The crying of x^2 + xy + y^2 = 49

October 25, 2009 10:00 PM

"Pynchon, postmodern author, is commonly said to have a non-linear narrative style. No one seems to have taken seriously the possibility, to be explored in this essay, that his narrative style might in fact be quadratic." Number theorist Michael Harris on Pynchon and conic sections.
posted by escabeche (60 comments total)

This makes about as much sense as Gravity's Rainbow.
posted by BuddhaInABucket at 10:02 PM on October 25 [4 favorites]

Gravity's Rainbow is a parabola, right?
posted by gimonca at 10:03 PM on October 25 [2 favorites]

Gravity's Rainbow, obviously dominated by the image of the parabola, and full of explicit references to the shape

Well, there you go.
posted by gimonca at 10:05 PM on October 25

If Pynchon's work is quadratic, doesn't that mean he's writing the simplest possible nonlinear novels? I like my fiction to be cubic at the very least.
posted by lukemeister at 10:21 PM on October 25 [1 favorite]

Every time I see Pynchon's name somewhere, I assume that he's dead.

This was a pleasant surprise, though. Great find.
posted by Limiter at 10:24 PM on October 25

lukemeister: It isn't that simple. Most people can't get through Gravity's Rainbow in polynomial time.
posted by Limiter at 10:25 PM on October 25 [15 favorites]

There is no difference between the behavior of a god and the 4 simultaneous 24-hour days existing within a single rotation.
posted by griphus at 10:25 PM on October 25 [6 favorites]

Stop putting literature in boxes.
posted by turgid dahlia at 10:35 PM on October 25 [5 favorites]

Philip K. Dick was better.
posted by bardic at 10:42 PM on October 25 [1 favorite]

conic sections

The Shadows of God are not God Herself.
posted by Blazecock Pileon at 10:42 PM on October 25

This is excellent. Really excellent.

It doesn't even matter if Pynchon didn't indend to write quadratically, or if he didn't rigorously prove his text to be quadratic before publishing. Fact is, his works are both spacious and rigid enough in meaning, alliteration and style to easily support the plausibility of Michael Harris' ideas.

And it's literature, beautiful ideas put forth beautifully. And I very much appreciate the beauty of what Harris is saying. Even though I may be imagining it all. (c.f. paranoia).
posted by krilli at 10:50 PM on October 25 [4 favorites]

No, no, no. Pynchon's narrative style has complexity O(n α(n)). It's a serious pain in the ass to analyze, but for all practical purposes, it's linear with a HUGE constant factor.
posted by erniepan at 10:56 PM on October 25 [1 favorite]

Gravity's Rainbow is a parabola, right?

It's late and I'm tired, but aren't rainbows circular arc segments?
posted by sebastienbailard at 10:57 PM on October 25 [1 favorite]

Sebastien, yes, but gravity's, the arc of a V2 rocket falling through the air, is a parabola.
posted by krilli at 10:59 PM on October 25

There's a lot of interesting discussion of David Foster Wallace and the role mathematics plays in his narrative style. One of the Bookworm podcasts (The one for Infinite Jest, I think) has a long discussion on this issue. It's also mindblowingly smart, listening to the two sharpest guys in the room talk about one of the best books of the FOREVER.
posted by GilloD at 11:19 PM on October 25

Ahem: See here.
posted by GilloD at 11:19 PM on October 25 [4 favorites]

tl;dr version:

His theory is that Pynchon patterned four of his novels after 4 conic sections:

V - intersecting lines (v shape)
Gravity's Rainbow - Parabola
Mason Dixon - Ellipse
Against the Day - Hyperbola
posted by empath at 11:36 PM on October 25

No, no, no. Pynchon's narrative style has complexity O(n α(n)).

If by that you mean it's intensely masturbatory, I fully agree with you.
posted by skwt at 11:51 PM on October 25 [16 favorites]

The parabolic shape of GR was apparent to me while I was reading it, with its amazing sense of paranoia and connectedness completely overwhelming me by the middle of the book, and then my sense of confusion as all the threads were disentangled on the downward slope. It completely blew me away -- I'd never read a book that did that before (or since), and the experience sticks with me even now, well over a decade later.

I also got the V structure of V., although it was a much simpler concept and didn't leave me as awestruck as GR.

M&D... I'd have to read that again. And ATD... Um... I tried, I really really did. Couldn't get through it, sadly.

I guess I'm a freak, but I love the recursive nesting doll narrative in Vineland best of any of his literary structure tricks. That he makes it SO clear which level of the narrative you're in at any moment, even sometimes shifting between them unexpectedly but still having it just work... Really stunning.

I'll have an actual verdict of his latest novel here one of these days. It's a fun read thus far...
posted by hippybear at 11:52 PM on October 25

If by masturbation you mean "something everybody likes but you don't want to admit it", then yes, skwt.
posted by krilli at 11:53 PM on October 25 [3 favorites]

Relax, it was a pun. I think he's an ok writer.
posted by skwt at 11:55 PM on October 25 [1 favorite]

AH I didn't understand the equation. See it now.

(Please note that my last comment was joky and self-humiliating – effectively, I loudly stated that I like to jerk myself off.)
posted by krilli at 12:00 AM on October 26

And yet the only book for which any of those claims is substantiated is—heh—the degenerate case of V., two separate narratives that intersect. He notes that Rushdie compared GR to a parabola but doesn't say why, and whether or not the book is dominated by images of parabolas has nothing to do with whether its narrative structure is parabolic. For that matter it seems to me that we lack any definite understanding of what a parabolic narrative would be: likewise hyperbolic, circular, elliptical, or punctual. (Points are also conic sections, after all.)¹

"Linear" is not actually a good metaphor for what are called linear narratives, which are basically those in which the order in which you encounter descriptions of the goings-on is the same as the order in which the goings-on go on. (This doesn't necessarily correspond to page numbers, of course: Hopscotch is a linear narrative, either way you read it.) It is true that if you think of the order of readerly time as the x-axis and the order of narrative time as the y-axis, you can express this relationship with the formula for a line (with positive slope). But then, y = log (x+1) also has that property. Would you call a book in which the amount of time that passes in the narrative gradually decreases over equal page-spans one with a logarithmic narrative? No. That is still a linear narrative. Indeed, y = x² also has that property. So you might think that a narrative that gets really hectic (and fast!) is a parabolic one. But really, as the term is used, that would still be a linear one.

Presumably this guy is using "linear narrative" in a different way than is customary, so it would be nice if he said what that was. So there are lots of hyperbolas in Against the Day: that doesn't show that it was structured hyperbolically or that it it is narratively hyperbolic as opposed to linear (two things that are utterly independent of one another, to the extent that the second can be given content). Nor does it help us understand whether Luc Sante and Louis Menand really were competent to pass judgment on the novel, though I'm rather inclined to think that they can still be trusted somewhat.

[1] The closest approximation to a punctual narrative might be one in which everything happens at the same time; however, that description could never distinguish punctual narratives from so-called "linear" ones with multiple viewpoints, since it takes time for anything to happen and as soon as you've got time, and happenings, on the scene, you're back in the world of linearity. The book that comes closest to a punctual narrative might be Life a User's Manual, though it, of course, has plentiful encrustation in the way of side- and back-story.
posted by kenko at 12:01 AM on October 26 [3 favorites]

erniepan: Since α(n) is < 5 for all practical values, doesn't O(n α(n)) come out to linear with a very SMALL constant factor?

I might be misreading that -- I know Ackerman's function, but haven't dealt with the inverse before.
posted by Limiter at 12:36 AM on October 26

A punctual narrative, by that definition, would be one in which everything happens at one time and is described at the same time. A photograph, say.

A parabolic narrative by that definition would be one which runs either backwards or forwards to some event, dwells for a bit, then reverses course. Use of Weapons has this structure. I can't think of others offhand, even though I have the feeling that it's pretty common.
posted by hattifattener at 1:14 AM on October 26

Seems like hyperbole to me.
posted by twoleftfeet at 2:33 AM on October 26 [3 favorites]

Well, quadratic functions are also nonlinear. So there you go.
posted by delmoi at 3:39 AM on October 26 [1 favorite]

"... O(n α(n)) ... " ... If by that you mean masturbatory

for those that didn't get the joke...
posted by delmoi at 4:06 AM on October 26 [3 favorites]

Metafilter: making Pynchon that much more intimidating.
posted by not_on_display at 4:06 AM on October 26 [7 favorites]

Hippybear, looking forward to your verdict. I just finished "Inherent Vice" and can only say "shaggy dog flashbacks, man," but then maybe the novel is just fractal weirdness?
posted by MonkeyToes at 4:44 AM on October 26

If it's that important to know, can't someone just ask the dude, "Hey Thomas, are your narratives quadratic or anything?"
posted by gorgor_balabala at 5:15 AM on October 26 [1 favorite]

"If it's that important to know, can't someone just ask the dude, "Hey Thomas, are your narratives quadratic or anything?""
posted by gorgor_balabala

Pynchon's infamous for his inaccessibility, both on and off the page.
posted by not_on_display at 5:19 AM on October 26

"Sebastien, yes, but gravity's, the arc of a V2 rocket falling through the air, is a parabola."

If you disregard air resistance.
posted by Eideteker at 6:16 AM on October 26

Is this guy reading the same Against the Day as I am? It clearly was modeled after the Riemann zeta function, there was even a scene where the protagonist is walking on the beach and notices that the path he is taking is curiously similar to the polar graph of the Reimann Zeta. Then there were graphic sex scenes that attempted to take place in four states at the same time ... so I don't know if you can draw any conclusions other than Pynchon spends all day in his room reading math textbooks.
posted by geoff. at 7:02 AM on October 26 [1 favorite]

Parabolic? Linear? Does it matter? Most of his novels are like a billion pages long, so you'll probably put them down halfway through and go have a beer and make out with someone and forget all about reading Pynchon because you've got other stuff to do. At least that's what usually happens to me, with Inherent Vice and The Crying of Lot 49, being exceptions.
posted by dortmunder at 7:44 AM on October 26

"DILL(v.o)
Lilly and Ian whop our asses
And stole are car. I should of help Beetlejuice…."
-Thomas Pynchon, from the introduction to his short story collection Slow Learner
posted by Damn That Television at 7:47 AM on October 26 [1 favorite]

So he's just making wild guesses based upon Pynchon's largely correct usage of mathematical terms?

I think mathematics actually has some bearing on constructing non-linear story lines. In particular, you want story lines that appear linear when restricted to various factors, like the events that befall one character or occur in one location, while globally the story cannot exist without close time-like curves. You could get more local time consistency if you allow more complex plots built on more complex digraphs. I'd think the challenge would be finding small plot digraphs that admit non-linear plots which are more highly non-obviously non-linear.
posted by jeffburdges at 7:51 AM on October 26

My narrative is pi to 42 digits.

Must we quantify everything? 11
posted by iamkimiam at 8:02 AM on October 26

This was great. I loved Against the Day, and, 'correct' or not, this reading of the book is really interesting.
posted by statolith at 8:15 AM on October 26

this reading of the book is really interesting.

What reading? There was a reading there?
posted by kenko at 8:27 AM on October 26

This is kind of fascinating, but I find I come around to the view I hold on musical works, as well, best put by Steve Reich:

The use of hidden structural devices in music never appealed to me. Even when all the cards are on the table and everyone hears what is gradually happening in a musical process, there are still enough mysteries to satisfy all.

posted by LooseFilter at 8:51 AM on October 26

and whether or not the book is dominated by images of parabolas has nothing to do with whether its narrative structure is parabolic

Not sure it has nothing to do with it; the author, after all, might be choosing imagery purposely to resonate with or draw your attention to the narrative structure.
posted by escabeche at 9:06 AM on October 26

I always had the impression of some sort of complex topology hidden away in Pynchon's novels, but couldn't really explain the sensation accurately. This is a completely awesome essay.
posted by Aquaman at 9:15 AM on October 26 [1 favorite]

Not sure it has nothing to do with it; the author, after all, might be choosing imagery purposely to resonate with or draw your attention to the narrative structure.

That would be extrinsic and have to do primarily with the author, who after all creates both the imagery and the structure. In any particular case the author might put in such imagery because he's also using such and such a structure, but in principle the two are independent.
posted by kenko at 9:18 AM on October 26

Reading the article, which was very interesting and strikes me as a perceptive and useful way to look at the books, and reading the comments here, it seems like it might be more useful to name what Harris is talking about something like "thematic structure" rather than "narrative structure." Narrative structure already has a technical definition, which involves the order in which plot events are presented, and that's not quite what Harris seems to be talking about. He's talking more about the themes of each book, the arcs described by the characters and events that befall them, and the thematic progression. It is, admittedly, a very fuzzy notion to imagine a kind of graph of some set of characters relations to their environment over time, but it works for me. I think this idea actually sheds a lot of light on why I find Pynchon so satisfying to read.
posted by rusty at 9:22 AM on October 26

Can't you just read for the fun of it?
posted by tommasz at 10:22 AM on October 26

Can't you just read for the fun of it?

Well now, see... That's why I picked up Gravity's Rainbow. A friend of mine read out loud to me the English Candy Drill sequence, which had me laughing so hard I was in tears, and I decided that I absolutely had to read the book. I did so without any other knowledge of what it was about, or what it was supposed to be... I read with no annotations (which probably would have helped AND hindered the process), and I was commuting 90 minutes a day to work on the lousy city bus system in Las Cruces NM, so I had lots of enforced reading time.

During the course of the book, I was transformed. The parabolic structure of the book isn't about the plot; it reflects exactly a description of the course of a V2 rocket which is in the book (which I'm too lazy to dig up now). Pynchon describes the rocket as being very crudely aimed, and that it starts off from some location within Germany, not necessarily public knowledge, and that as it burns its fuel its path becomes more and more deterministic and laden with destiny, but then it burns out its fuel and tops the parabola and as it falls back to earth, the location wherein it will land becomes less and less certain until finally it strikes at a fairly random place.

I had no idea when I was reading that, that I was ingesting what was about to happen to me. But the layers of paranoia and the deterministic threads of the novel weave themselves together SO tightly by the middle of the book, I was seeing patterns and significance everywhere I looked within my own life. But all those untangle in the second half, and by the end it's all just white noise with no deep intent.

I'd never read a book like that before, or since. I am sure it's not a book for everyone, but for me, at that point in my life, it undid me in a way no other book has before or sense.

So, yeah, I was reading for the fun of it, but "fun" transformed during the course of my experience.
posted by hippybear at 10:31 AM on October 26 [3 favorites]

Any rocket that is burning fuel for the first half but not for the second half is very unlikely to describe anything like a parabolic arc. Is that actually what Pynchon said?
posted by louigi at 12:12 PM on October 26

from chapter 7 of V ("He" is Sid Stencil):

"He had decided long ago that no Situation had any objective reality: it only existed in the minds of those who happened to be in on it at any specific moment. Since these several minds tended to form a sum total or complex more mongrel than homogeneous, The Situation must necessarily appear to a single observer much like a diagram in four dimensions to an eye conditioned to seeing the world in only three ... it was a neat theory, and he was in love with it.The only consolation he drew from the present chaos was that his theory managed to explain it."

I am reminded of Stencil's Theory of Situations a lot when reading TP.

I've re-read V, GR, Vineland, and IV (read it twice straight through) this year, and I'm now re-reading Against the Day (the first time, a few years ago, like GR the first time, was a bit too much *whooosh*). Like almost all of his books, it's better the second time around for me.
posted by mrgrimm at 1:05 PM on October 26

Can't you just read for the fun of it?

For some of us, literary criticism is "the fun of it."

And hippybear has it right, for me. A new Pynchon novel like Inherent Vice (TP has to know how much that sounds like Infinite Jest, no?) is as fun for me as reading Dan Brown or other popular mystery stuff is for mystery-novel fans.

Anyway, the essay was a little rough, but a very interesting analysis to me.
posted by mrgrimm at 1:21 PM on October 26

Mason & Dixon described a lovely parabola in the air when I flung it away from myself as hard as I could.
posted by Skot at 1:21 PM on October 26

What does that make Proust?
posted by broken wheelchair at 1:38 PM on October 26

broken wheelchair: What does that make Proust?

Ellipsis.
posted by shakespeherian at 1:51 PM on October 26

I think "Infinite Jest" was based on the Sierpinski triangle fractal.

Oh, wait, it's the same interview, I think. Hey, look, a transcript!
posted by Pronoiac at 2:02 PM on October 26

Is that actually what Pynchon said?

There aren't any good pull-quotes to make my point, but I'm sure you can find an e-book version of the novel online (I did), and do your own searches for words such as "fuel" "trajectory" "parabola" and many others to get an idea of what he did write. I just spent 20 minutes doing that, got a quick refresher on what he did say but can't possibly summarize the entire book's contents about the V2 rockets and such here.
posted by hippybear at 3:13 PM on October 26

If you really want to get into it, you might want to join pynchon-l.

Also amusing: True Tales of Conversational Vengeance.
posted by muckster at 3:48 PM on October 26

...his narrative style might in fact be quadratic.

That's why we call them story arcs, duh.
posted by ZenMasterThis at 4:30 PM on October 26

For some of us, literary criticism is "the fun of it."

I'm so sorry.

A new Pynchon novel like Inherent Vice (TP has to know how much that sounds like Infinite Jest, no?) is as fun for me as reading Dan Brown or other popular mystery stuff is for mystery-novel fans.

Um, Inherent Vice basically is "popular mystery stuff," and will absolutely appeal to mystery-novel fans. Pynchon's a (sometimes) great writer, but let's not over-mystify him here.
posted by dersins at 5:10 PM on October 26

Those amused by quadratic fiction should definitely read Scott Buchanan's Poetry and Mathematics, in which Lewis Carroll's Alice is examined as a geometric figure.
posted by 0rison at 9:07 PM on October 26

Is that actually what Pynchon said?

Okay, I still haven't found the exact passage within GR about this, but wikipedia (which I know isn't considered an actual source by many here on the blue), does describe the path flown by a V-2 rocket as approximately parabolic.

My understanding is that a V-2 flies a path similar to a bottle rocket, albeit with less tumbling-action and a bit more planning as to where it lands. (And achieving much greater altitude.)
posted by hippybear at 9:54 AM on October 27