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So you're me and you're in math class

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and you're learning about graph theory,

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a subject too interesting to be included

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in most grade school curricula.

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So maybe you're in some special program

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or maybe you're in college

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and were somehow not scarred for life

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by your grade school math teachers.

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I'm not sure why you're not paying attention

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but maybe you have an incompetent teacher

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and it's too heart-breaking to watch him

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butcher what could be a fun subject,

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full of snakes and balloons.

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Snakes aren't really all that relevant to the mathematics here.

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But being able to draw them will be useful later,

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so you should probably start practicing now.

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I've got a family of 3 related doodle games to show you,

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all stemming from drawing squiggles all over the page.

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The first one goes like this:

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draw a squiggle- a closed curve that ends where it begins.

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The only real rule here is to make sure

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that all the crossings are distinct.

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Next, make it start weaving-

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follow the curve around and

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that each crossing alternate going under and over

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until you've assigned all the crossings.

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Then put on the finishing touches, and voila!

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You try it again, adding a little artistic flair to the lines.

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The cool part is that the weaving always works out perfectly,

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when you're going around alternating over and under

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and get to a crossing you've already assigned,

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it will always be the right one.

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This is very interesting, and we'll get back to it later.

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But first I'd like to point out 2 things: one is that

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this works for any number of closed curves on the plane.

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So go ahead and link stuff up

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or make a weaving out of 2 colors of yarn.

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The other is that

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this doodle also works out for snakes on a plane

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as long as you keep the head and tail on the outside

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or on the same inside face.

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because mathematically it's the same as if they linked up

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or just actually link up the head and tail into an Ouroboros.

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For example, here's 3 Ourobori in a configuration

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known as the Borromean Rings

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which has the neat property that

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no 2 snakes are actually linked with each other.

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Also because I like naming things,

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this design shall henceforth be known as

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the "OuroBorromean Rings".

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But you are me, after all,

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so you're finding a lot to think about

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even with just drawing one line that isn't a snake.

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Such as, "What kinds of knots are you drawing?"

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"And can you classify them?"

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For example, these 3 knots all have 5 crossings

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but 2 are essentially the same knot and one is different.

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Knot theory questions are actually really difficult and interesting

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but you're going to have to look that one up yourself.

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Oh, and you should also learn how to draw rope

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because it's an integral part of knot theory.

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So integral, in fact, that

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if you draw a bunch of integral signs in a row,

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a sight which is often quite daunting to a mathematician,

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you can just shade it in, and TA-DA.

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But, being able to draw snakes is also super useful

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especially as this doodle game is excellent for

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producing Dark Mark tattoo designs.

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Also, this doodle game can be combined

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with the stars doodle game.

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For example, if this pentagram gets knighted,

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it will henceforth be known as "Serpentagram"

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Also notice that this snake is a 5 twist Mobius strip

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so you could also call it a "Mobiaboros"

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but we'll get back to one-sidedness later.

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Or, if you want to draw something super complicated

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like the 8th square star,

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

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Here's a boa that ate 8 8gons.

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The creativity that your mind is forced into

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during these boring classes, is both a gift and a burden.

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But here's a few authentic doodles using these techniques

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that I did when I was in college.

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Just to show you I'm not making all this up.

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These are from a freshmen music history class,

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because I happen to be able to find this notebook.

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But this is a doodle I actually did most often

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during my 9th grade Italian class.

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Language being another subject

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usually taught by unfathomably stupid methods.

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For example, these snakes are having trouble communicating

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because one speaks in Parseltongue

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and the other speaks in Python.

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And their language classes, much like math classes,

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focus too much on memorization and not enough on immersion

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But just pretend you're in math class,

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learning about graph theory so that I can draw the parallels

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because here's the 2nd doodle game

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which is very much mathematically related.

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Draw a squiggle all over the page

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and make sure it closes up.

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Pick an outside section and color it in.

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Now you want to alternate coloring

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so that no 2 faces of the same color touch.

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Curiously enough,

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much like the weaving game,

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this game always mathemagically works out.

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It also works really well if you make the lines spiky

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instead of a smooth curve

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and once again, it works with multiple lines too.

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It probably has something to do with

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the 2 colorability of graphs of even degree,

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which might even be what your teacher is trying to

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teach you at this very moment

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for all you're paying attention.

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But maybe you can chat with him after class about snakes

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and he'll explain it to you

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because I'd rather move on to the next doodle game.

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This is a combination of the last 2

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Step 1: draw a smooth closed curve

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Step 2: assign overs and unders

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Step 3: shade in every other face

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After that, it takes a little artistic finesse

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to get the shading right,

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but you end up with some sort of really neat surface.

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For example, this one only has one edge and one side

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but if you're interested in this,

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you should really be talking to

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your resident topology professor and not me.

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But here's the thing:

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if someone asked you 5 minutes ago

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what tangled up snakes, demented checkerboards,

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and crazy twisty surfaces have in common?

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what would you have answered?

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This is why I love mathematics:

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the moment when you realize that

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something seemingly arbitrary and confusing

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is actually part of something.

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It's better than the cleverest possible ending

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to any crime show or mystery novel,

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because that's only the beginning.

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Anyway, have fun with that.