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## Math for fun and glory

### Course: Math for fun and glory>Unit 1

Lesson 4: About pi and tau

# 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. Created by Vi Hart.

## Want to join the conversation?

• At , are raindrops actually spherical?
• The shape of a drop of water depends on its size. Smaller ones are spherical.
http://en.wikipedia.org/w/index.php?title=File:Raindrops_sizes.svg&page=1
• If √2 (1.4142135623730951) is a common geometric number is √3 (1.7320508075688772) one?
• sqrt 3 has a lot of interesting properties just like sqrt 2. sqrt 3, for example, is the length of the space diagonal of a unit cube. It is also double the ratio of the height of a equilateral triangle to the side length.

Hope this helped!
• Would a infinagon be considered a circle?
• Maybe, it looks like a circle but in some sense it might be, it depends on how you calculate it.
(1 vote)
• Lots of my friends say Pi is right. Who should I believe?
• Tau is a million times easier than Pi, so that's why people think Tau is better.
• At won't the diameter change when you "inflate" the circle?
• honestly whenever she says something like "oh no the teacher's walking around" my heart skips a beat
• If pi is 4,then tau is 8,and in other videos Vi wrote tau as 6.28...,so in this video she proved herself wrong.Why did she do that?
• She says "you may think it's 4", not "it is 4".
(1 vote)
• Computers have pixels. Does that mean that pi=4 on a computer?
• Yes, for a computer, pi does equal 4. But the computer's pi is not right.
(1 vote)
• The peaks aren't zero. They're 1/infinity. Am I correct? (Or maybe 1/(2^Infinity)😉
• The peaks have a height of zero (1/infinity = 0), and each segment also has a length of zero. Note, however, that 0*infinity is undefined, and it does not make sense to try to calculate it.
(1 vote)
• So the logic in this video is exactly the same as the logic in the more famous of Zeno's paradoxes, sometimes known as Achilles and the Tortoise. I never used that, so I'll explain it my way.

Say a man in archery class is shooting arrows at a target and every time he shoots he gets 1/2 of the distance between the last arrow and the target. The first time he shoots, his arrow lands halfway to the target, the second time halfway between those, so 3/4 to the target, and so on until infinity. Does the man's arrows ever actually hit the target? Realistically, yes, but technically, no. The arrow infinitely approaches the target and it seems like it hits it, but in a perfect mathematical reality with no gravity or wind or variables in any way, it never does, only gets infinitesimally close to the target without ever actually touching it.

So, the logic here is that the peaks of the triangles and the zigs and zags of the square around the circle infinitely approach their destination, but as the arrow never hit the target, if you keep zooming in infinitely into the fractal of the idea, you'll see that it never actually becomes flat, and the square will be forever lonely because it will never ever actually hug its perfectly circular love :'[
(1 vote)
• But -- if the length of the arrow is a fixed constant
and
the distance between the shooter and the arrow is a fixed constant
and the man/shooter is a fixed constant...

The arrow will eventually have to hit the target.

If I'm understanding your explanation/example -- it seems to suggest/assume that the length of the arrow is not a fixed constant and that the distance is not a fixed constant.

or -- did I miss something?