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.