Doodling in Math
Doodling in Math: Squiggle Inception How to draw squiggles like a Hilbert.
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- Maybe you like to draw squiggles when you're bored in class.
- Somehow, the wandering path of the line that goes
- in monotone droning of the teacher, perfectly capturing
- the way it goes on and on about the same things over and over
- but without really going anywhere in a deep display of artistic metaphor.
- But once you're a veteran of bored doodling
- you learn that some squiggles are better than others.
- Good squiggles really fill up the page, squiggling around themselves
- as densely as possible in a single line that doesn't cross itself.
- It's like the ideal would be to sit down at the beginning
- of your least favorite class, put your pencil on the page, and keep drawing a single line
- filling up more and more space until the bell rings.
- Which is basically what your teacher is doing, except with words.
- You might find yourself developing some strategies.
- For example, you're careful not to cut off a chunk of space
- because you might want to get back in there later.
- And if you leave only a little room to get to a certain section,
- then when you go there, you fill up a lot of it
- before you leave that section again.
- Or else, instead of a "DO"-odle you'll have an unhappy "DON'T"-dle.
- Or maybe you decide to make a meta-squiggle:
- A squiggle made out of squiggles.
- This can be done kind of abstractly, or extremely precisely.
- For example, let's say you're drawing this simple squiggle.
- Then, you draw that squiggle using that squiggle.
- But to make it fill up space nicely,
- you make the outside parts bigger.
- Then to make it precise, you make the number of squiggles
- always the same.
- It's easy to keep squiggling this squiggle all the way
- across the page if you keep the rhythm of it in your head.
- This one's like:
- [speaking rhythmically]
- Down a squiggle, up a squiggle, down a squiggle, up a squiggle...
- ...down a squiggle, up a squiggle, down a squiggle, up a squiggle, down a squiggle, up a squiggle.
- But after you've done that awhile, you decide to go a level deeper:
- A squiggle within a squiggle within a squiggle!
- That's right, we're going three levels down.
- This serious business can go something like this:
- Right a squiggle, left a squiggle, right a squiggle, left...whoop!
- .....right a squiggle, left a squiggle, right a squiggle, left...whoop!...
- ...right a squiggle, left a squiggle, right a squiggle, left...whoop!
- And the next one's even crazier!
- Like: And...Up a squiggle, down a squiggle, up a squiggle, down...woop!...
- .....up a squiggle, down a squiggle, up a squiggle, down...woop!...
- ...up a squiggle, down a squiggle, up a squiggle, down...woop!
- Wop...all the way over here! And...
- [speaking faster] Down a squiggle, up a squiggle, down a squiggle, up...woop!
- ...Down a squiggle, up a squiggle, down a squiggle, up...woop!...
- ...down a squiggle, up a squiggle, down a squiggle, up...woop!...
- Wop...all the way over here!
- Okay, but say you're me and you're in math class.
- This means that you have graph paper,
- opportunity for precision.
- You can draw that first curve like this:
- [rhythmic speaking] squig-uh, squig-uh, squig-uh, squig-uh,
- squig-uh, squig-uh, squig-uh, squig-uh...
- The second iteration to fit squiggles going up and down
- will have a line three boxes across on top and bottom.
- If you want the squiggles close on the grid as possible without touching.
- You might remind yourself by saying: Three uh-squig, uh-squig, a squiggle.
- ...Three uh-squig, uh-squig, a squiggle.
- The next iteration has a "whoop".
- And you have to figure out how long that's going to be.
- Meanwhile, other lengths change to keep everything close.
- And...two a squig, a squig, a squiggle...
- ...three a squig, a squig, a squiggle; three a squig, a squig, a squiggle, two...nine.
- Two a squig, a squig, a squiggle...
- ...three a squig, a squig, a squiggle; three a squig, a squig, a squiggle, two...nine.
- We could write the pattern down like this.
- So what would the next pattern be? Five.
- Two a squig, a squig, a squiggle...
- ...three a squig, a squig, a squiggle
- three a squig, a squig, a squigle...two
- Nine.
- Two a squig, a squig, a squiggle...
- ...three a squig, a squig, a squiggle...
- three a squig, a squig, a squiggle...two
- Nine.
- Two a squig, a squig, a squiggle...
- ...three a squig, a squig, a squiggle...
- three a squig, a squig, a squiggle...two
- Nine...and 15 all the way over to here
- And now...oh yeah, I can talk that fast! Totally!
- Okay, but let's not get too far from the original purpose
- which was to nicely fill a page with a squiggle.
- The nicest page filling squiggles have kind of the same density
- of squiggle everywhere.
- You don't want to be clumped up here
- but have left over space there,
- because monsters might start growing
- in the leftover space.
- On graph paper you can be kind of precise about it.
- Say you want a squiggle that goes through
- every box exactly once and can be extended infinitely.
- So you try some of those and decide that since the point
- of them is to fill up all the space, you call them
- space filling curves.
- Yes, that's actually a technical term, but be careful
- because your curve might actually be a...
- [rhymic speaking] ...snake, snake, snake, snake, etc...
- Also to make it nearter, you draw the lines on the lines
- and shift the rules so you go through each intersection
- on the graph paper exactly once
- which is the same thing as far as space is concerned.
- Okay, heres the space filling curve that a guy named Hilbert made up.
- Because Hilbert was awesome, but he's dead now.
- Here's the first iteration.
- For the second one, we're going to build it
- piece by piece by connecting four copies of the first.
- So here's one. Put the second a space away
- next to it and connect those,
- then turn the page
- to put the third sideways under the first
- and connect those.
- And then the fourth will be the mirror image of that
- on the other side.
- Now you've got one nice curve.
- The third iteration will be made out of four copies of the second iteration.
- So first build another second iteration curve out of four copies of the first iteration...
- ...one, two, three, four...
- Then put another next to it, then two sideways on the bottom.
- Connect them all up; there you go!
- The fourth iteration is made of four copies of the third iteration
- the same way.
- If you learn to do the second iteration in one piece,
- it will make this go faster.
- Then build 2 third iterations facing up next to each other
- and two underneath sideways.
- You can keep going until you run out of room.
- Or you can make each new version the same size
- by making each line half the length.
- Or you can make it out of snakes.
- Or if you have friends, you can each make an iteration
- of the same size and put them together
- or invent your own fractal curve so that you can be cool like Hilbert,
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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