sum of m independent quantities, in the same way.

There's another way of looking at areas. Again, with ABCDE, you can compute the usual area of ABCDE and, also, the area of its 2-twist ACEBD. These quantities are also independent, and if you know both of them you can work out the two areas I mentioned in the last paragraph. With hexagons, you'd look at the area of ABCDEF and the sum of the areas of ACE and BDF, and so on up the line.

I've known about this stuff for quite a while, but trying to extend the concept of area to G-gons, I've been forced to confront it again, and I understand it better now. For any choice of G, there are multiple independent measures of "area", in two different categories. There are "character-based" area measures - these are analogues of the stuff in paragraph three, but involve some complicated mathematics - and "element-based" measures.

I can describe the latter simply, I think. The thing about abelian groups like G is that you can add their elements together; for example, with Zn, the elements are remainders mod n, and you add them mod n. For V - whose elements are pairs of bits (00, 01, 10, 11) - you can add them component-wise mod 2. (That is, 10 + 01 = 11, 10 + 10 = 00, 10 + 11 = 01, and so on.) Now, a G-gon is a set of points in the plane, each labeled by one of the elements of G. Let Z be the center of gravity of those points, and let x be an element of G. Then the "x-area" of our G-gon is the sum of the areas of the triangles with vertices Z, v(y), v(x+y) as y ranges over the elements of G. For our pentagon ABCDE, the 1-area turns out to be the area of ABCDE, and the 2-area is the area of ACEBD. (Draw a picture: A = v(0), B = v(1), and so on.)

I'm beginning to see how these two different kinds of areas - character-based and element-based - are related to each other for G-gons; what I saw dimly for polygons is now coming into sharper focus. And that, friends, is another of the things abstraction is for.

(By the way, those misbehaving groups I mentioned before, like V? If G is one of those groups, every G-gon has area 0. Areas, I should say: all of them are 0, always. Those groups are *really* distinct from the rest.)