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##### Doodling in Math Class: Binary Trees : Thank you to all the people I know whose math classes do not require such pastimes! More videos/info: http://vihart.com/doodling Doodling Snakes + Graphs: ttp://www.youtube.com/watch?v=heKK95DAKms Doodling Stars: http://www.youtube.com/watch?v=CfJzrmS9UfY http://vihart.com

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- Okay, let's say you're me and you're in math class
- and you're supposed to be learning about
- exponential functions, but you're having
- trouble caring about exponential functions
- because unfortunately your math class
- is probably not terribly engaging.
- You're supposed to be drawing and labeling some axes
- so that you can graph this y equals 2 to the x thing
- and your teacher seems to think
- that drawing and labeling axes
- is the very essence of mathematics
- but you're bored and you can't help but wonder...
- Why?
- So you do what any conscientious student would do
- in this situation and start doodling
- and because you're me
- you like to play games with yourself when you doodle.
- Here's one game.
- You're drawing a line, but when it crosses
- one of the blue lines on your little piece of paper
- it splits into two lines.
- Maybe this line is like the neck of the mythical hydra
- where every time one of its heads gets chopped off
- by a blue line, it grows two more in its place.
- You want to see if you can get all the way
- to the bottom of the page following this rule
- because if you do, you can draw
- all of the little hydra heads at the end.
- But you don't get very far on your first try.
- You decide to try again, this time
- spacing things out a little more at the beginning.
- Unfortunately, things are filling up fast,
- though you got farther than last time.
- Maybe if you had more room.
- Or maybe if you sharpened your pencil more,
- you could get to the bottom of the page.
- Oh, and don't forget to draw and label your axes.
- If each broad swing of Hercules' sword chops off
- all the heads thus doubling your number, well,
- you can see where I'm going.
- I'm not going to try and teach you math,
- just how to wield it for doodling purposes.
- In this case, that's going to be a lot of heads.
- Good luck Hercules.
- But maybe drawing binary trees all straight like that
- is not an interesting game to hold your attention for long
- so you start drawing them in arbitrary shapes.
- Or less arbitrary shapes.
- Maybe you start drawing a binary tree
- that looks like a tree.
- And maybe you can't see this tree in very high quality
- because your camera, very much like your math class
- is fuzzy, unfocused, and all together not very good.
- Maybe you changed the rules slightly and
- make a ternary bush where each branch sprouts
- three more branches.
- Unfortunately, your math class is 45 minutes long
- and soon you need a more interesting doodle game.
- Say you go back to the game
- where your line splits at every level.
- Only this time, instead of trying to squish all the lines in,
- you let them hit each other.
- And when they crash, there's a firey explosion
- and the crashing lines end there.
- Maybe you turn your notebook sideways
- so that you can make sure
- you're getting the horizontal spacing right.
- Maybe, to go back to mythology,
- Hercules has a method where, instead of cauterizing
- the necks of the Hydra to keep them from growing back
- he's found that the necks stick together if they get too close
- And instead of growing new heads,
- they just fill up with blood.
- It might seem a little morbid for math class,
- but maybe, if the curriculum wasn't so appalling
- and the teaching methods wern't so atrocious,
- you wouldn't have to entertain yourself
- with these stories and games.
- Speaking of this doodle game,
- something very interesting is happening.
- Looks like your simple rules about splitting
- and crashing are creating Sierpinski's triangle.
- Which is a pretty awesome fractal.
- But the point is not to learn about fractals
- or cellular automata or Sierpinski,
- but to show that simple doodle games
- can lead to mathematical results
- so cool and beautiful that they're famous.
- At least, famous to people like me.
- And if you're good at inventing doodle games,
- you might even end up
- doing some real mathematics during your math class.
- Anyway, maybe you don't care about accuracy.
- Maybe you try the game again,
- only you don't keep track of spacing,
- and when you make a mistake and
- accidentally grow heads where you shouldn't
- you just roll with it.
- Now you've introduced an element of random error
- and you want to know
- how this will affect the final picture.
- It still looks like a pretty awesome doodle,
- and has many of the same elements,
- though it lacks the structure.
- Speaking of the structure,
- maybe because you're really super bored
- and your class is seemingly never going to end,
- you start looking at the number of necks at each level
- and trying to figure out the pattern.
- Maybe you haven't forgotten about powers of two.
- Anyway, I hope I've provided you with
- something entertaining to do next time you're bored.
- Good luck with your math class.