trigonometry like you've never seen it

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23 май 2024

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Комментарии : 235

@owensthethird 23 дня назад

I drew a unit piggy and subsequently defined the coswine function

@samueldeandrade8535 23 дня назад

Hahahahahaha.

@sethdurais2477 23 дня назад

*click* Noice!

@5alpha23 23 дня назад

that actually made me giggle, haha

@jeffreyjdesir 23 дня назад

good one!🍹

@kruksog 23 дня назад

Swine/coswine is the hamgent (a reach, I know. I'm sorry😅)

@zh84 23 дня назад

Fun fact: x²ª + y²ª = 1 for a positive integer has no rational solutions when a > 1 other than (±1, 0) and (0, ±1). For if it did, put both terms over the lowest common denominator, multiply out by that, and you have a counterexample to Fermat's Last Theorem. So all the squircles make their way through the everywhere-dense set of points which have both coordinates rational, without touching a single one of them!

@Zartymil 23 дня назад

This blows my mind

@aniruddhvasishta8334 23 дня назад

That's a crazy fact, wow!

@pelledanasten1615 23 дня назад

What if gcd=1?

@samwalko 23 дня назад

@@pelledanasten1615 Then the least common denominator is the product of the denominators, if i understand your question correctly.

@aniruddhvasishta8334 22 дня назад

@@samwalko I think he got lcm and gcd mixed up

@lurkmoar3926 22 дня назад

Hi Michael, The higher dimensional analogues of the 👉interiors👈 of squircles are called "superballs" and have been studied by Rush (myself) and Sloane, "An improvement to the Minkowski-Hiawka bound for packing superballs", Mathematika, June 1987, Volume34, Issue 1, pp 8-18 and by Elkies, Odlyzko and Rush (again myself) "On the packing densities of superballs and other bodies", Invent Math, December 1991, Volume 105, pp 613-639. I am a subscriber and I enjoy your lectures. 👍

@sbares 23 дня назад

Interestingly, it is known that Gamma(1/4) and pi are algebraically independent, so it follows that rho is transcendental.

@hybmnzz2658 22 дня назад

Where does this come from?

@onionbroisbestwaifu5067 22 дня назад

@@hybmnzz2658From “Modular Functions and Transcendence Questions” by Yuri Nesterenko, 1996.

@sbares 22 дня назад

@@hybmnzz2658 Modular forms of all places. I don't know the details, but apparently there is a theorem (Nesterenko's theorem) that for any z in the upper half plane, there is a set of three algebraically independent numbers among E2(z), E4(z), E6(z) (Eisenstein series) and q (=exp(2pi i z)). Setting z=i, we have E6(i)=0, so the other three must be algebraically independent. Their values are E2(i) = 3/pi E4(i) = 3Gamma(1/4)^8/(2pi)^6 and of course q(i) = exp(-2pi).

@PillarArt 22 дня назад

sup Queen's Uni 🍁

@sbares 22 дня назад

@@PillarArt ??

@godofmath1039 23 дня назад

Squircle sounds like an awesome Pokémon name ngl

@thomasblackwell9507 22 дня назад

CHARMANDER AGREES!!!

@user-en5vj6vr2u 22 дня назад

No arent you just thinking of squirtle???

@godofmath1039 22 дня назад

@@user-en5vj6vr2u Yes. I was thinking exactly of Squirtle. However, Squircle sounds like what you'd get if you'd evolved a Squirtle alongside Porygon, which would be awesome

@skyper8779 23 дня назад

mistake at 22:43 in dU. It is Beta(1/4, 1/4), the final value is Gamma^2(1/4)/2Gamma(1/2) = 3.70814

@user-gs6lp9ko1c 23 дня назад

Ah yes! That's it!

@emanuellandeholm5657 23 дня назад

Yes, it's Beta(1/4, 1/4). He got the numerical approximation correct tho. Python 3.12.3 (main, Apr 23 2024, 09:16:07) [GCC 13.2.1 20240417] on linux Type "help", "copyright", "credits" or "license" for more information. >>> from scipy.special import beta >>> 1/2 * beta(1/4, 1/4) 3.708149354602744

@NStripleseven 23 дня назад

Doesn’t look like it. The beta function (described in the correct form in the video) would have to be evaluated at (5/4, 5/4) to look like the integral we have. Though the resulting denominator would be gamma(5/2), not gamma(1/2).

@edvink8766 22 дня назад

@@NStripleseven Michael made a mistake with the change of variables from u to x. Where he wrote 5/4 in the integrant, it should've been 1/4 instead. The resulting expression is what the original comment above said. Michael's final expression doesn't even evaluate to the correct answer.

@gregsarnecki7581 22 дня назад

Furthermore, since Gamma^2(1/4) = (2.π^1.5)/AGM(rt2/2,1), so rho = π/AGM(rt2/2,1), providing a nice relationship between rho and π: π/rho = AGM(rt2/2,1)

@sucroseboy4940 23 дня назад

I was honestly shocked to see that this video had come out just now. Yesterday I spent a few hours, attempting to define the basic trigonometric functions for the squircle. Didn’t know other people had thought about this problem as well, although I’ll admit that isn’t the craziest idea in the world.

@lunaticluna9071 22 дня назад

theres a whole book on the topic: "Squigonometry: The Study of Imperfect Circles"

@sucroseboy4940 22 дня назад

@@lunaticluna9071 Yes I saw that. I didn’t know that before seeing this video, so I had a difficult time trying to figure it out on my own. I only managed to approximate the cosquine, as I’m not skilled enough to derive the formula myself

$@arthurfraco2970$

@arthurfraco2970 15 дней назад

I also was looking for something like that to model my data. Now that I found it I am afraid of using it and having to explain to peiple 😂

@bloom16night 21 день назад

"pi" formula for arbitrary n is rho = Г(1/2n)^2 / (n Г(1/n)) and it does approaches 4 as n->inf 😊

@davidemasi__ 17 дней назад

Do you know any proof of the fact it converges to 4?

@WasickiG 16 дней назад

@@davidemasi__ That appears to be equivalent to 4 - ζ(2)/n² + ζ(3)/n³ - 9ζ(4)/(16n⁴) + k ζ(5)/n⁵ ∓ … You only have to prove this holds :-)

@user-lr9ue4ro3i 23 дня назад

22:49 I think it should be 1/4-1

@filbranden 23 дня назад

My first instinct was that x=√cos(t) and y=√sin(t) would solve x⁴+y⁴=1 and would serve as a definition for the squircle, but then was surprised to see that the equivalent of π wouldn't match. I suspect the step of "taking" the definitions x'=-y³ and y'=x³ is what rules out the square root of the original trig functions to match the new functions, which is an interesting development.

@__christopher__ 22 дня назад

Well, the square roots only work for positive numbers (we don't want to have complex values for x and y), but then, the quircle is clearly symmetric, therefore the sign of the cosine times the square root of the absolute value of the cosine, and the sign of the sine times the square root of the absolute value of the sine should work quite fine, and indeed, every solution has to be some reparameterization of that. Indeed, I suspect the sign and absolute value make it non-analytic without appropriate reparameterization. Now we can try to calculate the derivatives of that, then we get (in the positive quadrant) x'(t) = -sin(t)/(2 sqrt(cos(t))) = -y^2/(2x), which looks quite different from the x'=-y^3 in the video. Note however that the differential equations in the video don't give a constant speed on the curve, since x'^2 + y'^2 = y^6 + y6, which clearly isn't constant. Therefore there is no reason to assume that the periodicity of the function tells us anything about the circumference of the curve. Now the fact that the number is between pi and 4 may be suggestive, but note that also the area of the unit circle is pi, and the area of the outer square is 4, so it could just as well be the area of the squircle. Or something completely different, of course.

@oscarlama 22 дня назад

@chloefisher1838 23 дня назад

Shouldn't the integral around 24:00 be of x^(1/4 - 1) (1-x)^(1/4-1) not 5/4?

@krisbrandenberger544 20 дней назад

Yes

@chessematics 23 дня назад

Time to introduce poqueuepine

@Durglass 23 дня назад

What

@OpRaven-62 22 дня назад

I'm assuming it's a joke, as the last word would be roughly pronounced like "po-cue-pine" or, in other words, "porcupine", an animal with spikes on it's back. @@Durglass

@coryschwartz1570 15 дней назад

Time is a flat squircle

@haldanesghost 15 дней назад

“The squine “ I fucking lost it 😂

@e.s.r5809 22 дня назад

I *should* be revising fluid dynamics, but 'squine and cosquine' was the nerdiest reason I've ever laughed myself incapacitated.

@giorgiobarchiesi5003 23 дня назад

Showing that all squircles with n > 1 never pass through points having rational coordinates would prove the Fermat’s theorem for even powers, right?

@ingiford175 22 дня назад

First time I saw something like that was in a sci fi book about what was riddle an equation for a square (Piers Anthony) back in the 80's. The answer was take the equation for a circle, and instead of 2nd power it would be 2*n power, and let n go to infinity, and the shape will approach a square.

@sylowlover 23 дня назад

My favourite way to define a squircle is to take a polar definition of a square and circle, i.e. r=1 and r=sec(theta), and average them, giving r=1/2+sec(theta)/2, bounded between -pi/4 and pi/4, and extending to the other 4 sides. This will however not be smooth.

@strangelaw6384 14 дней назад

is there a way of treating that smoothening such that it is like taking a limit?

@Nolrai12 23 дня назад

You forgot to link the book!

@get2113 22 дня назад

All solutions to the linear system are given by a matrix exponential. That should cover all solutions of |x|^a + |y|^a = 1, 1

@BryndanMeyerholtTheRealDeal 16 дней назад

I wonder what the squine and cosquine, as well as the other four functions (tan, csc, sec, cot) but for the squircle, look like?

@florentvaladier5099 23 дня назад

It should be 1/4 - 1 instead of 5/4 - 1 to have -3/4

@MooImABunny 22 дня назад

a different parameterization of the curve would be x(s) = ±√cos(s), y(s) = ±√sin(s) This means the squine and cosquine functions are just remappings of these sqrt trig functions. There is some mapping s = g(t) so that sq(t) = √sin(g(t)) csq(t) = √cos(g(t))

@rocky171986 23 дня назад

It should be gamma(5/2) in the denominator

@user-gs6lp9ko1c 23 дня назад

Yes, and that results in rho = 0.3090..., which makes the arguement that it should be between pi and 4 problematic! Edit: Some folks pointed out the error was made at 22:43, where the exponents should be 1/4 -1.

@krisbrandenberger544 20 дней назад

No, the denominator is correct. The numerator should be (Gamma(1/4))².

@Ben-wv7ht 18 дней назад

Will you do a video about lemniscatic trigonometric functions ? I love your way of introducing functions that are not usually taugth at school !

@alexeylvov6304 23 дня назад

A cross section of a sq-ube with x+y+z = 0 is a circle: {x^4+y*4+z^4 = 1, x + y + z = 0} => x^2 + y^2 + z^2 = const.

@schmitzbeats6102 23 дня назад

Ever since I learned about the squircle I wondered about the squine and cosquine.

@orionspur 23 дня назад

Pearls before squine?

@august3101 16 дней назад

Use the Jacobian elliptic function sn(u,m) with period K(Elliptic integral of first kind) with m= (say) 0.99 Use sn to replace sin and shifted sn(by K/4 ?) to replace cos . Then plot cn+i*sn with the understanding (1,0) and (0,1) are scaled differently (K>2*pi). Gives nice squarish squig.

@GameJam230 22 дня назад

Squigonometry is my new favorite math unit

@benheideveld4617 15 дней назад

Beautiful episode!!

@YuanAurion 23 дня назад

I feel like the squircle family should include going downwards from x^2n+y^2n=1, with some sort of deflated circle between n=1 and .5, a diamond at n=.5 with rho_0.5=4sqrt2, to gaunt diamonds below .5 and finishing with a + shaped thingy at n=0 with rho_0=2.

@crimsnblade8555 12 дней назад

Crazy, that's exactly what I was thinking about a few days ago, was thinking of doing a line integral on a squircle and find the limit as it approaches a square, so I needed to parameterize it

@TedHopp 20 дней назад

One way to think about the scaling of the sine and cosine parameterization is that the value of the parameter equals the path length along the curve. The same idea can be used to define a natural value for rho, rather than arguing by analogy to the "role" of π/2 in standard trigonometry.

@StuartSimon 17 дней назад

I actually found another video that defines the generating object as the square with vertices at (+-1, +-1) taken in all combinations. The interesting thing is that the ratio of squine to cosquine is actually the tangent for all angles; no "squangent" function need be defined.

@mza3764 20 дней назад

Thank you Michael for making this video, i sent the suggetion to your email more than a year ago, and I am happy that you finally found the time to look at it.

@mza3764 20 дней назад

Please make a video about how this is related to fractional derivatives

@Czeckie 22 дня назад

-3/4 is not equal to 5/4-1

@Alan-zf2tt 23 дня назад

Fantastic! I am glad Michael put it in ordinary speak rather than professional math researcher speak what with Γ and Β functions knocking about. Plus a bit of real analysis (complex analysis too?) when the integrand has a denominator tending to ∞ as absolute value of t tends to 1 And that is also a good place to start 🙂

@fupengmou3317 16 дней назад

the coefficients in the Mclaurin at 15:50 also show up in the expansion of a similarly defined functions, sm, which originates from the curve x^3+y^3=1. The coefficient x, x^9 and x^13 is exactly the same for this sq and that sm, but there is a difference in x^5 term. By the way, in the expansion of sm, the coefficient in the term x^17 is 189611/56576000, I don't know if this is also the same for sq here. If they match, maybe this is a good place to dig.

@vukapic7403 15 дней назад

0:13 can you say which math magazine you found this in? Sounds like a collection of quite fun publishing content to get a hold off!

@jewishjewom12ify 22 дня назад

If you take for N>=1, the curves x^2 + y^(2N) =1 and put cos_N, sin_N to be the x and y coordinates, you get another version of trigonometry, this time with sin_N’=cos_N. Sin_N can also be described as the unique solution to the ODE y’’ = -N y^(2N-1) with y(0)=0 and y’(0)=1. I’m currently leveraging properties of these functions to solve problems in high dimensional sub-Riemannian geometry for my PhD thesis.

@reinerwilhelms-tricarico344 23 дня назад

This was very instructive. It's quite surprising that these functions hadn't been defined long ago, like in the 18th century. It reminds a bit of the famous Jacobi elliptic functions sn(t) and cn(t) , which all come from the elliptic integrals. Not too long ago I wasn't really aware how hard the problem is to compute the exact circumference of an ellipse, when it's so easy to do for a circle. I wonder if you could sometime make a similar presentation about cn(t) and sn(t). It's important stuff, for instance for exactly computing the duration of oscillation of a simple pendulum, and not just for the small amplitude approximation - Because that's where you need elliptic integrals.

@lunaticluna9071 22 дня назад

they're actually related , see "Squigonometry: The Study of Imperfect Circles"

@user-wp1uw8fv6y 22 дня назад

For anyone who didn't get the condition "dx/dt = -y^3 and dy/dt =x^3", let me add more description here. The relation is come from 'extension of angle'. A parameter t is the generalized angle on an arbitrary curve, which is double area of a region made by the curve, x-axis, and a line from (0, 0) to (x, y). The differential form is "dt = x dy - y dx" [*], representing 'double area of a triangle' made by a line from (0, 0) to (x, y) and a vector (dx, dy). For trigonometric functions, they are defined by a unit circle, "x^2 + y ^2 = 1" so that "x dx + y dy = 0". Using [*] above, we get "dx/dt = -y" and "dy/dt = x", where t means 'ordinary angle'. Next, for hyperbolic functions, basic curve is now changed to "x^2 - y ^2 = 1" so that "x dx - y dy = 0". In the same way, we get "dx/dt = y" and "dy/dt = x", where t now implies 'hyperbolic angle'. Following that logic, squigonometric functions can be defined by "x^4 + y^4 = 1" and [*], and it derives the differential equation in the first line, based on the parameter t ― 'squircle-angle'.

@Chris-op7yt 22 дня назад

since you can have a circle with exact area of one or two etc., either there's something wrong with PI or how we use it, or it's not irrational after all.

@lurkmoar3926 21 день назад

By the way, |x|^(1/2) + |y|^(1/2) = 1 defines a non-convex closed curve around the origin whose length is 2(2 + 2^(1/2) ArcSinh(1) ) corresponding to 2 Pi, the circumference of a circle. The quantity corresponding to Pi is half that, or 2 + 2^(1/2) ArcSinh(1), which is about 3.246450480, surprisingly close to genuine Pi.

@davidemasi__ 17 дней назад

Definitely the most interesting video I've seen in a while

@m9l0m6nmelkior7 22 дня назад

I'd say it could be interesting to look at the "squirclexponential" : sqexp(z) = csqt(z/i)+isquirt(z/i) ? or would there be a problem with that…

@divyakumar8147 19 дней назад

it was fun thanks prof

@Pablo360able 22 дня назад

The denominator in the rho integral is raised to the 3/4, so how do you come to (1-x)^(5/4-1)? Should it not be (1-x)^(1/4-1)? Where does the extra 1-x come from?

@SeekingTheLoveThatGodMeans7648 19 дней назад

I asked Wolfram Alpha to draw graph x=(sqrt(abs(sin(t)))*sgn(sin(t))), y=(sqrt(abs(cos(t)))*sgn(cos(t))) for t=-pi to pi and got something that kind of looks like a squircle. But of course it isn't your same x^4 + y^4 = 1 squircle. (How close is it, though? I wasn't able to get WA to show them on the same graph.)

@Icenri 23 дня назад

Please now do elliptical squares. Science needs rectangular trigonometry, will explain later!

@samueldeandrade8535 23 дня назад

Hahahahahahaha.

@mohammedfarhaan9410 22 дня назад

hey so i had two ideas: 1. could we similarly define the properties of squiggles with replacing n with rational, irrational and complex numbers to x^n +y^n=1 2. could we somehow relate this to fourier series?

@FoodNerds 16 дней назад

Mr. Penn when you described the squircle I immediately thought of Apple 🍎 products and how Jobs designed product with this kind of shape. Q: Would an iPhone be an example of a 3 dimensional rectangil - or would it be cirtangle? In other words a rectangle with curved corners.

@user-et9ub3dc3j 22 дня назад

Reminds me of an article (Mathematical Recreations?) in Scientific American from back in the 1960s that examined general values of n. However, it did not use the term sqircle. Rather, it concerned itself with the curvature at the intersection with the coordinate axes. ~~~~Arthur Ogawa

@purplerpenguin 17 дней назад

Are there any actual use cases for squircles or squigonometry?

@LewisBavin 23 дня назад

Where's the Squangent

@sucroseboy4940 23 дня назад

That would just be squine/cosquine

@spacespark6922 22 дня назад

Or the sequant

@HisZotness 17 дней назад

@@spacespark6922Squeecant? 😂

@AleksyGrabovski 22 дня назад

In a limit this will give us rectangular trigonometry, which sounds like an oxymoron but here it is...

@SmithnWesson 5 дней назад

Wow. Nice. If g is a generator of a multiplicative group, then perhaps (g^k + g^(-k)) / 2 could be thought of as cos(k) on the group. But cosine on a multiplicative group is a weird thing to even think about. Perhaps, for example, think about the cyclic group mod 7. Or cyclic group mod 11. Then what is the cosine on those groups? And what is pi on that group? And what is sqrt(-1) on that group?

@trumanburbank6899 12 дней назад

Very cool video. How about a rectangularish ellipse? A rectipse?, lol.

@joelproko 22 дня назад

Rather than limiting ourselves to x^(2n)+y^(2n)=1, we can get a continuous deformation by using abs(x)^q+abs(y)^q=1, where q is any positive real number (2>q>1 has curves that bulge out less at 45° than a circle, q=1 is a square, 0

@VideoFusco 23 дня назад

I'm probably missing something, but I feel like this video is more complicated than it needs to be. If t always represents the angle in radians that is formed in O then, alongside the fundamental relation csq(t)^4+sqn(t)^4=1, it can be written that sqn(t)/csq(t) =sin(t)/cos(t). These are just two equations in the two unknowns csq and sqn which allow us to express the cosquine and squine functions through the cosine and sine functions.

@CatherineKimport 23 дня назад

I can't decide which part of my brain this video makes happier, the math nerd part or the word nerd part

@giuseppepapari7419 16 дней назад

25:00 I do not get this point. Did we prove that t plays the role of arc length?

@robmahurin5411 20 дней назад

I wanted `t` to be a sort of an arc length parameter, but it isn't. If you define a little distance `ds = √(dx^2 + dy^2)`, the speed with the parameter `ds/dt` is pretty close to unity on the straight segments, but drops by at least 15% in the corners. As the squircles get square-ier, the discrepancy in the curvy part gets more extreme, but the curvy part occupies less of the shape overall. The period in `t` approaches the perimeter of the shape as the shape approaches a square, but the period in `t` is only equal to the perimeter of the shape for the circle. I also wanted `t` to be some kind of an angle parameter, but it isn't that either. Only for the circle itself is the period in `t` independent of the size of the shape. For the squircles, making the shape bigger (by e.g. choosing (1.1,0) instead of (1,0) for the initial condition) makes the period in `t` shorter, so that the same "squangle" takes you more of the way around the bigger shape. Given these two misinterpretations, the relationship between the period in `t` for the high-order "unit squircles" and the perimeter of the shape feels like a coincidence or a conspiracy. Why is it "natural" for the period in `t` to approach the perimeter of the square?

@Chiborino 22 дня назад

I don't think I've ever hated a word more than I hate "cosquine," it just does not feel good to hear or say

@angeldude101 23 дня назад

Needing to add 1 just so you could subtract 1 in the integral is reminding me of why I think the pi function should be more common than the gamma functions.

@curtiswfranks 4 дня назад

All of these are intimately connected to the Lₚ-norms and, thus, common non-Euclideän geometries.

@wyboo2019 22 дня назад

havent finished the video, but my silly and probably-not-very elegant solution was to: 1. convert x^4+y^4=1 to polar coordinates 2. solve for r 3. convert to parameterization in the normal way (r cos(t), r sin(t))

@amirharoush5210 18 дней назад

the coefficients related (up to alternating sign and factorial) to OEIS A153301

@amirharoush5210 18 дней назад

although, I did found recurrence relation, but couldn't solve it (because of m+1 term) f(n,m)=(m+1)f(n-1,m+1)+(2n-m+2)f(n-1,m-3) f(0,1)=1

@tomholroyd7519 22 дня назад

Calling Matt Parker! We found somebody who can say "cosquine" with a straight face!!

@racheline_nya 17 дней назад

i'm so confused at 22:46... i think there's a mistake you start from integrating du/(1-u^4)^(3/4), which is equal to (1-u^4)^(-3/4)*du. so substituting x=u^4, we see that we're integrating (1-x)^(-3/4)*du, and then substituting du=(constant)*x^(-3/4)*dx, we see that we're integrating (1-x)^(-3/4)*x^(-3/4)*dx. -3/4 is 1/4-1, not 5/4-1, so you should get 1/2*B(1/4,1/4), which is 1/2*Γ(1/4)^2/Γ(1/2). in fact, you correctly got Γ(1/2) in the denominator, and even explained that it's a quarter plus a quarter, but in the numerator you didn't use the quarters and instead used 5/4. you can check out an approximation to these squigonometric things at www.desmos.com/calculator/ur3fpdxm2i, though i computed the coefficients of the taylor series in google sheets (i was too lazy to write python code and too curious about whether i could do it in google sheets) so the precision may be slightly limited by that.

@talberger4305 23 дня назад

24:01 gamma of 2.5 not 0.5

@mikeholt2112 23 дня назад

Both z and w are 1/4 so the denominator is correct but the numerator should read gamma(1/4)^2. Final calc of 3.708 is correct. Typo

@tomholroyd7519 22 дня назад

I'm not sure I'm ready for a hyperbolic squircle. Horrorcycle maybe?

@giuseppepapari7419 16 дней назад

12:48 should be 3csq(t)^2sq(t), no cube

@gwalla 20 дней назад

It seems like the 2n in the definition of the squircle is unnecessary. If you define it as xᵐ+yᵐ=1 then all that really changes is the true circle case is at m=2 instead of n=1, and you can just use 2 as your lower bound for m. I was wondering why you were giving it a positive lower bound at all, since you still get well-defined shapes at lower m until you hit m=0: as m gets smaller, you eventually get rotated square with a diagonal of length 2 at m=1 (n=1/2), then below that you get tetracuspids (the m=0 case is impossible because x⁰+y⁰=1 obviously has no solutions). But then I realized that this technique wouldn't work on those because they have sharp corners, so the squigonometric functions would not be differentiable everywhere. If you limit it to circles and above, you don't have to deal with discontinuities until you get to the limit as n approaches infinity, which is itself fortunately trivial to define without any need for calculus at all.

@Necrozene 23 дня назад

If that is a blue square, then I am a pink squircle. lol

@Kapomafioso 23 дня назад

Why can't we just use this parameterization? x(t) = sqrt(|cos t|) sign(cos t) y(t) = sqrt(|sin t|) sign(sin t) It satisfies the system of differential equations (you can check that x' x^3 + y' y^3 = 0) and it parameterizes the squircle x^4 + y^4 = 1

@neonlines1156 23 дня назад

How did you come up with that???

@Kapomafioso 23 дня назад

@@neonlines1156 if you know that cos and sin satisfy cos^2+sin^2=1, then what satisfies (...)^4+(...)^4=1? An obvious choice is sqrt(cos) and sqrt(sin), at least in the interval of the argument between 0,pi/2. The absolute values and signs are just to make it parameterize the full unit squircle.

@almostme80 23 дня назад

@@Kapomafioso very clever!

@Kapomafioso 23 дня назад

@@almostme80 thanks...I think what Michael was getting at (but didn't say explicitly) is, that while you can easily make up some parameterization, it might not be analytic (and mine, indeed, can't possibly work as analytic functions that would describe the whole squircle). You can see that immediately from the absolute values and signs, but also the series expansion for my y(t) features t^1/2 as the first term...

@frobeniusfg 23 дня назад

This parametrisation is not natural (in a sense of differential geometry) as t is not the arclength of the curve.

@krisbrandenberger544 20 дней назад

Hey, Michael! @ 20:03 Should be sq^4(z), not sq^4(t).

@scottmiller2591 23 дня назад

The once and future king, the superellipse, has entered the chat.

@AlvinFS27 21 день назад

Also the superhyperbolic

@PawelS_77 22 дня назад

I think we're missing the point here, because (at least this is how I understand it) the trig functions should be defined as functions of angle, not some arbitrary parameter. So the "value of pi" in this case should be the same as the normal value of pi, because angles work the same regardless of the shape of the object in question. Unless we're talking about the length of the object, which is a different thing (for a unit circle the length is the same as the full angle, but not for a squircle).

@kinoseidon 22 дня назад

For the sake of completeness, what are the first five terms of the Taylor series for the cosquine function?

@blue5659 17 дней назад

At 24:00 shouldn't it be 5/2, not 1/2?

@mehdimabed4125 21 день назад

It's funny this video comes out now, I'm currently focusing on superellipses, and in fact I struggle for some weeks now finding the coordinates of the intersection points of a striaght line and a superellipse, does anyone has an idea ? Thanks :)

@sicko_the_ew 20 дней назад

That relation looks to me like the Commutator? (As in Lie Algebras.) So this whole construct might be deeply interconnected with that topic? Just a thought. I'm at a point where I can do some pattern recognition, but still hit an out-of-depth error anywhere beyond that. Anyway, my ears are standing straight up as I watch this. :-)

@ricardoguzman5014 23 дня назад

As n-->∞ and n an even whole number, the graph of x^n +y^n=1 approaches a square.

@sukursukur3617 11 дней назад

Is there a mathematical expression of that: How is a square different from a circle?

@ezrakornfeld8436 11 дней назад

I thought this was geometry but you called that pink rectangle a square!

@pietrocoletti3995 22 дня назад

I actually prefer to use σ, where σ=2ρ

@wolliwolfsen291 22 дня назад

24:01 Why is z+w=1/2 when both are 5/4?

@timgchannel3328 22 дня назад

What is the difference between a squircle and a squarish superelipse?

@shruggzdastr8-facedclown 22 дня назад

Matt Parker covered the squircle on his Stand-Up Maths channel a year or two ago

@massimo888888 22 дня назад

What if n

@trueriver1950 23 дня назад

Exactly what is the motivation for this?

@sucroseboy4940 23 дня назад

I don’t think there is any. Perhaps it’s helpful in the same way as circular trig, if for some reason you’re working with squircles a lot

@lexyeevee 23 дня назад

it's interesting

@JanJannink 22 дня назад

A missed opportunity to define the squeacant and the cosqueacant !

@allmight801 23 дня назад

Let's say you have two vectors AB and AC where A B C are coordinates of points in space. If you have those two vectors for example AB=(1,2,3) and AC=(2,3,4) can you find the points A B and C?

@maxhagenauer24 23 дня назад

Not really because vectors can start anywhere. The for example means you go over 1 on the x-axis, over 2 on the y-axis, and over 3 on the z-axis. But you can start it at any point you want because it doesn’t matter where a vector is, it's just some magnitude and direction being applied but it doesn't matter where it's being applied.