Thursday, September 25, 2008

Poetry primer

I promised a couple of folks who'd attended this past week's CWPA with me that I'd provide a layperson's guide to the math-based poetry I've begun to write. You'll find just such a guide below, but first I wanted to list the links to posts that have the most to do with math and poetry, for easy access:

  1. "I sing the verses eclectic": this is the first post in which I showcase math-themed poetry written by my Calc I students from Fall 2007. This was almost immediately followed by...
  2. "Round two," the second such post.
  3. "Ars poetica" is the post in which I laid out my idea for (G,φ)-grams, examples of which are explained more fully below.
  4. "Six: being a modular G-gram in 14 lines" is the first post containing a G-gram, "Six." (Its structure is explained in layperson's terms below.)
  5. In "Deep thoughts and more dilettantish dalliances" I printed the first non-modular G-gram. Its structure is a bit harder to describe, but I'll attempt to draw a rudimentary roadmap in a later post (when I'm not so tired!).
So what's, uh, the deal?

Here's the idea: anyone who as a kid played at all with codemaking and codebreaking knows how one assigns numbers to the letters of the alphabet: "A" corresponds to 1, "B" to 2, and so forth, until "Z" corresponds to 26. We'll be making heavy use of this correspondence, so keep it in mind. Mathematicians might denote this correspondence as a certain kind of function (the fancy-schmancy term for it is homomorphism), let's call it φ, that takes each letter over to its corresponding number:

φ(A) = 1, φ(B) = 2, ..., φ(Z) = 26.

And yes, this φ is precisely the φ occurring in the term "(G,φ)-gram."

It might help you to think of the number associated with a particular letter by φ as a the "value" of that letter. Next we're going to compute the values of words by "adding up" the values of the letters the word comprises, but the way we perform addition is going to be a bit wonky.

In what way? We're going to "wrap around" so that our sums never go past 26. For instance, if we add 13 + 4 = 17, since this number is less than or equal to 26, we're golden. 13 + 8 = 21 is still less than or equal to 26, so we're fine. But if we add 13 + 18 = 31, we've gone 5 steps past 26 (27, 28, 29, 30, 31), so the number 31 is really going to be the same thing as 5, by our reckoning. Similarly, 32 = 6, 33 = 7, ..., 52 = 26. And what then, past 52? We start wrapping around again. That is, 53 = 26 + 26 + 1, so 53 should be equal to 1 again. 54 = 2, and 55 = 3, and so on.

By the way, the motivations one might have for counting like this are many. For instance, classical computers do all of their computations in binary arithmetic, a form of counting in which the only digits allowed are 0 and 1; in such arithmetic if you add 1 to itself you have to wrap around to 0, so that 0 + 0 = 1 + 1 = 0, and 1 + 0 = 0 + 1 = 1. We're doing the same thing, only we don't wrap around right away, we wait until we get to 26.

The set of numbers {1,2,3,...26}, along with the "operation" of wrap-around addition, forms an object mathematicians call a group. Groups are often denoted by the letter G, and the "G" in the term "(G,φ)-gram" refers to precisely this sort of object. There are simply oodles of interesting examples of groups. The group G we've described here is called the modular group of order 26 and is a member of one of the most useful families of groups there is.

Now let's apply this weird kind of "wrap-around" arithmetic to words. Take the word "WORD," for example. Since φ(W) = 23, φ(O) = 15, φ(R) = 18, and φ(D) = 4, the value of this word is 23 + 15 + 18 + 4 = 60. But since 60 = 26 + 26 + 4, 60 really should be 4 by our reckoning. The upshot is that φ(WORD) = 4.

So what in the heck does this have to do with the poem "Six"?

If you're blessed with a great deal of patience, apply φ to each of the lines of the poem, one at a time. (Spaces and punctuation don't count for anything, so you've only got to worry about the letters.) If you do all of your addition right you'll find that the "sum" of any given line is the number 6. Kinda cool, huh?

Notice also that there are 14 lines to the poem, and therefore if we want to compute the sum of the entire poem, we only have to add 6 + 6 + ... + 6, 14 times. This sum is

6 x 14 = 84 = 26 + 26 + 26 + 6 = 6,

in our wrap-around arithmetic. Thus the whole poem has the value 6 as well!

Finally, note that the title of the poem has the value φ(SIX) = 19 + 9 + 24 = 52 = 26 + 26 = 26. But what happens if we add 26 to any number between 1 and 26? You get back whatever number you started with! For instance, 17 + 26 = 17, since by taking 26 steps, you wrap around to 17 again. This means that even if we tack the value of the poem's title onto the value of the poem itself, the result doesn't change: 6 + 26 = 6. (26 acts exactly like 0 in this regard, and for that reason mathematicians usually call it "0" and talk about the set {0,1,2,...,25} instead.)

I hope this makes some sense.

As I mentioned above, I'll try to describe "Inverse" in layperson's terms in a later post.

Before I tackle that task, though, I'll be posting a 12-part essay inspired by my conversations with the writing folks at CWPA this past week. In these next several posts I'll be using this space as a proving ground for some of my recent thoughts on writing and writing pedagogy, and likely on college teaching in general. I hope that this window onto my process will prove meaningful to some of my readers, students and faculty alike, and that all will feel free to chime in with thoughts of their own.

I'll try to get the first post in this series out tomorrow, depending on the status both of the first of the presidential debates and of my sobriety, the both of which are sure to be inextricably linked.

For now, however, I'm off to bed, as it's late and I've got another full, fun day tomorrow.

1 comment:

Kaz Maslanka said...

Fun stuff --
Thanks Doc!