Doodling in Math Class: Snakes + Graphs

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 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 incompetent teacher

and it's too heart-breaking to watch him

butcher what could be 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 3 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

that 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 2 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 2 colors of yarn.

The other is that

this doodle also works out 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 3 Ourobori in a configuration

known as the Borromean Rings

which has the neat property that

no 2 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 kinds of knots are you drawing?"

"And can you classify them?"

For example, these 3 knots all have 5 crossings

but 2 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 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 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 5 twist Mobius strip

so you could 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 8th square star,

combining snakes and stars is a great technique for that too.

Here's a boa that ate 8 8gons.

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 freshmen 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 9th 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 that I can draw the parallels

because here's the 2nd 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 2 faces of the same color touch.

Curiously enough,

much like the weaving game,

this game always mathemagically 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 2 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 2

Step 1: draw a smooth closed curve

Step 2: assign overs and unders

Step 3: 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 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 5 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.