Doodling in Math Class: Squiggle Inception

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Doodling in Math Class: Squiggle Inception : How to draw squiggles like a Hilbert.

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Maybe you like to draw squiggles when you're bored in class.
Somehow, the wandering path of the line that goes
in monotone droning of the teacher, perfectly capturing
the way it goes on and on about the same things over and over
but without really going anywhere in a deep display of artistic metaphor.
But once you're a veteran of bored doodling
you learn that some squiggles are better than others.
Good squiggles really fill up the page, squiggling around themselves
as densely as possible in a single line that doesn't cross itself.
It's like the ideal would be to sit down at the beginning
of your least favorite class, put your pencil on the page, and keep drawing a single line
filling up more and more space until the bell rings.
Which is basically what your teacher is doing, except with words.
You might find yourself developing some strategies.
For example, you're careful not to cut off a chunk of space
because you might want to get back in there later.
And if you leave only a little room to get to a certain section,
then when you go there, you fill up a lot of it
before you leave that section again.
Or else, instead of a "DO"-odle you'll have an unhappy "DON'T"-dle.
Or maybe you decide to make a meta-squiggle:
A squiggle made out of squiggles.
This can be done kind of abstractly, or extremely precisely.
For example, let's say you're drawing this simple squiggle.
Then, you draw that squiggle using that squiggle.
But to make it fill up space nicely,
you make the outside parts bigger.
Then to make it precise, you make the number of squiggles
always the same.
It's easy to keep squiggling this squiggle all the way
across the page if you keep the rhythm of it in your head.
This one's like:
[speaking rhythmically]
Down a squiggle, up a squiggle, down a squiggle, up a squiggle...
...down a squiggle, up a squiggle, down a squiggle, up a squiggle, down a squiggle, up a squiggle.
But after you've done that awhile, you decide to go a level deeper:
A squiggle within a squiggle within a squiggle!
That's right, we're going three levels down.
This serious business can go something like this:
Right a squiggle, left a squiggle, right a squiggle, left...whoop!
.....right a squiggle, left a squiggle, right a squiggle, left...whoop!...
...right a squiggle, left a squiggle, right a squiggle, left...whoop!
And the next one's even crazier!
Like: And...Up a squiggle, down a squiggle, up a squiggle, down...woop!...
.....up a squiggle, down a squiggle, up a squiggle, down...woop!...
...up a squiggle, down a squiggle, up a squiggle, down...woop!
Wop...all the way over here! And...
[speaking faster] Down a squiggle, up a squiggle, down a squiggle, up...woop!
...Down a squiggle, up a squiggle, down a squiggle, up...woop!...
...down a squiggle, up a squiggle, down a squiggle, up...woop!...
Wop...all the way over here!
Okay, but say you're me and you're in math class.
This means that you have graph paper,
opportunity for precision.
You can draw that first curve like this:
[rhythmic speaking] squig-uh, squig-uh, squig-uh, squig-uh,
squig-uh, squig-uh, squig-uh, squig-uh...
The second iteration to fit squiggles going up and down
will have a line three boxes across on top and bottom.
If you want the squiggles close on the grid as possible without touching.
You might remind yourself by saying: Three uh-squig, uh-squig, a squiggle.
...Three uh-squig, uh-squig, a squiggle.
The next iteration has a "whoop".
And you have to figure out how long that's going to be.
Meanwhile, other lengths change to keep everything close.
And...two a squig, a squig, a squiggle...
...three a squig, a squig, a squiggle; three a squig, a squig, a squiggle, two...nine.
Two a squig, a squig, a squiggle...
...three a squig, a squig, a squiggle; three a squig, a squig, a squiggle, two...nine.
We could write the pattern down like this.
So what would the next pattern be? Five.
Two a squig, a squig, a squiggle...
...three a squig, a squig, a squiggle
three a squig, a squig, a squigle...two
Nine.
Two a squig, a squig, a squiggle...
...three a squig, a squig, a squiggle...
three a squig, a squig, a squiggle...two
Nine.
Two a squig, a squig, a squiggle...
...three a squig, a squig, a squiggle...
three a squig, a squig, a squiggle...two
Nine...and 15 all the way over to here
And now...oh yeah, I can talk that fast! Totally!
Okay, but let's not get too far from the original purpose
which was to nicely fill a page with a squiggle.
The nicest page filling squiggles have kind of the same density
of squiggle everywhere.
You don't want to be clumped up here
but have left over space there,
because monsters might start growing
in the leftover space.
On graph paper you can be kind of precise about it.
Say you want a squiggle that goes through
every box exactly once and can be extended infinitely.
So you try some of those and decide that since the point
of them is to fill up all the space, you call them
space filling curves.
Yes, that's actually a technical term, but be careful
because your curve might actually be a...
[rhymic speaking] ...snake, snake, snake, snake, etc...
Also to make it nearter, you draw the lines on the lines
and shift the rules so you go through each intersection
on the graph paper exactly once
which is the same thing as far as space is concerned.
Okay, heres the space filling curve that a guy named Hilbert made up.
Because Hilbert was awesome, but he's dead now.
Here's the first iteration.
For the second one, we're going to build it
piece by piece by connecting four copies of the first.
So here's one. Put the second a space away
next to it and connect those,
then turn the page
to put the third sideways under the first
and connect those.
And then the fourth will be the mirror image of that
on the other side.
Now you've got one nice curve.
The third iteration will be made out of four copies of the second iteration.
So first build another second iteration curve out of four copies of the first iteration...
...one, two, three, four...
Then put another next to it, then two sideways on the bottom.
Connect them all up; there you go!
The fourth iteration is made of four copies of the third iteration
the same way.
If you learn to do the second iteration in one piece,
it will make this go faster.
Then build 2 third iterations facing up next to each other
and two underneath sideways.
You can keep going until you run out of room.
Or you can make each new version the same size
by making each line half the length.
Or you can make it out of snakes.
Or if you have friends, you can each make an iteration
of the same size and put them together
or invent your own fractal curve so that you can be cool like Hilbert,
who was like: "Mathematics? [scoff sound] Pff...I'm going to invent meta-mathematics! Like a boss."