This is a short invited talk I gave in April, 2013 to the University of New Mexico Hobbit Society’s “Intellectual Hooliganism” Colloquium. Although I’ve never gotten around to formally expanding on these brief remarks, in the years since I presented this talk, the concept put forth here has become a core tenet of my personal writing philosophy.
What is the mathematical concept of fractal dimensionality, and how does it relate to literature? What traits do the works of Roger Zelazny, master fantasist, and Raymond Chandler, king of hard-boiled noir fiction, share in common? And what can those similarities teach us about the craft of storytelling? A common characteristic in the works of beginning and early career writers is the desire to create “plotty” stories that trick the reader with unforeseen twists. But the mastery of craft doesn’t begin from the top down. Instead, it’s grown from the bottom up: from paragraphs, sentences, even the author’s choice of individual words — the atoms of storytelling. And it’s at this scale, the micro level, where intriguing and engaging tales are born.
I’d like to begin by thanking the UNM Hobbit Society for the invitation to participate in this year’s Intellectual Hooliganism colloquium. I’m honored to be a part of this. Thank you, sincerely, for having me.
Now. Rather than presenting myself as somebody with a surfeit of wisdom and experience to bestow, I’d like to share some thoughts — as a fellow traveler — that I’ve been mulling for some time. I’m going to present an unfinished hypothesis, because I still don’t have it entirely figured out yet.
I’m going to start by talking about mathematics. But bear with me. I’ll bring it back to fiction.
In 1967, the Polish/French/American mathematician Benoit Mandelbrot published, in the journal Science, a paper titled, “How Long is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension.” Mandelbrot is perhaps best known to us now as the namesake of the eponymous, or perhaps infamous, Mandelbrot Set: that infinitely fascinating, infinitely zoomable alien blob that rules over the esoteric realm of fractal mathematics. But this paper marked one of his earliest forays into the study of fractal geometry, and in fact it came several years before he coined the term “fractal”. The paper outlines his thoughts regarding an old idea that mathematicians had been kicking around for a long time: the concept of a fractional dimension (sometimes also known as the Hausdorff dimension).
We’re all familiar with the idea of dimensionality. A piece of string, say, is essentially one-dimensional because it basically has just a single dimension: length (but not really). A piece of paper is essentially two-dimensional because it has length and width but almost no depth. So is the humble narrator of Flatland, who happens to be a sentient square. This table, say, is three dimensional, because it has finite length, width, and depth. (I’m cheating a bit with the string and the paper, but you get my point.) But it turns out that there are objects — or perhaps it’s better to say, there are concepts — in nature whose dimension is not an integer: not simply 1, 2, or 3, or 4. A coastline is one such object. Mandelbrot’s paper picked up on an earlier observation by Lewis Fry Richardson, who noted that the measured length of a coastline or geographical border increases as the unit of measurement decreases. If you measure the coastline of Britain with a hundred-mile-long measuring stick, you get one answer; if you measure it by laying down a succession of inch-long sticks, you get a different, and larger, answer because the path you traced through space would be more curvilinear. Oddly, the measurement appears to increase without bound as the unit of measurement gets smaller and smaller.
Why is this?
The culprit is called self-similarity. An object is said to be self-similar if its overall structure can be seen repeated within itself at smaller and smaller scales. The string is a trivial example: a long string is made up of nearly identical bits of shorter string. A uniform cube can be divided into a bunch of smaller identical cubes. And if you were to imagine floating in space hundreds of miles above Great Britain, and then you zoomed in on a particularly crinkly bit of coastline, you’d find that the crinkles contained crinkles. And, if you kept zooming, you’d find that those crinkles contained still more crinkles, and so on and so forth, on and on and on. Not forever, of course, because the material world is not infinitely divisible — it’s made of atoms. (But that’s okay. So are stories. Which I’ll get to in a moment.)
This is seen frequently in nature. River networks, fault lines, coastlines, mountain ranges, Romanesco broccoli, clouds, snowflakes, ocean waves, and even the circulatory system inside your own body: all exhibit an approximate form of self-similarity to greater or lesser degrees. (And in fact, computer modelers sometimes make use of fractals to generate synthetic landscapes.) Self-similarity is an important concept not only in mathematics but also in physics. For example, the evolution of a supernova blast wave plowing through the interstellar medium can be described in terms of self-similarity.
It’s the self-similarity of an object, such as a coastline, that gives rise to its fractal dimension. And I’ll argue here, today, that self-similarity — of a particular type — can be a great thing for storytelling, too. A while ago I had a revelation when I realized that I happen to enjoy writing that exhibits a noninteger fractal dimension. I’m still processing this, but that was a watershed moment in my writing.
We all enjoy stories that take us to unexpected places. That’s why we read. But I think we tend to overestimate the importance of large-scale, structural surprises. The twist ending. The solution to the murder mystery that nobody anticipates. (In another medium, film, M. Night Shyamalan has tried to build a career replicating the singularly successful (commercially and storytelling-wise) ending to The Sixth Sense. And in my humble and completely unqualified opinion, that has been hampered his later work, to the extent that his films have become parodies of themselves. A truly successful twist ending is a rare thing, but his work mistakes one tool in the toolbox for a defining trademark.) But I’m convinced that this is emphasizing surprise on entirely the wrong scale. We’re looking at the map, the entire island of Great Britain, when we should be looking at the grains of sand on a particular stretch of beach in a particular cove. We should be looking not just at the big picture, but at the smallest scale– at the atoms of story. The words and sentences.
One thing that we writers are taught, and correctly so, is to remove clichés from our writing. And what is a cliché? A cliché is an idea or expression so overused that a reader can anticipate exactly where it’s going. (We should take care to distinguish this from a trope, however.) At one end of the scale, plots can be clichéd (there’s a reason many short fiction venues maintain lists of story concepts they will NOT accept). At the opposite end of the scale, sentences can be clichéd, too. A clichéd sentence contains a turn of phrase so overused that a reader can stop reading and recite the rest of it. That is to say, it’s a turn of phrase entirely devoid of surprises.
Consider the opening to Roger Zelazny’s story, “King Solomon’s Ring:”
King Solomon had a ring, and so did the guy I have to tell you about. Solomon’s was a big iron thing with a pentagram for a face, but Billy Scarle’s was invisible because. . .Roger Zelazny, “King Solomon’s Ring”
Well, because why? Excluding those of you who already know the story, how do you think that sentence ends? Any predictions? OK. Here’s the answer.
King Solomon had a ring, and so did the guy I have to tell you about. Solomon’s was a big iron thing with a pentagram for a face, but Billy Scarle’s was invisible because he wore it around his mind.Roger Zelazny, “King Solomon’s Ring”
Let’s try another one. Zelazny’s Hugo-award winning novel, Lord of Light, starts like this:
His followers called him Mahasamatman, and said he was a god. He preferred. . .Roger Zelazny, Lord of Light
Again, don’t cheat if you already know the answer– he preferred what? Take a moment a try to predict where that line is going. Ready?
His followers called him Mahasamatman, and said he was a god. He preferred to drop the Maha- and the – atman, however, and called himself Sam.Roger Zelazny, Lord of Light
The point I’m trying to make is that the uniqueness and creativity of the story isn’t reserved for a handful of plot twists, or even for the succession of eyeball kicks in the worldbuilding. It’s present at the lowest level of the storytelling. It resides in the atoms. Zelazny’s prose was surprising at multiple levels: word choice, sentence structure, character voice, worldbuilding, plot. In that way, Lord of Light and “King Solomon’s Ring” are self-similar: they have a nontrivial fractal dimension.
High-concept worldbuilding makes Lord of Light a very fine novel. But it’s the craftsmanship at the lowest level, the atomic level, that makes it one of the all-time great science fiction novels. All in my humble opinion, of course.
Notice, too, that when I say a piece of writing exhibits self-similarity, that’s not the same as saying it’s repetitive or monotonous. It means the writing has an interesting texture at multiple levels. Like a coastline, it’s crinkly at large, medium, and small scales. It has a noninteger fractal dimension.
This self-similar craftiness isn’t restricted to genre fiction. Raymond Chandler, author of the Philip Marlowe detective novels, had a rather low opinion of science fiction. But his prose crackles in a similar way. He didn’t rely on his mystery plots to do the heavy lifting. (And that’s a good thing, too — like much noir fiction, his stories rely on a finite set of plot elements that get rearranged for each novel. Nor does the characterization provide any surprises whatsoever — his characters, particularly women, fall into a limited set of categories that get recycled from book to book.) Instead, Chandler established himself as one of the great American writers on the virtue of his sentence-level craft. It’s just as crinkly as the coastline of England.
For instance, here’s a line from the opening pages of Farewell, My Lovely. This is Philip Marlowe’s first impression when he meets an ex-con with the wonderful name Moose Malloy:
Even on Central Avenue, not the quietest dressed street in the world, he looked about as inconspicuous as a tarantula on a slice of angel food.Raymond Chandler, Farewell, My Lovely
Isn’t that fantastic? And if you think about how that sentence works, it’s doing double duty — it’s practically self-similar because it tells you just as much about the narrator as it does about Moose Malloy.
The Philip Marlowe novels were a revelation to me. They’re chockablock with descriptions that are perfect yet utterly unpredictable. Here’s another famous example:
She was a blond. A blond to make a bishop kick a hole in a stained-glass window.Raymond Chandler, Farewell My Lovely
And my all-time personal favorite:
The girl gave him a look that ought to have stuck at least four inches out of his back.Raymond Chandler, The Long Goodbye
I find that last one deeply fascinating. The idea of it is so strong, so powerful, and so unique that when I first read this sentence, Chandler himself might as well have clobbered me with a claw hammer. It’s an almost perfect description, capturing the true essence of something almost indescribable. But at the same time I’ll admit the line itself is almost a little clunky. Part of me — and I know this his high heresy, daring to revise the blessed Chandler — wants to trim that sentence just a wee bit. And yet, even in this not-quite-perfect form, it’s one of the greatest sentences I’ve ever read. It’s atomically crinkly.
Again, the point I’m trying to make is that Chandler established himself as a master craftsman not through plot or characterization (though some of his setting descriptions are pure poetry), but through his sentence-level work. The Philip Marlowe novels are self-similar because they contain surprises at every single level: sentence, scene, and plot. (Though after you’ve read a few the plots become less surprising.) Recall that Chandler was famous for his somewhat-cynical advice for countering writer’s block: if you’re uncertain what should happen next, have somebody burst in wielding a gun. (Or, if you’re Tim Powers, instead of a gun-wielding thug, it’s a clown on stilts with his hair on fire. He actually does that in The Anubis Gates.)
I’m not saying that we should strive to craft impenetrably complex Mandelbrot Set-stories. That wouldn’t do any good. Rather, there’s a lot to be gained from contemplating the shape of a tale at every scale. How you tell a story is just as important what story you’re telling. The atoms are just as important as the coastline.
Don’t follow in the mold of M. Night Shyamalan. Instead, examine your storytelling as if you were Benoit Mandelbrot.