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Doodling in Math Class: Infinity Elephants

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Doodling in Math Class: Infinity Elephants : More videos/info: http://vihart.com/doodling Doodling Snakes + Graphs: http://www.youtube.com/watch?v=heKK95DAKms Doodling Stars: http://www.youtube.com/watch?v=CfJzrmS9UfY Doodling Binary Trees: http://www.youtube.com/watch?v=e4MSN6IImpI http://vihart.com

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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!
Awesome.
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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Doodling in Math Class: Infinity Elephants