The Novelist and the Gasket

You know who loves fractals?

Well, yes, everybody.

(I mean, can you blame them? Fractals are gorgeous monsters.)

In particular, I’m thinking of the late novelist David Foster Wallace. His writing was brilliant and baffling, wise and excessive – not to mention heavily math-inflected.

Consider this interview with Michael Silverblatt, about Infinite Jest:

MS: Something came into my head that may be entirely imaginary, which seemed to be that the book was written in fractals?

DFW: Expand on that.

MS: It occurred to me that the way in which the material is presented allows for a subject to be announced in a small form… and then it comes back in a second form containing the other subjects… and then comes back again… and I don’t know this kind of science, but I said to myself, “This must be fractals.”

DFW: I’ve heard you were an acute reader. That’s one of the things structurally going on, it’s actually structured like something called a Sierpinski gasket, which is a very primitive kind of pyramidal fractal. Actually, though, what was structured as a Sierpinski gasket was the draft that I delivered to [my editor] in ’94, and it went through some, I think, mercy cuts, so it’s probably kind of a lopsided Sierpinski gasket now.

What is DFW going on about? Well, the gasket is this critter, a mascot of fractal geometry:

Since we’re talking about a highly digressive writer, it’s worth wandering down the garden path that leads to the gasket. Its coolest feature is that it has a non-integer number of dimensions.

See, when you double the dimensions of a 1-dimensional shape, like a line segment, it creates two copies:

When you double the dimensions of a 2-dimensional shape, like a triangle, it creates four copies:

And when you double the dimensions of a 3-dimensional shape, like a pyramid, it creates eight copies:

In this sense, “dimension” is the answer to the question: “What happens when you double the shape’s measurements?”

Does it result in a single doubling? 1 dimension.

A double doubling? 2 dimensions.

A triple doubling? 3 dimensions. (Either that, or you’re watching James Harden. Who is also – the record should reflect – three-dimensional.)

But look! When you double the gasket, it neither doubles, nor double-doubles. Instead, something in between happens. You get three copies:

That makes the gasket bigger than one-dimensional objects (of which there’d be two copies). Thus, to trace out the gasket with 1D string, you’d need an infinite length.

But it’s smaller than two-dimensional objects (of which there’d be four copies). To cover the gasket with 2D paper, an infinitesimal scrap would suffice (as long as your scissor skills are on-point).

The gasket, it turns out, has dimension log₂(3), which is about 1.58.

Anyway, back to Wallace.

What makes Infinite Jest a gasket? In the LA Review of Books, Kyle McCarthy quotes Wallace’s rather unexplanatory explanation (“Its chaos is more on the surface; its bones are its beauty”). Then she offers one that I find more satisfying (if less specific to Infinite Jest). She writes:

The best novels already have a fractionalized quality – each chapter, and indeed every paragraph and sentence, reproduce in miniature its central conflict and arc.

I love this, for two reasons:

First, it’s truer to the nature of a fractal than Silverblatt’s description. It’s not that the pieces of a fractal “announce” themselves in advance, giving cameos that foreshadow later starring roles. It’s that every piece of a fractal recapitulates the structure of the whole; that its entire nature can be found everywhere, in microcosm.

Second, it suggests that every novelist is a maker of fractals.

We see fractals everywhere in nature: forked lightning, fluffy clouds, branching trees, expanding lichens. And what is a novel, if not our most naturalistic art form? Novels branch like trees; they fork like lightning; they grow like lichens; and they’ve got fluff to match the coziest cloud.

Doesn’t it make sense, then, that a novel would be a kind of literary fractal?

Who loves fractals? Anyone who reads novels. Which is to say, once again: everybody.

Thanks for reading! My new book (with a chapter on Wallace’s mathematical obsession) is Change is the Only Constant: The Wisdom of Calculus in a Madcap World.