Doodling in Math: Snakes + Graphs More videos/info: http://vihart.com/doodlingDoodling Stars: http://www.youtube.com/watch?v=CfJzrmS9UfYDoodling Binary Trees: http://www.youtube.com/watch?v=e4MSN6IImpIhttp://vihart.com
Doodling in Math: Snakes + Graphs
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- 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.
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