Forms and Formulae: Self-Avoiding Walk


A picture of the Sun peeking over the spine of The Princeton Companion to Mathematics as it rests on top of The Princeton Encyclopedia of Poetry & PoeticsThis is the fourth in a series called ‘Forms and Formulae‘ in which I write about articles in the Princeton Companion to Mathematics using poetic forms covered by articles in the Princeton Encyclopedia of Poetry and Poetics. This post’s mathematics article is entitled ‘The General Goals of Mathematical Research‘ and the poetic form is alba, which is a kind of song; I recorded it [direct mp3 link] using my robot choir and some newfound musical knowledge, and there are many notes on that after the lyrics below.

Here are some extracts from the article on the alba, explaining the features that I ended up using:

A dawn song about adulterous love, expressing one or both lovers’ regret over the coming of dawn after a night of love. A third voice, a watchman, may announce the coming of dawn and the need for the lovers to separate. An Occitan alba may contain a dialogue (or serial monologues) between lover and beloved or a lover and the watchman or a combination of monologue with a brief narrative intro.

The alba has no fixed metrical form, but in Occitan each stanza usually ends with a refrain that contains the word alba.

…the arrival of dawn signaled by light and bird’s song…

The watchman plays an important role as mediator between the two symbolic worlds of night (illicit love in an enclosed space) and day (courtly society, lauzengiers or evil gossips or enemies of love)

I based the song on section 8.3 of the article, entitled ‘Illegal Calculations‘. In retrospect, using the word alba in each refrain (are these even refrains?) doesn’t make much sense, since I’m not writing in Occitan, and the casual listener will not know that alba means ‘dawn’ in Occitan. But hey, it kind of rhymes with the start of ‘self-avoiding walk‘. How can I not rhyme an obscure foreign word with an obscure mathematical concept?

Introduction:
Mathematicians struggle even today to learn about the average distance between the endpoints of a self-avoiding walk. French physicist Pierre-Gilles de Gennes found answers by transforming the problem into a question about something called the n-vector model when the n is zero. But since this implies vectors with zero dimensions, mathematicians reject the approach as non-rigorous. Here we find that zero waking up next to its cherished n-vector model after a night of illicit osculation.

Zero:
I am just a zero; I am hardly worth a mention.
I null your vector model figure, discarding your dimension,
and every night I’m here with you I fear the break of day,
when day breaks our veneer of proof, and we must go away.

Here by your side
till alba warns the clock.
Fear’s why I hide
in a self-avoiding walk.

N-vector model:
Let the transformations of De Gennes show your place.
Never let them say we’re a degenerate case.
When I’m plus-two-n there’s just too many ways to move,
But you’re my sweetest nothing and we’ve got nothing to prove.

Here by your side
till alba warms the clock.
Fear can’t divide;
it’s a self-avoiding walk.

Watchman:
The sun has come; your jig is up. It’s time for peer review.
You think your secret union has engendered something new.
You thought you would both find a proof, but is it you’re confusing
The sorta almost kinda-truths the physicists are using?

That’s not rigorous,
says alba’s voice in shock.
All but meaningless
to the self-avoiding walk.

Zero and N-vector model together:
If you say that our results don’t matter,
then go straight to find a better path.
For as long as you insult our data,
Is it wrong to say you’re really math?

Hey there, Rigorous
at alba poised in shock,
you are just like us,
in a self-avoiding walk.

All voices are built-in Mac text-to-speech voices, some singing thanks to my robot choir (a program I wrote to make the Mac sing the tunes and lyrics I enter, which still needs a lot of work to be ready for anyone else to use.) Older voices tend to sound better when singing than the newer ones, and many new voices don’t respond to the singing commands at all, particularly those with non-US accents. So for the introduction I took the opportunity to use a couple of those non-US voices. These are the voices used:

Introduction: Tessa (South African English) and, since I also can’t fine-tune Tessa’s pronunciation of ‘Pierre-Gilles de Genne’, Virginie (French from France)

Zero: Junior

N-vector Model: Kathy

Watchman: Trinoids

Most of the bird noises come from the end of Jonathan Coulton’s ‘Blue Sunny Day‘, and I can use them because they’re either Creative Commons licensed or owned by the birds. The two peacock noises are from a recording by junglebunny. Free Birds!

As I mentioned, I’ve been learning about songwriting from John Anealio, and since the Forms and Formulae project sometimes requires me to write songs, I’m putting the new knowledge into practice sooner than I expected. This song uses several musical things I’ve never tried before, which is quite exciting, but it also means I probably didn’t do them very well, because there’s only so much I can learn in a couple of months of half-hour weekly lessons. I welcome friendly criticism and advice. The new things are:

Chords/guitars: People are always going on about chords, as if they’re the most important things in a song, even though I mainly pay attention to the melody. I happened to have a few lessons on chords around the time I was starting this song, so for most of the song I found a chord progression (using the Smart Guitar on the iPad version of GarageBand), figured out a tune to go with that, and only then wrote lyrics, which is interesting since I usually write words first, then a tune, then wonder what I need chords for. Technically, there were chords in The Numbers Are Not Enough as well, but I didn’t know they were chords, because they were arpeggiated. Or, if you’re a pirate, arrrpegleggiated.

A bridge: I heard there was such a thing, probably from Tom Smith’s songwriting workshop,  and decided I may as well try it out, so I spent a whole lesson asking questions about them, and then did my best. The bridge (the part that begins with ‘If you say that our results don’t matter…’) started out as lyrics with a tune, and then I hamfistedly added chords to that (following some advice about what bridges often do with chords) and I don’t think it worked very well. So I transposed the tune (that’s music speak for dragging all the MIDI notes up or down) so it would work better with some chords that I’m told are often used in bridges. Now I think it’s kind of okay.

Harmonies! I love harmonies, and in this song there are in fact two characters that sing together in some parts, so it was time I tried making some. I learnt a little bit about how they work and how they relate to chords, but not a whole lot. I think they sound cool anyway, though.

Cello: Previously, my instrumentation has been pretty much limited to triangles, meows, drum loops from Wikipedia, and Wilhelm screams. This time I tried to use the smart drummer in GarageBand, but it never seemed to do what I wanted it to do. Then as I was watching my footage of The Doubleclicks’ concert on JoCo Cruise Crazy 4 so I could upload it, I was inspired to add some cello. I used some cello figures that followed the right chords in the the ‘smart strings’ on Garageband for iPad, and added a few simple bits of my own. This is without having studied orchestration at all, so let’s hope they didn’t ruin it. I’m working on the assumption that there’s not much of value there to ruin, so I may as well experiment.

The article in The Princeton Companion to Mathematics was quite long, and I read the whole thing, despite the fact that I knew what the song would be about as soon as I read the subheadings and the text of section 8.3. I may not be writing these on a regular schedule, partly due to the long articles and the added difficulty of writing songs, but I’m really enjoying this project. My bachelor’s degree is in mathematics, and I last studied mathematics formally in 2002, and these days I tend to skip over the complicated equations in a text. The Princeton Companion to Mathematics uses a lot more words, and in a much more engaging way, than your average mathematics textbook. I read them all, and the equations they lead up to, and so far it’s like a pleasant stroll down memory lane, stopping to smell roses I’d forgotten about or never seen before. Maths is really cool, still!

The next Forms and Formulae will be an allegory based on the article, ‘From Numbers to Number Systems’. Once I’d read the article on allegory, I read the article on numbers and number systems as if it were already an allegory, and it really doesn’t seem like I’ll need to invent much. That article could easily be a tale of erasure, exclusion, and privilege.

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