So you're me and you're in math class,
and you're learning about graph theory.
A subject too interesting to be included in most grade school
curricula, so maybe you're in some special program,
or maybe you're in college and somehow
not scarred for life by your grade school math teachers.
I'm not sure why you're not paying attention,
but maybe you have an incompetent teacher,
and it's too heartbreaking to watch him butcher what
could have been such a fun subject full of snakes
and balloons.
Snakes aren't really all that relevant to the mathematics
here, but being able to draw them will be useful later.
So you should probably start practicing now.
I've got a family of three related doodle games
to show you, all stemming from drawing squiggles
all over the page.
The first one goes like this.
Draw a squiggle.
A closed curve that ends where it begins.
The only real rule here is to make
sure all the crossings are distinct.
Next, make it start weaving.
Follow the curve around, and at each crossing, alternate going
under and over until you've assigned all the crossing.
Then put on the finishing touches, and voila.
You try it again, adding a little artistic flair
to the lines.
The cool part is that the meeting always
works out perfectly.
When you're going around alternating over and under,
and get to a crossing you've already assigned,
it will always be the right one.
This is very interesting, and we'll get back to it later.
But first I'd like to point out two things.
One is that this works for any number of closed curves
on the plane, so go ahead and link stuff up or make
a weaving out of two colors of yarn.
The other is that this doodle also
works for snakes on a plane, as long as you keep the head
and tail on the outside or on the same inside face,
because mathematically it's the same as if they linked up.
Or just actually link up the head and tail,
into an Ouroboros.
For example, here's three Ouroborii
in a configuration known as the Borromean Rings, which
has the neat property that no two snakes are actually
linked with each other.
Also, because I like naming things,
this design shall henceforth be known as the Ouroborromean
Rings.
But you are me, after all, so you're
finding a lot to think about, even with just drawing
one line that isn't a snake.
Such as what kind of knot are you drawing?
And can you classify them?
For example, these three knots all have five crossings,
but two are essentially the same knot, and one is different.
Knot theory questions are actually
really difficult and interesting,
but you're going to have look that one up yourself.
Oh, and you should also learn how to draw rope,
because it's an integral part of knot theory.
So integral in fact, that if you draw
a bunch of integral signs in a row--
a sight which is often quite daunting to a mathematician--
you can just shade it in, and tada.
But being able to draw snakes is also super useful,
especially as this doodle game is
excellent for producing Dark Mark tattoo designs.
Also, this doodle game can be combined with the stars doodle
game.
For example, if this pentagram gets knighted,
it will henceforth be known as Sir Pentagram.
Also notice that this snakes is a five twist Mobius
strip, so you could also call it a Mobiaborros,
but we'll get back to one sidedness later.
Or if you want to draw something super complicated,
like the eighth square star, combining snakes and stars
is a great technique for that to.
Here is a boa that ate eight eight-gons.
The creativity that your mind is forced
into during these boring classes is both a gift and a burden,
but here's a few authentic doodles using these techniques
that I did when I was in college, just
to show you that I'm not making all this up.
These are from a freshman music history class,
because I happen to be able to find this notebook--
but this is a doodle I actually did most often
during my ninth grade Italian class-- language
being another subject usually taught
by unfathomably stupid methods.
For example, these snakes are having trouble communicating,
because one speaks in parseltongue,
and the other speaks in Python.
And their language classes, much like math classes,
focus too much on memorization, and not enough on immersion.
But just pretend you're in math class,
learning about graph theory, so I can draw the parallels.
Because here's the second doodle game,
which is very much mathematically related.
Draw a squiggle all over the page
and make sure it closes up.
Pick an outside section and color it in.
Now you want to alternate coloring
so that no two faces of the same color touch.
Curiously enough, much like the weaving game,
this game also always mathematically works out.
It also works really well if you make the lines
spiky instead of a smooth curve.
And once again, it works with multiple lines too.
It probably has something to do with the two-colorability
of graphs of even degree, which might even
be what your teacher is trying to teach you
at this very moment, for all you're paying attention.
But maybe you can chat with him after class about snakes
and he'll explain it to you, because I'd rather
move on to the next doodle game.
This is a combination of the last two.
Step one, draw a smooth, closed curve.
Step two assign overs and unders.
Step three, shade in every other face.
After that, it takes a little artistic finesse
to get the shading right, but you end up
with some sort of really neat surface.
For example, this one has only one edge and one side.
But if you're interested in this,
you should really be talking to your resident topology
professor, and not me.
But here's the thing, if someone asked you five minutes ago,
what tangled up snakes, demented checkerboards,
and crazy, twisty surfaces have in common,
what would you have answered?
This is why I love mathematics.
The moment when you realize that something seemingly
arbitrary and confusing is actually part of something.
It's better than the cleverest possible
ending to any crime show or mystery novel,
because that's only the beginning.
Anyway, have fun with that.