Okay, let's say you're me and you're in math class

and you're supposed to be learning about

exponential functions, but you're having

trouble caring about exponential functions

because unfortunately your math class

is probably not terribly engaging.

You're supposed to be drawing and labeling some axes

so that you can graph this y equals 2 to the x thing

and your teacher seems to think

that drawing and labeling axes

is the very essence of mathematics

but you're bored and you can't help but wonder...

Why?

So you do what any conscientious student would do

in this situation and start doodling

and because you're me

you like to play games with yourself when you doodle.

Here's one game.

You're drawing a line, but when it crosses

one of the blue lines on your little piece of paper

it splits into two lines.

Maybe this line is like the neck of the mythical hydra

where every time one of its heads gets chopped off

by a blue line, it grows two more in its place.

You want to see if you can get all the way

to the bottom of the page following this rule

because if you do, you can draw

all of the little hydra heads at the end.

But you don't get very far on your first try.

You decide to try again, this time

spacing things out a little more at the beginning.

Unfortunately, things are filling up fast,

though you got farther than last time.

Maybe if you had more room.

Or maybe if you sharpened your pencil more,

you could get to the bottom of the page.

Oh, and don't forget to draw and label your axes.

If each broad swing of Hercules' sword chops off

all the heads thus doubling your number, well,

you can see where I'm going.

I'm not going to try and teach you math,

just how to wield it for doodling purposes.

In this case, that's going to be a lot of heads.

Good luck Hercules.

But maybe drawing binary trees all straight like that

is not an interesting game to hold your attention for long

so you start drawing them in arbitrary shapes.

Or less arbitrary shapes.

Maybe you start drawing a binary tree

that looks like a tree.

And maybe you can't see this tree in very high quality

because your camera, very much like your math class

is fuzzy, unfocused, and all together not very good.

Maybe you changed the rules slightly and

make a ternary bush where each branch sprouts

three more branches.

Unfortunately, your math class is 45 minutes long

and soon you need a more interesting doodle game.

Say you go back to the game

where your line splits at every level.

Only this time, instead of trying to squish all the lines in,

you let them hit each other.

And when they crash, there's a firey explosion

and the crashing lines end there.

Maybe you turn your notebook sideways

so that you can make sure

you're getting the horizontal spacing right.

Maybe, to go back to mythology,

Hercules has a method where, instead of cauterizing

the necks of the Hydra to keep them from growing back

he's found that the necks stick together if they get too close

And instead of growing new heads,

they just fill up with blood.

It might seem a little morbid for math class,

but maybe, if the curriculum wasn't so appalling

and the teaching methods wern't so atrocious,

you wouldn't have to entertain yourself

with these stories and games.

Speaking of this doodle game,

something very interesting is happening.

Looks like your simple rules about splitting

and crashing are creating Sierpinski's triangle.

Which is a pretty awesome fractal.

But the point is not to learn about fractals

or cellular automata or Sierpinski,

but to show that simple doodle games

can lead to mathematical results

so cool and beautiful that they're famous.

At least, famous to people like me.

And if you're good at inventing doodle games,

you might even end up

doing some real mathematics during your math class.

Anyway, maybe you don't care about accuracy.

Maybe you try the game again,

only you don't keep track of spacing,

and when you make a mistake and

accidentally grow heads where you shouldn't

you just roll with it.

Now you've introduced an element of random error

and you want to know

how this will affect the final picture.

It still looks like a pretty awesome doodle,

and has many of the same elements,

though it lacks the structure.

Speaking of the structure,

maybe because you're really super bored

and your class is seemingly never going to end,

you start looking at the number of necks at each level

and trying to figure out the pattern.

Maybe you haven't forgotten about powers of two.

Anyway, I hope I've provided you with

something entertaining to do next time you're bored.

Good luck with your math class.