Once in while, a new idea clobbers the brains of physicists and mathematicians.  Suddenly, they are forced to look at the universe in new and sometimes counter-intuitive ways.  Relativity.  Quantum physics.  Game theory.  Chaos theory.

Abstruse stuff like that isn’t just parlor entertainment for members of Mensa.  It has practical applications.  Nuclear energy.  Space travel.  Computers.  Even if we don’t get binomial distribution and math equations look to us like the number and letter stickers that my son has randomly plastered on his bedroom door, we should pay attention.  Physics explains strange phenomena.  Math orders what seems chaotic.

In the 1960’s, a French mathematician named René Thom came up with a way of understanding and describing sudden change.  It was a breakthrough because ordinary math presumes that things are fixed and solid like number lines, triangles and cubes.  Even the math of motion, calculus, conceives of moving bodies following smooth curves.  Throw a stone or fire a rocket into the air and it follows a predictable and elegant path up and back down.

But not everything that happens is so smooth and elegant.  The car ahead of you brakes sharply.  A corn kernel in a pan of hot oil suddenly pops open.  An old wall cracks.  Such events may seem unpredictable, measurable only as probabilities.  But every sudden change occurs in a particular moment.  Think of a dome with a marble perfectly balanced on its apex.  That marble isn’t going to stay there for long.  At a certain moment, the marble will suddenly start to roll in one direction or another.  Wouldn’t it be nifty to know when that moment will occur?

Okay, maybe knowing the precise second when a marble will begin to roll off a dome isn’t all that exciting.  But what if math could tell us ahead of time when an airplane engine will fail, the day on which a depressed individual will commit suicide, or when prisoners will riot, or when a government will go to war?

René Thom came up with math which might predict such sudden changes, which he called catastrophes.  His insights stirred hope that math could predict the previously unpredictable even in soft-science realms such as biology, psychology and social science.  Catastrophe theory, it was thought, might illuminate things as diverse as animals’ territoriality, corporate decisions to raise prices, and humans’ behavior in paying taxes.

The key was Thom’s application of topology which, in plain English, considers what happens to invariant points when the set of which they are a part bends or stretches around that point.  According to Thom, there are seven types of unstable curves, simple or three-dimensional, that show how a point will move “suddenly”—but predictably—according to how various forces are warping around it.

The seven types of unstable curves Thom named fold, cusp, swallowtail, butterfly, hyperbolic umbilic, elliptic umbilic, and parabolic umbilic.  Don’t stress.  There won’t be a quiz.  Just have a look at this cusp catastrophe diagram that predicts when a tax scofflaw will suddenly decide to pay his or her taxes:

What you see above is that the space around a point (“A”) can bend and fold as different pressures effect it.  When the fold becomes extreme enough, point “A” suddenly leaps from one spot to another.  Point “A”, the tax scofflaw, in a startling instant becomes Point “B”, a dutiful taxpayer.

What Thom showed was that such behavior is not random, meaning that it will either happen or not.  It will happen, math proves it, but to you and me the change that happens will look sudden and without immediate cause.

So, what has all of that to do with characters in fiction?  Well, consider this: characters change.  But how?  When we speak of character arcs, we are—literally—using the old-school terminology and thinking of calculus, which is to say the presumption that change is a slow and smooth process.  It isn’t.  Not always.

If you have ever had trouble writing a scene, hacking at it over and over, trying this and that, tweaking to no avail, and then suddenly have muttered aloud, “Ah, fuck it”, and hit delete…well, then you know what I’m talking about.  A sheet of paper crumpled and tossed into the waste basket is, math-wise, a catastrophe.  Sudden.  Violent.  Without warning.  However, it is the product of conflicting forces in your head curving around you to extremes and…BOOM!  You hit delete.  What forces might those be?  Lousy draft tolerance versus looming deadline pressure?  Whatever those forces are, you are going to change certainly and suddenly.

So it is with characters.

When they arrest our attention, changes in characters arrive without apparent warning.  They are sudden.  They are extreme.  A character who behaves in one way all at once behaves in another.  If forces acting upon that character are present and convincing, though, then as readers we don’t think, “Huh?”, but rather “Wow!”

Erika Johansen’s YA fantasy The Queen of the Tearling sets in motion the story of princess Kelsea Raleigh Glynn, who is reared in hiding in a forest by two mentors until her nineteenth birthday, when a company of soldiers called the Queens Guard arrives to escort her to her throne.  However, her emergence from hiding is not welcomed by all.  The kingdom of Tear’s regent, her vile and obsequious uncle, wants her dead in order to preserve the status quo, in particular Tear’s subjugation by its powerful neighboring Kingdom, Mort, and Mort’s evil queen.  (Get it?  “Tear”?  “Mort”?)

The journey to her country’s capital is fearsome, with the traveling party spied upon by hawks and trailed by ninja-like assassins called Caden, and briefly Kelsea’s capture by a legendary (though perhaps patriotic) masked brigand named The Fetch.  Kelsea is not brave, nor strong.  She is terrified for her life and burdened by her heritage.  (Her mother was a weak queen.)  However, thanks to the captain of her guards, Mace, she finally arrives at the capital…on the day on which Tear’s annual tribute to Mort, a shipment of two hundred and fifty people, including children, chosen by lottery are being loaded into stout cages to be transported to Mort as slaves.

Up to this point Kelsea has been a plain-faced and fearful girl, utterly dependent on her guards for her survival.  At this moment, though, she is called upon to become a queen and claim her birthright before her people.  She obtains Mace’s pledge that he will defend her to the death, then asks him a question:

“What are those cages made of, Larazrus?”

“Mort iron.”

“But the wheels and undercarriage are wood.”

“Tearling oak, Lady.  What are you getting at?”

Staring down at the table full of blue-clad officials in front of the Keep, Kelsea took a deep breath.  This was her last moment to be anonymous.  Everything was about to change.  “The cages.  After we empty them, we’re going to set them on fire.”

Kelsea’s transformation into The Queen of Tear is all the more dramatic for being, apparently, sudden.  But actually, forces have been acting upon her all along.  On the one hand she is a young woman of nineteen, raised in isolation.  The reality of Tear is, to her, new and dangerous.  On the other hand, her education by her mentors was to prepare her to rule.  Her guards regard her with pessimism, but also with hope.  The day of the tribute itself is an unbearable sight of weeping mothers and heartless enforcers.  Fear and courage war within Kelsea and—BOOM—courage wins.

The word catastrophe carries an implication of disaster, destruction and defeat, but in fact René Thom showed us that catastrophe is only an instant shift from one place to another; in terms of character development, a “sudden” transformation from one state of being to another.

Let’s make this practical:

• What is the greatest change your protagonist must make? Who and what forces your protagonist to stay the same?  Who and what urges your protagonist to change?  Build up both forces.  Set them in irreconcilable opposition to each other.
• What’s the greatest risk of change? Turn that risk from a possibility into a certainty.
• What makes staying the same impossible? What makes the status quo a kind of living death?
• What is the most dangerous—and important—moment for your protagonist to say, “Ah, fuck it!” What will your protagonist do at the moment that is fearful and irrevocable?  Who will be terrified because of that act?  Who will cheer?

Catastrophe theory shows us that change can be sudden, yet it is inevitable.  It must happen and when it does it surprises us with the force of our hope, and lifts us with the buoyancy of its drama.  If as a writer your process depends on random lightning bolts of inspiration that’s fine, but why not plan character catastrophes and bring them about?

See, math isn’t so hard, is it?

What change might your protagonist go through, and what opposing forces can push your protagonist to a “sudden” change?

Illustration from “A Cusp Catastrophe Model of Tax Behavior” by Ioannis Katerelos and Nikolaos Varotsis, Panteion University of Athens, Greece, originally published in Nonlinear Dynamics, Psychology and Life Sciences, Vol. 21, No. 1, pp 89-122, ©2017 Society for Chaos Theory in Psychology & Life Sciences.