Thursday, September 30, 2010

The Inflection Point

I know, you were thinking, what this art blog needs is more math. Well, I've got ya covered. Let's talk about what I was thinking about during calculus class, when I should have been thinking about calculus.

A little terminology for you. This isn't tough stuff, trust me. You may think you're bad at math, but you're not. I'm not going to get into the underlying equations and whatnot, which, sure, that's tough. We're just going to look at the visual part here, which is the math equivalent of eating your pudding without eating your meat.

Here's a line (mathematicians call it "the graph of a function", not a "line"):

As you can see, on the left it is called decreasing, because as you travel right along the line, you also move down. On the right, it is called increasing, because as you travel right, you move up. The minimum is called a minimum because it's the lowest vertical point on the graph. It separates the decreasing and increasing regions.

Overall, this kind of line is called a convex function.

Now, let's look at another line:

This is called a concave function. Instead of having a minimum in the middle, it has a maximum.

Now, let's say you have a function with convex and concave regions:

That point in the middle where one kind of curve turns into another kind is called an inflection point.

That's all the terminology we need. Let's sum it up:

Perhaps you are objecting, Hey Maidman, that big middle part, the "increasing" part? What the hell is the difference between "convex" and "concave" in that part?

Good question. It took me ages to keep this straight in my head. To the left of the inflection point, it's "convex increasing." To the right it's "concave increasing." This is the difference between convex and concave:

If you draw a straight line between any two points in a convex function, that straight line will be above the curve. If you draw it between any two points in a concave function, the straight line will be below the curve.

I'm glad I could help you out there.

Anyway, what I was thinking in calculus class was this - that this is a good simple model of the complexity of lines in artwork. And this helped me to make quantitative sense of something I liked about Picasso - his marvelously complex and fluid line. I'm not going to show you a good Picasso, because that would be distracting. Let's look at a crappy Picasso - in fact, a dashed-off picture of an ass:

Really look at that outline on the right. It would have been so simple to produce a single curve, a unified sweep of the hand. But Picasso cannot do that; his hand has fidelity to his eye, and he represents the separation of anatomical structures. Or, in our current framework, he does this:

The single curve is represented by means of five and a half complete convex and concave regions separated by five inflection points.

Maybe you're wondering, Uh, wouldn't *anyone*'s art look like that if you were anal enough to analyze it that way? The answer is no. Let's look at a drawing we've studied before, by Matisse, whom I also like very much:

Look at the line of the right side of the back. You can do it mentally by now - one inflection point separating one convex region from one concave region. We both know backs are much more complicated than that. There is a huge distance between Matisse's complexity here and Picasso's complexity in a similarly simple drawing.

There are only three artists I think of as having "perfect line": Da Vinci, Ingres, and Picasso. When I bring up this bizarre classification, the idea of "the perfect line," people usually bring up their own favorite draughtsmen, and say, well, what about so-and-so?

I'm not discounting so-and-so. So-and-so is really very talented, and has a strong personal style and vision. All I'm saying is that I, personally, have a sense that there is some quality, unique to these three draughtsmen in the history of art. This decomposition of line into functional categories helps to narrow down what I'm talking about, but it does not define it. Math is only one facet of the perfect line - it's just the part I happen to be thinking about today. Let's look at a couple more drawings, both by super-talented and worthwhile artists. Here's a man sketched by Seurat:

Notice that upper arm. It is a single concave region. Its complexity comes not from shifts in convexity/concavity, but rather from a breaking of the curve into a number of short nearly straight line segments:

For me, this is not as satisfying a degree of complexity as we find in the Picasso.

Now let's look at Annibale Carracci:

Examining the same region of the arm as we did in the Seurat, we find five inflection points - the same number we saw in the Picasso, but in a much smaller length of line:

This, to my eye, is too dense - too much variation in this particular system of analysis. It may be accurate, but it is not perfect - it does not have that scent of truth that I see in the middle-ground of complexity taken by Da Vinci, Ingres, and Picasso. Is it fair that we looked at a soft female outline with Picasso and a muscly dude with Carracci? Absolutely not. Life isn't fair.

So this perfection I am describing, when analyzed through a strictly mathematical lens, can be described as having a quality of "some, but not too many, inflection points." Does this refined description define the perfect line? It still doesn't. I tend to think that it is undefinable.

Here's an aside - all this science and math I throw at art on this blog isn't an attempt to demystify art: it's an attempt to carve away ignorance, mostly my own. If ignorance is a state of lacking knowledge, and mystification is a state of beholding mystery - then I want to distinguish between the knowable and the unknowable. The science and math doesn't take away the mystery - it helps light our way to the boundary of the real mystery.

So our refined description of line doesn't tell us what the perfect line is, but it isn't useless either - it brings us closer to understanding a category for which neither a word nor a definition exists. That's not nothing.

What does it buy us? Well, I can tell you what it bought me. It was 1994 when I sat in calculus class, doodling the arm of the girl in front of me while thinking about the mathematical description of complex line.

In 1998, I started to draw seriously. And I marched in armed with this conception of the mechanical underpinning of the complex line. I would look at the edge of the model, and say, "Aha - here is a maximum! Here a minimum! Here is an inflection point, and this is the character of the curve between these two points."

This is something I did with metal-nib pen and an inkwell around that time (a very fun medium which I highly recommend):
I thought it was pretty neat. But I also thought I was able to make rapid progress, because I had this one useful way of looking at a complex phenomenon and breaking it down into simple enough phenomena that I could hope to grasp and represent each one with fidelity. Better still, perceiving the complicated, I could decide what to keep and what to throw away, consciously. I simplified nature by choice and not by being overwhelmed.

So - this is a system of analysis which is probably most useful to somebody with a bit of a geometer's edge, like me. But it demonstrates, I think, the general idea that deeply unromantic tools furnish the art student with the means to struggle toward those depictions of beauty and grace they may hope to reach.

Do I think this way anymore? Of course not. The entire tool has become automatized; it is a subliminal process which I don't need to consciously check on to benefit from. Stanislavsky says that the actor should learn all his fussy techniques, and get soaked in them to the bone, so that when he hits the stage, he can forget all of it, and simply act - but act in the context of continuing to be informed by the technique he has absorbed.

When I was absorbing these basic techniques, I called myself an art student. When I had them licked, I called myself an artist. I am still learning techniques, of course, but not the very foundational ones that you need in order to make a picture at all. This doesn't mean I know all of them - I'm excited to find other fundamental techniques I've overlooked. But, you know, at some point you have to either stop making pictures, or call yourself an artist, I guess. For me, learning those techniques was the dividing line.


  1. Very interesting. But it seems that from your description of perfect lines and inflection points that these concepts serve the artist more than the non-artist who views the art. Would you agree?

    Also, as fodder for your suggestion box, perhaps one day you can write a post about when more inflection points are appropriate and when fewer inflection points are appropriate. Thanks.

  2. Chris -

    Thanks for the thought; absolutely, this is of more practical use for the artist than the non-artist art viewer, although I think it is of at least some interest to the avid art viewer. Primarily, perhaps, it serves to flesh out what is actually meant by such vague statements as, "Oh, he's an artist with a very *mathematical* outlook." This sort of thing annoys me when I hear it - I want detail and substance. I'm hoping to provide, in this blog, a little bit of the real nitty-gritty, the machine language, of making art.

    To quickly respond to your request: skinny people and muscley men - more inflection points. Consider Egon Schiele. Fat people and curvy women - fewer. Turn your attention, if you will, to Klimt's drawings.

    Of course, a more detailed reply could be given, but, as you say, you'll have to wait on "one day..." with, perhaps the addendum, "...when hell freezes over." It's just not so interesting, right?

  3. Sorry, that last bit was blithe. If it were so boring, you wouldn't have asked; and I appreciate your asking. I'll keep it in mind and if I find something that meets my exacting, yet scattershot, standards of interesting, I will totally write it down.

  4. In the book Only the Paranoid Survive, former Intel CEO Andy Grove wrote about what he called Strategic Inflection Points. This basically refers to a crisis when the fundamental rules of the game have changed. The challenge is to adapt to the new reality or perish.

    I hope this comment is not too random, but
    your graph explaining what an inflection point is reminded me of that.

  5. Andrew - not at all! When I was getting ready to write this post, I thought, "What was the name of that Gladwell book? It wasn't 'The Inflection Point,' was it?" Since it wasn't, I figured I was safe. I guess Grove beat me to the punch though. Humph.