What is the area of a Squircle?

This is the GeoGebra file Sam Hartburn made for me:

Here is The Coding Train's first part of the Super Shape series.

I mentioned the these old videos of mine:

Lemniscate on MathWorld

This is a really good write-up about Gauss and the Arithmetic-Geometric Mean.

This is a nice related problem about the area v perimeter of a squircle.

CORRECTIONS
- If you feel a sense of déjà vu: yes, the two sections at 16:13 and 16:45 cover the same content. I tried saying it two different ways and we accidentally left both in the edit. So, let's call that 'buy one get one free' sale on learning.
- At 21:42 I say "geometric mean" instead of "arithmetic mean". And that is probably not the only time. These words have lost all meaning to me.
- The Patreon credits stop early! Long story. I'll explain on Patreon.
- Let me know if you spot any other mistakes.

Crazy amount of editing and meshing of footage by Alex Genn-Bash
Maths graphics by Sam Hartburn and Matt Parker
Kitchen by Carrie and Nina
Music by Howard Carter
Design by Simon Wright and Adam Robinson

MATT PARKER: Stand-up Mathematician

[VY_Canis_Majoris]

He has done it, he has officially achieved both quality and quantity.

[Nikolas_Davis]

Matt is the only person I know who can successfully bully himself.

[KratosElGreatos]

In America, we pronounce it "Lemnookie" because it rhymes with cookie.

Edit: This is also done to avoid confusion with the American band "Limp Bizkit", a common mistake in the late 90's which ultimately led to a scheduling mishap that resulted in 1 unexpectedly interesting and educational concert and 1 very disappointed maths conference audience. Although some argue that pronouncing it "Lem-nookie" only increases the associative confusion...

[jackdog]

I’m 20 years old with an A level in further maths and thinking “the value of pi for a square is 4” kinda just blew my mind.

Update: I am now a 23 year old studying a Master’s degree in Data Science and this is still a cool insight to me.

[MuradBeybalaev]

33:04 I love how that backpack gag, as presented, holds no extraordinary nature at all but so many levels of "dad editing magic" deep it does feel gratifying somehow.

[andrycraft]

What I learned today: Matt lives with an identical clone of himself and he's also in possession of some sort of trans-temporal device with which he communicates with his future self. What a man!

[TheCodingTrain]

Amazing video, thank you for the mention!

[justpaulo]

Matt : "- Let me know if you spot any other mistakes.
"

Well, Future Matt at 23:00 decides to skip the 1st decimal digit of the arithmetic mean (it should be 1), and it ends up saying that 1.1981 is super close to 1.981 never noticing the mistake.

I guess it's time for Future² Matt to enter in action.

[michihaba]

I love how Future Matt could flawlessly edit that whole multiverse-Matt stuff but made a simple editing error between 16:00 - 17:30 :D

[derekhasabrain]

I need more videos where the creator interacts with other versions of themselves, even past and future. I'm sure that took a HEFTY amount of scripting and even more tries to get the timing right between them like, it legit just sounds like a normal conversation between the three of them. You outdo yourself on EVERY video, I swear. I'm planning on making an official channel and you are definitely one of my muses

[PaulPaulPaulson]

Me: Obviously, he's gonna use polar coordinates and averages the distances to the center
Matt: Let's take the average between 2 and infinite as the exponents

[rubenlarochelle]

Matt: "So squircles are a midpoint between squares and circles, but at what arbitrary point will we settle?"
Me: "I'd say at the point where the area equals the average of the two, that would be nice..."
Matt: "Oh no, that would not."

[Nylspider]

Me: existing
Matt: so have you ever wondered what the area of a Squircle is?"
*And that's the moment I started wondering what the area of a Squircle was*

[storyspren]

The cirare, or perhaps the Parker Squircle? Hear me out: the Parker Square and the Parker Circle may have been examples of giving it a go, but the true average of them, the Parker Squircle, is an example of vastly improving on something.

[vipinx]

The quality of this video is actually insane. Like, it’s good. The math is super cool (and explained very well), the skits are funny, and the production quality is through the roof. Hats off to you, this is now one of my favorite videos of all time.

[EngineerWhen]

Matt: "There's a whole family of so-called superellipses..."
Me: so this shape is clearly a superellipse with degenerate foci!
Matt: *squircle*

[johnchessant]

The AGM of 1 and 1/sqrt(2) (instead of sqrt(2)) is used in a very fast π calculation algorithm, called the Gauss-Legendre algorithm. The number of correct digits of π roughly doubles in each iteration.

[Verlisify]

That icing F looks like Matt invented a new constant symbol

[ssvis]

"The gamma function is not friendly." There's an understatement.
I also like his comment on perimeters. Look at fractals and their boundaries. You can have infinite perimeter length with finite area.

[TheDrawnator]

I don't think people talk enough about how good your video descriptions are, you even link the videos you mention only once throughout the 34 minutes, I appreciate this a lot