Doodling in Math: Infinity Elephants More videos/info: http://vihart.com/doodlingDoodling Snakes + Graphs: http://www.youtube.com/watch?v=heKK95DAKmsDoodling Stars: http://www.youtube.com/watch?v=CfJzrmS9UfYDoodling Binary Trees: http://www.youtube.com/watch?v=e4MSN6IImpIhttp://vihart.com
Doodling in Math: Infinity Elephants
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- So you're me, and you're in math class yet again
- because they make you go, like, every single day.
- And you're learning about, I don't know,
- the sums of infinite series.
- That's a high school topic, right?
- Which is odd, because it's a cool topic,
- but they somehow manage to ruin it anyway.
- So I guess that's why they allow infinite series in the curriculum.
- So, in a quite understandable need for distraction,
- you're doodling and thinking more about
- what the plural of "series" should be
- than about the topic at hand.
- "Serieses," "seriese," "seriesen," and "serii?"
- Or is it that the singular should be changed?
- One "serie," or "serus," or "serum?"
- It's like the singular of "sheep" should be "shoop."
- But the whole concept of things like
- 1/2 +1/4 +1/8 +1/16 and so on, approaching one,
- is useful if you wanna draw a line of elephants
- each holding the tail of the next one:
- normal elephant, young elephant,
- baby elephant, dog-sized elephant, puppy-sized elephant...
- All the way down to Mr. Tusks and beyond.
- Which is at least a *tiny* bit awesome,
- because you can get an infinite number of elephants in a line
- and still have it fit across a single notebook page.
- But there's questions like,
- "What if you started with a camel, which,
- being smaller than an elephant,
- only goes across a third of the page?"
- How big should the next camel be
- in order to properly approach the end of the page?
- Certainly you could calculate an answer to this question,
- and it's cool that that's possible,
- but I'm not really interested in doing calculations,
- so we'll come back to camels.
- Here's a fractal.
- You start with these circles,
- in a circle,
- and then keep drawing the biggest circle
- that fits in the space in-between.
- This is called an "Apollonian Gasket."
- And you can choose a different starting set of circles,
- and it still works nicely.
- It's well known in some circles
- because it has some very interesting properties
- involving the relative curvature of circles
- which is neat and all,
- but it also looks cool
- and suggests an awesome doodle game.
- Step one:
- draw ANY shape.
- Step two:
- draw the BIGGEST circle you can within this shape.
- Step three:
- draw the biggest circle you can
- within the space left.
- Step four:
- see Step three.
- As long as there is space left after the first circle,
- meaning don't start with a circle,
- this method turns any shape into a fractal.
- You can do this with triangles,
- you can do this with stars, and don't forget to embellish!
- You can do this with elephants, or snakes,
- or profile one of your friends.
- I choose Abraham Lincoln!
- Okay, but what about other shapes besides circles?
- For example, equilateral triangles
- say, filling this other triangle, which works
- because the filler triangles are in opposite orientation
- to the outside triangle (and orientation matters).
- This yields our friend, "Sierpinski's Triangle,"
- which, by the way, you can also make out of Abraham Lincoln.
- But triangles seem to work beautifully in this case.
- But that's a special case,
- and the problem with triangles is that
- they don't always fit snugly.
- For example, with this blobby shape,
- the biggest equillateral triangle has a lonely hanging corner.
- And *sure*, you don't have to let that
- stop you in this fun doodle game,
- but I think it lacks some of the beauty of the circle game.
- Or, what if you could change the orientation of the triangle
- to get the biggest possible one?
- What if you didn't have to keep it equillateral?
- Well, for polygonal shapes,
- the game runs out pretty quickly, so that's no good.
- But for curvy, complicated shapes,
- the process itself becomes difficult.
- How do you find the biggest triangle?
- It's not always obvious which triangle has more area
- especially when your starting shape is not very well defined.
- This is an interesting sort of question,
- because there IS a correct answer,
- but if you're going to write a computer program
- that fills a given shape with another shape,
- following even the *simpler* version of the rules,
- you might need to learn some computational geometry.
- I'm certain that you could move beyond
- triangles, to squares, or even elephants,
- but the circle is great because
- it's just so fantastically *round*.
- Oh, just a quick little side doodle challenge:
- A circle can be defined by three points,
- so draw three arbitrary points, and then
- try to find the circle that they belong to.
- So, one of the things that intrigues me about the circle game
- is that whenever you have one of these sorts of "corners,"
- you know there's going to be
- an infinite number of circles heading down into it.
- Thing is, for every one of those infinite circles,
- you create a few more little corners
- that are gonna need an infinite number of circles,
- and for every one of those and so on.
- You just get an incredible number of circles breeding more circles,
- and you can see just how dense infinity can be.
- Though, the astounding thing is that this kind of infinity
- is still the smallest countable kind of infinity, and
- there are kinds of infinity that are just mind-bogglingly infinite-er.
- But wait, here's an interesting thing:
- if you call *this distance* "One Arbitrary Length Unit,"
- then *this distance* plus *this,* dot dot dot...
- is an infinite series that approaches ONE.
- And this is another, different, series that still approaches one.
- And here's another, and another,
- and as long as the outside shape is well defined,
- so will the series be.
- But if you want the "simple" kind of series
- where each circle's diameter is
- a certain percentage of the one before it,
- you get straight lines. Which makes sense
- if you know how the slope of the straight line is defined.
- This is good, because it suggests a WONDERFUL,
- mathematical, and doodle-able way to solve our camel problem
- with no calculations necessary.
- If instead of camels, we had circles,
- we could make the right infinite series just by drawing an angle
- that ends where the page does, and filling it up.
- Replace circles with camels, and voila!
- Infinite Saharan caravan,
- fading into the distance,
- no numbers necessary!
- Well, I have an infinite amount of
- information I'd like to share with you in this last sentence.
- Maybe it'll still fit in the next five seconds
- if I say the next phrase twice as fast,
- (and the next phrase twice as fast as that)
- (and the next... *high pitched garble*)
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