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Rhapsody on the Proof of Pi = 4: Correction: when I mark where pi is on the graph, I meant pi/2! Note: If this video were supposed to be teaching you, I'd probably have to make it boring and say that in one sense of limits, spoiler alert, you actually do approach a circle and a line, solving the apparent paradox by saying that the invariant of length does not hold over infinity. Luckily I am an artist, and this is a Rhapsody, and instead of "learning," you get to actually think, if you like.
相關課程
- Say you're me in your math class,
- and you're supposed to be graphing functions.
- As if there were some deep relationship
- between y and x that your teacher just
- won't stop gossiping about.
- But, like most gossip, you really don't
- care about y's unhealthy dependency on x.
- Really y, get a life.
- Luckily, your friend passes you a note.
- You wait till the teacher is facing the board
- and sneakily open it up.
- And it's this proof that pi = 4
- that you've seen billions of times on the Internet.
- But because you're bored and
- at least it's not graphing, you take
- another look.
- It's like this: start with a circle,
- with diameter 1 and circumference pi.
- Then draw a square around it.
- The length of a side of the square is 1,
- so the perimeter of the square is, of course, 4.
- Now you start approaching a circle by
- zig-zagging this perimeter.
- This shape still has perimeter 4.
- And we do it again, and it's still 4.
- And again, and we approach a circle.
- While the perimeter never changes.
- So there, 4, pi equals 4.
- Obviously there's something wrong here.
- The circumference of a circle is pi but
- pi is more like 3.14 numbers and less like 4.
- So somehow what this process approaches is something that looks like a circle but is not a circle.
- I mean, what does circle mean anyway?
- There's this loopy line thing, and then there's this solid disc shape.
- well there related an all there not the same mathematical object
- this troll proof so much FUN because you repeat this zigzag move all the way to Zigfinity
- it approaches the shape of a circle in a disc seance
- an it approaches the area of a circle but it dose not approach a circle
- its all crinceled up an if you imagined if you put a astral in this thing
- an deflated it added as much air in it as you can wile keeping the primmer as 4
- all the infinite ringcals would smooth out an you get a circle with a perimeter 4 an diameter 4 over pi
- in fact you can inflate the square to get the same thing
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