Does pi = 4? (A Good Explanation)

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[Gorbinex]

Hey, I don't know what's been up, but if you ever see this I hope things are going alright for you. Thank you for sharing yourself with us all this time and I hope that you are well wherever you are right now, or if not, that things will get better. You have my best wishes!

[ZyTelevan]

Great job! I think that in general, it's really valuable to not only debunk fallacious arguments, but also debunk fallacious counter-arguments that happen to come to a right conclusion and offer valid counter-arguments instead.

[Zosso-]

i’m actually blown away by the quality of this. it’s so simple, yet no less enlightening. thank you!

[maggomor]

This is a stellar explanation for a problem that has been bothering me for a while. In particular I enjoyed that you first debunked some pseudo-explanations. The problem with online explanations of math is often that they give plausibly sounding but ultimately wrong intuitions to explain a correct answer. Most often I see that with the Monty Hall problem. Here, it had a similarly good application. Thank you, truly! You have put my mind at ease.

[rri]

that's the same problem as trying to measure the length of a coastline.

[BRLN]

I am an (applied) mathematician on my way to my phd and this particular question bothered me for years. This answer is not fully 'crystal clear to me', but good enough to find my peace. So Thank you a thousand times and greetings from a subscriber from Berlin.

[autumn]

Ok, this was actually very informative! I've tried to figure this out on my own, but I never understood why it didn't work. Separately, I wondered if it was possible to have a line identical to, say, y=x, but the slope at any given point would be completely different, say, 2, or -1.

But if the direction of the velocity matters, then it is impossible to have identical lines with different slopes, since in order to be identical both the position and direction would have to be the same. If the directions are the same, the slopes must also be the same.

I find it really interesting how one misunderstanding can cause you to be misinformed in some other place that seems unrelated. As soon as you think it doesn't matter, that it's all hypotheticals anyway, you open up a door elsewhere for fallacies to leak through, like how believing that .999... =/= 1 makes calculus a nightmare, even though such a distinction between two practically identical numbers is never directly applicable in real life.

[RijiVids]

Thanks for the great video, and I don't think a video of this length is too long. In fact, I like longer math related videos so that I can really immerse into them, although short videos like these are great too, good job!

[benburdick]

This video reminded me of something that my calculus teacher said that stuck with me: If you have a function, and you want to make something that fits it really well, one way to garentee that is to make sure the derivatives are all the same. Hence, how you build a taylor polynomial.

[markgross]

It’s important to note that while the approximations do not approach the same surface area/perimeter, they do approach the same volume/area.

[quintopia]

Good to see my initial intuition about the limit of the sequence of tangents was justified.

[stevel]

Pretty good explanation. The way I understand the argument: pi equals 4 only when there is motion. (Sober aunt cant do any better than drunk aunt etc.)
So we would need to experiment.

[Luke-utep]

basically this looks like the difference between making an open ended perimeter (in terms of length) fit inside set parameters versus taking a fixed length and fitting it to a shape. for instance say you wanted to measure the inside of a bottle if you line just the edge with as little paper as possible (internal approximation) you are trying as best as you can to match that internal bottle circumference, however if you took an obnoxiously large length of paper and folded it back and fourth such that it concertinaed against the edges it would technically be an internal approximation but as you have already started with a size you have decided it will be you aren't approximating the true length, more fitting the answer you want to a later asked question.

TL:DR the difference is having an unfixed length to attempt to accurately as possible measure the internal approximation of a shape vs having a predetermined value that you will fold an infinite time until it is forced to match visually

[GretgorPooper]

My favorite explanation is the one that shows that, only because the area of an iteratively constructed shape tends to the area of another shape when the number of iterations tends to infinity, that doesn't imply the perimeter does the same thing. It is possible to approximate the area without approximating the perimeter.

The reason we know that the rhombus example works, then, is because it approximates the perimeter itself. We can prove (1) that its perimeter is always at most that of the circle, and (2) for any given number that's lower than the perimeter of the circle, there is a number of iterations that will make the perimeter of that shape higher than that number. Those two put together show that the perimeter tends to pi.

[ntecleo]

was not expecting an explatation using velocity vectors...
amazing stuff!

[BigPapaMitchell]

I prefer to think about these as infinite sums, because some of the points you raised are countered when you take the limit of these curves. For instance, with the triangle one: The limit of this shape will mean you can't find a point on the zigzag line that is a positive distance away from the original line, and yet we know the value will approach two.

[johnchessant]

Did you delete a bunch of your old videos? Like the ones about dimension, Pujol, polling, email, confusing geography, etc. I miss them.

[Lucas]

Earlier today I was thinking about how long it's been since I've watched one of his videos, sad to see he isn't uploading anymore

[Asdayasman]

5:48 MY GOD

IT'S A FOURDGET SPINNER

[asherPrice]

A beautifully clear illustration of the issue; bravo...