An Abundance of Katherines (23 page)

author’s note

The footnotes of the novel you just read (unless you haven’t finished reading it and are skipping ahead, in which case you should go back and read everything in order and not try to find out what happens, you sneaky little sneakster) promise a math-laden appendix. And so here it is.

As it happens, I got a C-minus in pre calc despite the heroic efforts of my eleventh-grade math teacher, Mr. Lantrip, and then I went on to take something called “finite mathematics,” because it was supposed to be easier than calculus. I picked the college I attended partly because it had no math requirement. But then shortly after college, I became—and I know this is weird—kind of
into
math. Unfortunately, I still suck at it. I’m into math the way my nine-year-old self was into skateboarding. I talk about it a lot, and I think about it a lot, but I can’t actually, like,
do
it.

Fortunately, I am friends with this guy Daniel Biss, who happens to be one of the best young mathematicians in America. Daniel is world famous in the math world, partly because of a paper he published a few years ago that apparently proves that circles are basically fat, bloated triangles. He is also one of my dearest friends. Daniel is pretty much entirely responsible for the fact that the formula is real math that really works within the context of the book. I asked him to write an appendix about the math behind Colin’s Theorem. This appendix, like all appendices, is strictly optional reading, of course. But boy, is it fascinating. Enjoy.

—John Green

the appendix

Colin’s Eureka moment was made up of three ingredients.

First of all, he noticed that a relationship is something you can draw a graph of; one such graph might look something like this:

According to Colin’s thesis, the horizontal line (which we call the
x
-axis) represents time. The first time the curve crosses the
x
-axis corresponds to the beginning of the relationship, and the second crossing indicates the conclusion of the relationship. If the curve spends the intermediate time above the
x
-axis (as is the case in our example), then the girl dumps the boy; if, instead, the curve passes below the
x
-axis, that means that the boy dumps the girl. (“Boy” and “girl,” for our purposes, contain no gender-specific meaning; for same-sex relationships, you could as easily call them “boy1” and “boy2” or girl1” and “girl2.”) So in our diagram, the couple’s first kiss is on a Tuesday, and then the girl dumps the boy on Wednesday. (All in all, a fairly typical Colin-Katherine affair.)

Since the curve crosses the
x
-axis only at the beginning and end of the 
relationship, we should expect that at any point in time, the farther the curve strays from the
x
-axis, the farther the relationship is from breakup, or, put another way, the better the relationship seems to be going. Here’s a more complicated example, the graph of my relationship with one of
my
exgirlfriends: 

The initial burst came in February when, all in a matter of hours, we met, a blizzard started, and she totaled her car on an icy highway, breaking her wrist in the process. We suddenly found ourselves snowed in at my apartment, she an invalid doped up on painkillers, and me distracted and intoxicated by my new jobs as nurse and boyfriend. That phase ended abruptly when, two weeks later, the snow melted, her hand healed, and we had to leave my apartment and interact with the world, whereupon we immediately discovered that we led radically different lives and didn’t have all that much in common. The next, smaller spike occurred when we went to Budapest for vacation. That ended, moments later, when we noticed that we were spending about twenty-three hours of each romantically Budapestian day bickering about absolutely everything. The curve finally crosses the
x
-axis somewhere in August, which is when I dumped her and she threw me out of her apartment and onto the streets of Berkeley, homeless and penniless, at midnight.

The second ingredient in Colin’s Eureka moment is the fact that graphs (including graphs of romantic relationships) can be represented by functions. This one will take a bit of explaining; bear with me.

The first thing to say is that when we draw a diagram like this,

each point can be represented by numbers. That is, the horizontal line (the
x
-axis) has little numbers marked on it, as does the vertical line (the
y
-axis). Now, to specify a single point somewhere in the plane, it’s enough to just list two numbers: one that tells us how far along the
x
-axis the point lies, and the other that tells us where it’s situated along the
y
-axis. For example, the point (2,1) should correspond to the spot marked “2” on the
x
-axis and the spot marked “1” on the
y
-axis. Equivalently, it’s located two units to the right and one unit above the location where the
x
- and
y
-axes cross, which location is called (0,0). Similarly, the point (0,-2) is located on the
y
-axis two units below the crossing, and the point (-3,2) is situated three units to the left and two units above the crossing.

Okay, so functions: a function is a kind of machine for turning one number into another. It’s a rulebook for a very simple game: I give you any number I want and you always give me back some other number. For instance, a function might say, “Take the number and multiply it by itself (i.e., square it).” Then our conversation would go something like this:

 
ME: 1

YOU: 1

ME: 2

YOU: 4

ME: 3

YOU: 9

ME: 9,252,459,984

YOU: 85,608,015,755,521,280,256

 
Now, many functions can be written using algebraic equations. For example, the function above would be written

which means that when I give you the number
x
, the function instructs you to take
x
and multiply it by itself (i.e., to compute
x
²
) and return that new number to me. Using the function, we can plot
all
points of the form (
x,f(x)
). Those points together will form some kind of curve in the plane, and we call that curve the “graph of the function.” Consider the function
f(x) = x
²
. We can plot the points (1,1), (2,4), and (3,9).

In this case, it might help to plot the additional points (0,0), (-1,1), (-2,4), and (-3,9). (Remember that if you take a negative number and multiply it by itself, you get a positive number.)

Now, you can probably guess that the graph will be a curve that looks something like this:

Unfortunately, you’ll notice that this graph doesn’t do a particularly good job of representing relationships. The graphs that Colin wants to use for his Theorem all need to cross the
x
-axis twice (once for when a couple starts dating, and once for the dumping), whereas the graph we drew only touched it once. But this can easily be fixed by using slightly more complicated functions. Consider, for example, the function
f(x) = 1 -x
²
.