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's curricula so maybe you're in some
special program or maybe you're
in college and were 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 in
competent 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 that 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 crossings.
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
weaving 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've linked up.
Or just actually link up the head and tail into a
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 knots are you drawing? And, can you
classify them? For example, these 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 to look that one up yourself.
Oh, and you should actually learn how to
draw a 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 ta-da. 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
SerPentagram. Also notice that this snake is a
five twist mobius strip so you can also call it a
mobiaboros 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, too. Here's a boa that
ate eight eighth-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 I'm not making all this up. These are from a freshman
music history class because I happened 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
Parseltounge and the other speaks in
Python and their language
classes, much like math classes focused 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-color-ability of
graphs of even degree, which
might even be what your teacher is trying to teach you about at this very moment, for all your 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 only has one edge and only 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 checker boards 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.
(
Previously.)
posted by tomswift at 7:42 PM on February 22, 2011