Stand-up Maths - 2021-09-01
Check out https://www.kiwico.com/standupmaths and get 50% off your first month of any crate! Maybe enjoy making a pencil sharpener! Not that I'd know what that's like... This is the GeoGebra file Sam Hartburn made for me: https://www.geogebra.org/m/pffgbck8 Here is The Coding Train's first part of the Super Shape series. https://www.youtube.com/watch?v=z86cx2A4_3E I mentioned the these old videos of mine: Ellipses and perimeters https://www.youtube.com/watch?v=5nW3nJhBHL0 Generalising fibonacci https://www.youtube.com/watch?v=ghxQA3vvhsk Lemniscate on MathWorld https://mathworld.wolfram.com/Lemniscate.html This is a really good write-up about Gauss and the Arithmetic-Geometric Mean. https://homepage.univie.ac.at/tomack.gilmore/papers/Agm.pdf This is a nice related problem about the area v perimeter of a squircle. https://www.johndcook.com/blog/2019/04/11/dido-problem/ Cheers to my Patreon supporters for whom I have dedicated one digit of Gauss's constant each. It was their inspiration who made me do the 'multiple Matts' thing. You can also help support and shape the videos I make! https://www.patreon.com/standupmaths 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 Website: http://standupmaths.com/ US book: https://www.penguinrandomhouse.com/books/610964/humble-pi-by-matt-parker/ UK book: https://mathsgear.co.uk/collections/books/products/humble-pi-signed-paperback
He has done it, he has officially achieved both quality and quantity.
... and biscuits
@Xnoob Speakable And it gain self-aware
you missed entertainment. so he achieved qualntiment
Would that be qualinty or quanlity?
It's funny
LEMNISCATE is an anagram of CENTESIMAL (relating to the division of hundredths). Strange to see two long mathematical words being anagrams. The only other couplet (anagram: octuple) I can think of is TRIANGLE-INTEGRAL.
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.
"Pi is... about 3"
Yeah, the revelation that 4 is pi, but for squares took me by surprise.
I have 2 AS levels in maths (in addition to an A-level), and I've always been confused about how that happened but never bothered to question it.
@ibrahim Maybe we need to take the arithmetic geometric mean of pi based on circumference (8) and pi based on area (4) to get the true value of pi for a square?
pi for a square = 5.8271641
@terry boland "The diameter of a square is its diagonal"
based on wikipedia, yes. Based on the video, no.
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."
My gut for finding the 'midpoint' of two shapes is to draw a ray from origin at an angle of π/4, find the points of intersection on the square and circle, and then find the N value that places the Squircle on that point. Thjs seems calcuable, I will try tomorrow
I love how everyone is going "that would be a nightmare" when the title of the video is "what is the area of a squircle".
If your squircle is defined by the average area of a circle and a square, you've already solved it.
No need for N, no need for biscuits, the area of a squircle is half pi plus two, times the radius squared. Video done.
If I did my calculations correctly, that squircle would be the one where n is approximately equal to 3.16203795521946.
@Tom de Kleryeah this comment section is insanely funny
@saucylegs Using the desmos calculator, and a rather simple search algorithm, I got 3.1620379552 (this was the most precision I could get out of it) so your number looks right.
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
Honestly, way too many videos do this and it just drags things out longer than they need to be. I'm all for videos being entertaining but, like this video for instance could've been done in probably five minutes without the multiple sponsorships, built-in ads (how many ads were in this video is beyond me) and the talking to each other and the patreon credit sequence (we're turning into a Christopher Nolan film by this point). A myriad of physics, math, and graphic design videos have content creators talking to their doubles. It's a trope by now.
Matt is the only person I know who can successfully bully himself.
Nick Lucid does a pretty good job of bullying Question Clone.
Looks like someone haven't known about HAACHAMA and Amelia Watson. They bully themselves quite hilariously for almost a year now.
I feel personally attacked.
eminem
what abotu Ryan Reynolds?
In America, most people take a course called “Precalculus” in which one part focuses on graphing in the polar coordinate system. They introduce a handful of curves, including the the lemniscate, along with cardioids or rose petal curves to graph. Its not so much a ‘grand finale’ as in my experience they just give the equation and what it looks like, then ask you to try and graph it in different scaled versions or orientations, and thats it. We don’t really learn any properties besides how to graph them in polar.
By America do you mean the continent or the United States? I'm asking because as a Canadian, I haven't had precalc yet but I feel like I probably should have. Maybe it is a thing in Canada but not in the province I'm in. Maybe it's just Quebec being Quebec.
@MasicBemester They more than likely meant the United States. I can't speak for other countries, but my experience in the United States closely reflects the original comment.
@CarsonT i see
Yes, we know. America has horrible math and science education.
@Schmetter Ling crazy how almost every top stem college in the world is in the us then
"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.
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.
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
I now have interest in what a lemniscate is.
I've also learned that my phone keyboard dictionary knows "lemniscate"
@SgtKOnyx
Isn't it what you get when you sprinkle citrus juice on a flat fish? Yum!
@True River that pun there really threw me for a loop or two, I'd call it a lemniscate pun
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...
it's not a lemons-cake???
Contrary to popular belief, the maths conference wasn’t disappointed because they saw a concert instead of the pre-planned presentation, but rather that the concert was Limp Bizkit.
IS THIS FOR REAL ?
@Arin Roday Was my joke that convincing? 🤣
That's not why you pronounce it that way. Be honest now! :-)
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
@pvic he repeated the same points about area and perimeter of lemniscate
I thought I have a deja vu
@brachypelmasmith I assumed that was some kind of intentional joke
that is not an error! it is stylized editing: and quite brilliant in my opinion: he looks at the board in silence, then we glance at it as well 😄
@phpART nope, both past and future Matt have spoken (via the video description):
“If you feel a sense of Deja 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.”
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.
I love how you phrased it xd
The formula for a polygon perimeter is (sides)2r
A CIRCLE PERIMITER IS 2PI R
A CIRCLE HAS PI SIDES
Loved the lemniscate biscuit skit, Matt!
Oh, and as a curiosity, another way to define the unit square in the xy plane is given by the equation |x + y| + |x - y| = 2. Pretty cool equation if you ask me, but it wouldn't be practical for defining a squircle, I think.
Actually, |x+y|^2+|x-y|^2=2 defines a circle, which makes sense when you think about it — it has to be a conic section, for starters. Not sure whether the shapes you get for exponents between 1 and 2 are the same as some of the inner squircles here; I suspect so, but don’t know how to prove it, and you get radically different behavior for extreme values.
Fun fact! There's a special type of woodworking jig that can drill "square" holes, and it actually drills squircle-shaped holes!
Are you sure it's squircle and not a square with quarter-circle corners?
@Aleister Lavey Tbh it gets close enough that a quick nibble with a chisel squares the hole right up. :)
But then most woodworkers don't even bother wasting their money on the jig and just drill a normal hole and do more nibbling with a chisel to make it square, so... LMFAO.
It's just an expensive reinvention of the wheel. Could probably see justifying its expense if one routinely drilled square holes so much that the tiny extra bit taken out in the drilling step saves time on the chiseling step.
@vsm1 Pretty dang sure! It's a weird mechanism that uses a single cutting bit to drill a squircle-shaped "square" hole that takes only a minimal amount of nibbling with a chisel to square it up properly.
Holes drilled by those bits are squares with rounded corners, not squircles.
@Calyo Delphi When I worked at a commercial builder's yard involving a lot of timber-related stuff there was a custom jig to 'drill' square/rectangular mortices into/thru fenceposts to accept the rails. It wasn't very complicated: a Forstner bit (to remove the main material) surrounded by a square chisel enclosure (to square off the mortice) mounted into a drill press. It was one of the easiest repetitive jobs in the yard, provided you had the depth stop correctly set & didn't have waney wood to deal with
This was a tour-de-force. The effort required must have been enormous, but soooo worth it!
I come back to this every now and then. Having made several personal comedy videos with free budget editing software I found off of the internet, this is one of my favourite videos on YouTube ever. Great humour, great editing honestly, very creative, and it’s an interesting topic. But it was just really amusing to watch you get creative with the edits and fourth wall breaks.
I can only wish I had a math teacher like you when I was a child, I might have learned some math. Though I can't follow all or even very much, honestly very little, it still fascinates to hear someone knowledgeable talking about their subject of expertise. Very entertaining presentation, and I also feel like I am learning something!
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!
The "clone" is most likely Matt from another part on the timeline visiting present-Matt.
You telling me this temporal device is trans?
Mathematicians are essentially modern superheroes...
Yoo pfp pals
Nah, these are three people and this is just the way they live.
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
I was really inspired to find the formula for the area of a squircle / superellipse with a=b given any radius from this video but I got stuck after a lot of attempts and decided to set it aside for about 6 months. Returning recently, I finally figured out how to do it with some methods I've seen in other trig and Beta function related integrals I've worked with over that time to get A = (4r² Γ²(1+1/n))/Γ(1+2/n).
The quality of this video is actually insane. Like, it’s sooooo 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.
25:21 Matt's VFX are always so good that it's hilarious when he does stuff like this ahah
Your interaction edits are so hilarious. Their entertainment value doesn't take away from the learning value either.
Oh Matt, I tried to define the Sqwuircle and its area and perimeter about 20 years ago, but you made it so much more fun an interesting. Love your work!
I loved the interplay between the different Matts, that was hilarious!
By the way, before watching the video I thought of how I would approach finding the area of a squircle, but I realized by the end of the video that we were talking about different squircles. I took a square and rounded off the corners with quarter circles, keeping the four sides as straight parallel lines.
If we assume the distance across the center of the squircle, from one side to the other is length “c”, and each curved corner has a radius “r”, then the Area of that squircle would be A=c^2 -0.858r^2.
I even took a go at the perimeter, though that was just as easy: P=4c -8r +(pi)r^2.
It would seem that there are many ways to define a squircle, but knowing now that your squircle is different than mine, I’m curious to know why you chose the squircle you did.
A small observation I found interesting, the harmonic mean between n=1 (which leads to the square on it's side) and "n=infinity" if we assume we can use infinity as a number (which leads to a square) is 2 which gives a circle. So in certain way a circle is the midpoint between a square and a square, which intuitively makes a lot of sense to me.
Amazing video, thank you for the mention!
Go Dan! I was so happy to hear the mention!
I think you two should collaborate. You could make a program that scrolls the names of patreon supporters.
Yeah, just discovered your channel becuase of him. Subbed.
the godfarther of mutimedia design undergraduates
what a coincidence I knew both of you guys
To me, the perfect squircle would be the one where you take the distance from center to the "corner" exactly inbetween √2 and 1.
When you sketch the computation of the AGM of 1 and sqrt(2), in the second iteration you've missed the 1 after the decimal point of a_2: instead of 1.198156948 it reads 1.98156948.
Also, what's up with the number of matching digits in the k-th iteration being equal to k^2?
I was slightly confused by that because Future Matt said the numbers were really close, but the missing digit makes them appear much farther away.
@Ryan Ratcliff Yeah, it also made the larger number bigger even though applying the arithmetic mean to it should do the opposite.
Great video!
But you forgot another type of square-circle hybrid, one based on the polar equation for a square somehow mixed with a circle (I call it the Squarcle). I have no idea what the area or perimeter is though (unless a circle or a square. Small hint for future-future Matt... Matt's....?).
Personally I think it looks best when the (unit-ish) square radius is raised to the power of 1/e (Euler's nr).
(That's just my opinion which I got just now from a quick test of a small Processing snippet (basically Java) I wrote a while ago :P).
Unit-ish square radius (not the diagonal, which would be a bit more than unity at sqrt(2)) simply raised to a power between 0 and 1 (0 = Circle, 1 = square).
Also I'm hoping mentioning Euler get's all three future Matt's attentions.
Or not. Oh well :P
28:30 I'm glad you explained that was a pencil sharpener because, interesting fact, it looks like a coffee maker 😂
I want a Dr. Seuss-style book of Matt trying to convince people to try his lemniscate biscuits, corny rhymes and all.
They need to be lemon lemniscate biscuits.
They should be stacked, but they probably wouldn't be stacked vertically - they'd be leaning lemon lemniscate biscuits.
I hope they have a smooth texture - I wouldn't want lumpy leaning lemon lemniscate biscuits.
Do they glow in the dark? That would make them lumpy leaning lemon luminescent lemniscate biscuits.
I will not eat them on a fox. I will not eat them in a box.
Oh my god this is brilliant! For my first job I had to draw the 2d cross section of multiple copper wires squashed into cylinder (the wiring within an electric car engine motor). These shapes began as circles and were squashed slowly into squircles as you added more wires, or reduced the cross sectional area of the cylinder (I know technically they would be compressed to hexagons, but a squircle provided near-accurate predictions and was more simple).
I still remember my moment of pride, completing my first coding task ever, allowing a dynamically chosen number of wires to change the picture on the screen, slowly becoming more square as you added more.
Better yet, these wires had insulation and copper components, so drawing a compressed squircle within a squircle within a cylinder was quite the challenge! Ahh how I miss my old job.
For practical applications I use the diagonal. You make a function where you give the diagonal and it gives you back the exponent. Then you enter the ari.mean between the radius and half the square's diagonal.
The sheer joy Matt emanates during the lemniscate biscuit bit is immense <3
Another interesting thing about the squircle is that, as an implication of by Fermat's last theorem, at least one coordinate of all points along its perimeter must be irrational, except at the 4 points where it intercepts an axis.
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.
Wait, pi is the AGM of 1 and 1/sqrt(2)? EDIT: Looked it up on Wikipedia. There's more bits to the formula. Still cool, though
wtf damn
I think the AGM algorithm would be a great pi day calculation
@Marcus Orban way too accurate for pi day
Would that allow a raspberry Pi to easily calculate 1e15 (2^50) digits in a few hours?
I've always wondered if there was a name for what I now know is called the arithmetic geometric mean! I noticed I could always get the iterative process to center in on a final value, but I never knew it was an established process!
please tell me you have a video planned for dealing with the cirare and calculating that area... also, thank you for allowing me to spend my morning wrapped in calculus and trigonometry trying to see how far I could get with the area of the squircle using only the math that I have learned: got it down to sqrt(2) times the integral from 0 to pi/2 of sqrt(csc(x)) dx, using only trigonometric substitution and other similarly simple algebra.
Oh boy the ending scene got me laughing 🤣 top notch work Matt, really clever
To compute the "nightmare" complete elliptic integral of the first kind K(k), you precisely use the arithmetic geometric mean. Explicitly: K(k) = pi/(2AGM(1,sqrt(1-k^2)). So actually it is not really more complicated than what Matt presents afterwards.
Your brown paper explanation has objectively surpassed Brady's - yours is in much better taste!
you could say it's .. the icing.. on the cake biscuit.
"Objectively"->"Objectivity"
Hmm
@Amelia Kamel🌹 What the bot is this bot doing on a math vid
@Drenz 152 It's a rather recent bot.
Use a logarithmic slider so that you can more easily get closer to a square :)
..100℅ effort for this amazing, knowledgeable and interesting episode.
Recently I made a crt shader in hlsl and to do the rounded corners I needed the squircle! it was such perfect fit too. Man, squircles are great.
This video is so good. Had my brain hurting with all the arithmetic, but the fun editing kept me in
Really excellent video, both in terms of content and production. I love your channel, Matt.
Stand-up Maths - 2021-09-01
Sorry the Patreon credits cut off! They are all here https://youtu.be/s7YOx_3qTJc so you can find your Gauss digit. Or, here, just take this one: 8
And thanks to KiwiCo for supporting me along with every other nerdy channel. Best use my link just to be safe: https://www.kiwico.com/standupmaths
Asher S. - 2021-09-28
Heart a real squircle; https://youtu.be/jRhrbb9Hhq0
Anvilshock - 2021-10-14
You know what would have been nice before going down that rabbit hole to find the formula for the area? A definition of what a squircle actually is, as in, in what way it is a mix of the two bodies, to make the viewer understand the challenge in the first place. After all, if it was as simple as a square with a corner radius, that wouldn't have quite the need for half an hour of everybody's time, now would it?
Mauricio Micoski - 2021-10-16
Very well done! I can see it was a lot of work, as you had to use 3 Matt-Power to get it done in time 😄
Hari - 2021-12-01
You are missing a 1 on the right hand side at https://youtu.be/gjtTcyWL0NA?t=1391
ShadowsinChina - 2022-01-22
@jonny h. keep trying. 🤠 what if are important questions and math takes sweet time to land.