trigonometry like you've never seen it
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- เผยแพร่เมื่อ 23 พ.ค. 2024
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I drew a unit piggy and subsequently defined the coswine function
Hahahahahaha.
*click* Noice!
that actually made me giggle, haha
good one!🍹
Swine/coswine is the hamgent (a reach, I know. I'm sorry😅)
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!
This blows my mind
That's a crazy fact, wow!
What if gcd=1?
@@pelledanasten1615 Then the least common denominator is the product of the denominators, if i understand your question correctly.
@@samwalko I think he got lcm and gcd mixed up
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. 👍
Squircle sounds like an awesome Pokémon name ngl
CHARMANDER AGREES!!!
No arent you just thinking of squirtle???
@@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
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
Ah yes! That's it!
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
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).
@@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.
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)
Interestingly, it is known that Gamma(1/4) and pi are algebraically independent, so it follows that rho is transcendental.
Where does this come from?
@@hybmnzz2658From “Modular Functions and Transcendence Questions” by Yuri Nesterenko, 1996.
@@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).
sup Queen's Uni
🍁
@@PillarArt ??
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.
theres a whole book on the topic: "Squigonometry: The Study of Imperfect Circles"
@@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
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 😂
"pi" formula for arbitrary n is rho = Г(1/2n)^2 / (n Г(1/n)) and it does approaches 4 as n->inf 😊
Do you know any proof of the fact it converges to 4?
@@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 :-)
22:49 I think it should be 1/4-1
you can parametrize the segment from 0 to pi/2 rad of any squircle xⁿ+yⁿ=1 as (cosª(t),sinª(t)) where a=2/n. Any other value of the angle results in at least one complex value, but you can cover the four quadrands with combinations of absolute values of the parameter functions.
So x⁴+y⁴=1 first quadrant can be parametrized as (|cos(t)|½,|sin(t)|½), the second as (-|cos(t)|½,|sin(t)|½), the third as (-|cos(t)|½,-|sin(t)|½) and the fourth as (|cos(t)|½,-|sin(t)|½).
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.
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.
Time to introduce poqueuepine
What
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
Time is a flat squircle
Shouldn't the integral around 24:00 be of x^(1/4 - 1) (1-x)^(1/4-1) not 5/4?
Yes
“The squine “
I fucking lost it 😂
I *should* be revising fluid dynamics, but 'squine and cosquine' was the nerdiest reason I've ever laughed myself incapacitated.
Showing that all squircles with n > 1 never pass through points having rational coordinates would prove the Fermat’s theorem for even powers, right?
You forgot to link the book!
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.
It should be 1/4 - 1 instead of 5/4 - 1 to have -3/4
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.
is there a way of treating that smoothening such that it is like taking a limit?
Beautiful episode!!
It should be gamma(5/2) in the denominator
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.
No, the denominator is correct. The numerator should be (Gamma(1/4))².
All solutions to the linear system are given by a matrix exponential. That should cover all solutions of
|x|^a + |y|^a = 1, 1
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))
-3/4 is not equal to 5/4-1
Will you do a video about lemniscatic trigonometric functions ? I love your way of introducing functions that are not usually taugth at school !
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.
Pearls before squine?
Squigonometry is my new favorite math unit
I wonder what the squine and cosquine, as well as the other four functions (tan, csc, sec, cot) but for the squircle, look like?
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
Ever since I learned about the squircle I wondered about the squine and cosquine.
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.
Please make a video about how this is related to fractional derivatives
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!
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.
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.
Definitely the most interesting video I've seen in a while
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.
it was fun thanks prof
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.
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'.
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 🙂
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.
Are there any actual use cases for squircles or squigonometry?
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…
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.
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.
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.
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?
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.
they're actually related , see "Squigonometry: The Study of Imperfect Circles"
Please now do elliptical squares. Science needs rectangular trigonometry, will explain later!
Hahahahahahaha.
Where's the Squangent
That would just be squine/cosquine
Or the sequant
@@spacespark6922Squeecant? 😂
In a limit this will give us rectangular trigonometry, which sounds like an oxymoron but here it is...
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
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?
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
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.)
I'm not sure I'm ready for a hyperbolic squircle. Horrorcycle maybe?
I can't decide which part of my brain this video makes happier, the math nerd part or the word nerd part
Very cool video. How about a rectangularish ellipse? A rectipse?, lol.
25:00 I do not get this point. Did we prove that t plays the role of arc length?
Calling Matt Parker! We found somebody who can say "cosquine" with a straight face!!
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.
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?
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.
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.
the coefficients related (up to alternating sign and factorial) to OEIS A153301
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
If that is a blue square, then I am a pink squircle. lol
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?
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 :)
For the sake of completeness, what are the first five terms of the Taylor series for the cosquine function?
What is the difference between a squircle and a squarish superelipse?
All of these are intimately connected to the Lₚ-norms and, thus, common non-Euclideän geometries.
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.
I thought this was geometry but you called that pink rectangle a square!
Right at the beginning in the top left of the board he missed the u in squigonometry and it’s bothering me 😭
At 24:00 shouldn't it be 5/2, not 1/2?
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
How did you come up with that???
@@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.
@@Kapomafioso very clever!
@@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...
This parametrisation is not natural (in a sense of differential geometry) as t is not the arclength of the curve.
Is there a mathematical expression of that: How is a square different from a circle?
12:48 should be 3csq(t)^2sq(t), no cube
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
A missed opportunity to define the squeacant and the cosqueacant !
Matt Parker covered the squircle on his Stand-Up Maths channel a year or two ago
As n-->∞ and n an even whole number, the graph of x^n +y^n=1 approaches a square.
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. :-)
The once and future king, the superellipse, has entered the chat.
Also the superhyperbolic
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))
24:01 Why is z+w=1/2 when both are 5/4?
Is there no way to generalise this, at least in the first quadrant, to real powers rather than just even integer powers ?
What if n
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.
24:01 gamma of 2.5 not 0.5
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
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).
I actually prefer to use σ, where σ=2ρ
COSQUINE and SQUINE I'm dead 😂
Exactly what is the motivation for this?
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
it's interesting
Hey, Michael! @ 20:03 Should be sq^4(z), not sq^4(t).
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?
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.
No because you have only two equations
I think you can't. You have a 9 variables vs 6 equations sistem. Infinite values satisfie that.
Next video on reclipse would.......
Is the selection of x' and y' going to be unique?