6:42 It is a *quadratic mean* also (very well) known as *RMS* (Root Mean Square) by Electrical & Electronics Engineers. The quadratic mean is popular closer to the highest value (Max) or greater than the centered arithmetic mean. The geometric mean, lesser than the arithmetic mean, is near the lowest value (Min), and the harmonic mean is even closer. The error graph of those means drives us to conclude that the larger axis *_b_* has more influence on the perimeter of an ellipse than the minor axis *_a_* , mainly as eccentricity increases. We also can realize that such means are the main trunk line in the search for the perimeter of an ellipse: - The first Ramanujan approximation and the first Parker approximation are some kinds of playing around with weighted arithmetic, quadratic and geometric means... yes, they can all be weighted by multiplier coefficients; - The second Ramanujan approximation, excellent by the way, is a combination of weighted arithmetic mean and the use of *_h_* has some relation to a weighted quadratic mean; - The second Parker lazy approximation is a weighted arithmetic mean, relatively good compared to the quadratic one.
I think statisticians use it to calculate things like variance, too! Iirc cuberoot( (a^3 + b^3) / 2) helps get the skew (of a sample of size n=2). I wonder what the skew of a "radius" would be like
I see "root mean square" in a lot of audio plugins, as a way of detecting peaks in the audio (or as an alternative? I donno. It's usually a choice between "peak" and "RMS")
Very useful in machine learning - most models (mostly neural nets) are trained by taking the derivative of the "mean squared error" and following the gradient in the direction that lowers the error. Mean squared error is nice because it's differentiable - well, I guess the absolute value of the error is differentiable when the error is nonzero, but I think you'd be likely to overshoot using gradient descent on absolute value of the error.
@@josephbrandenburg4373 "RMS" in an electrical context is often a way of getting some sort of "average" because arithmetic mean in a sinusoid (AC signal) doesn't work and it ends up being useful in some areas. considering a lot of audio equipment is analog (and in odd waveforms) it would make sense to use RMS as sort of an average loudness
If you actually want the answer to "why don't we have a formula", it is simply that the perimeter of an ellipse is the line integral of its parametrisation: an ellipse is the set {(a cos(t), b sin(t)): 0
@@qborki Well, I am going to make an assumption here, because I do not know this with absolute certainty, but from what I do know, its math we are talking about. I am pretty sure there is an exact definition of the "usual" function. Its probably just the one you wont understand unless you have a certain level of math knowledge.
I am almost 60 years old. I love mathematics and I never, never imagen if somebody could make me laugh watching a math video. Well you did. Mathematics are so amazing, fun and funny too. Thank you so much for this 20 mins. Cheers!
6:45 thats called the 'root mean squared' value. Read the words in opposite order and you will know why. Very useful in kinetic theory of gases as well as calculations of alternating current.
@@danieljensen2626 they are much worse. edit: If i did my math correctly, then something traveling between Uranus and Earth will have that 75 ratio. But also i feel like at this point just calling it 4a is pretty accurate
There's a comet called Ikeya-Seki. It has an eccentricity of 0.999915. If I calculated correctly, that's 77 times more long than wide. But I think most comets are not that bad. For Hale-Bopp it's 11 something.
13:06 Well, because an object in free fall isn't really tracing out a parabola but instead a highly eccentric elliptic orbit around the earth's gravitational centre, you might in fact need such high eccentricity
I never thaugh about that. It's only a parabola if the force feild is an infinite plane, but on a sherical one, it's an extroardinaraly eccentricical elipse. My whole life is a lie.
@@jackys_handleFor most human-scale projectile motion, the difference is so insignificant that it doesn't make a difference. Local gravitational anomalies, like a mountain or heavy mineral deposit nearby, are going to be more significant, than accounting for the difference between an ellipse and a parabola as the shape of its trajectory.
I wonder if we flatten out an ellipse, since those simple calculations usually tends to treat earths surface as flat, will we actually find a parabola?
Its a parabola if its eccentricity is >= 1 (or is it just greater than? I forget) but an ellipse otherwise. IOW if it is a closed orbit its an ellipse, if it's orbit is open its a parabola.
I just realized that my math teachers frightened me in knowing formulas of perimeter, area and volume of nearly anything, omitting to tell that one was missing.
Did Ramanujan prefer "their" as a pronoun, or did you just disrespectfully choose the pronoun that was more comfortable for you? Oh, my... I shouldn't have assumed "you" to be the correct term either.... nevermind...
Considering the quality and amount of output, with very little formal training, and dying way too young, Ramanujan must be the greatest mathematician of all time.
I am NO mathematician, but programming, while accidentally seeing this. The information density of your beautyful feature is high AND entertaining, while i can learn in ease. I was browsing 20 unnecessary Sites to veryfy a typo in a book of Physics and found this comprehensive while deep and refreshing channel of yours. THANKS a LOT for occupying my screen, talking with purpose. I secretly like Maths in awe and i see you love it too. Being rewarded.
And ... then an even wilder Ramanujan appears. This formula C = π(a+b) ((12 + h)/8 - √((2 - h)/8)) fits much better than Ramanujan's (which is C = π(a+b) (3 - √(4 - h)), when expressed in terms of h). We're onto his game!
Another approach is to use the integral formula for the curve length. This integral can't be presented as a well-defined function, so you have to use a Simpson rule, for instance. With the Simpson rule, you can also estimate an error.
@@JosephEaorle but it would be exact, so the claim that there is no exact equation is false; there is no simple, exact equation; but there is an exact equation.
For further Reference on the subject one should consider the Extensively studied field of Elliptic Integrals [ en.wikipedia.org/wiki/Elliptic_integral ] and for Numerical Calculation of the Integrals one could use Adaptive Gaussian Quadrature schemes like Patterson methods [ en.wikipedia.org/wiki/Gaussian_quadrature ] which provides Much Better results than Simpson Rule, or for a simply Naive but much Better than Simpson calculation one could take Romberg Integration schemes.
Title: Why is there no equation for the perimeter of an ellipse? Trick answer: There is, but it involves an infinite series. Plot Twist Just like the equation for the perimeter of a circle.
This is where I ended up in my reasoning as well, which I guess was the point of the video. My intuition was telling me that pi was to circles what some other unknown constant would be to ellipses, and then my intuition also wondered if each ellipse might have its own unique "pi"-like constant.
@@geshtu1760 So, given a/b [which is consistent with his setting b=1, and by the way it makes more sense to use b/a -- and set a=1 -- because b can go to zero, unless you prefer that a can go to infinity] -- okay, given a/b, the perimeter equals 2*pilike(a/b)*avg(a,b)? Or perhaps 2*pilike(a/b)*a? Then the complications of figuring out the formula for pilike(a/b) are exactly the complications that he walks thru in the video. So, yes.
"And who's having an ellipse which is 75 times as wide as it is high?" As it turns out, there is the Hale-Bopp comet which, according to Wikipedia: Semi major axis = 186 AU eccentricity = 0.995086 Semi major / Semi minor = 203.5 Incidentally, Haley's Comet is pretty eccentric, but still below 75: Semi major axis = 17.834 AU eccentricity = 0.96714 Semi major / Semi minor = 30.4
I actually discovered *4(a + b) - ln(4a + 1)* at ~10AM on 08/04/2021 as my own Approximation! It only ever reaches 1.6813% (-When b = 1) error and eventually approaches -0.0297% error- 0.000% error.
If I had a nickel for every time Matt Parker called an ellipse an "eclipse", I'd have two nickels. Which isn't a lot, but it's weird that it happened twice.
Definitely more than twice - he did it twice just between 5:00 and 5:30. Using Keppler's approximation and the duration of this video (21 min), I'd say, he could've confused ellipses with eclipses as many as 84 times.
There's actually some deeper math hiding beneath the surface here. The elliptic integral (which is a non-elementary integral that calculates the circumference exactly) is related to elliptic functions and elliptic curves (which were used to prove Fermat's last theorem).
@@revcrussell Right, an integral who's solution can only be written as an infinite series... You can also write an integral equation for Pi, but that doesn't really get you anywhere.
When I was doing my GCSEs, I was doing Graphic Design, and I was building my design, a diorama using concentric elliptical curves of clear plastic with designs drawn on them to create an interesting parallax image. I ran into an issue though, I didn't know how long I needed to cut my plastic sheets. I knew how I would work it out if they were half-circles, but not if they were half-ellipse. So I asked my teacher how to work out the circumference of an ellipse, and tbh, he was stumped - so together we looked it up, and we discovered that it was a lot harder to do than we first thought it would be
Excellent, Excellent reporting! Wow! Ramanujen's brilliance was in finding something that freaking simple to do such a fantastic job. That kind of accuracy is good enough to land a probe on a comet. I enjoyed your improved lazy approximation, and I REALLY enjoyed the nice vocalist who sang Total Elipse of the Chart.
And with his mathematical insight, I've got something he didn't have, I've got a quantum computer. ................................................ so even though I only know juuust enough mathematics to be hazardous I can outsource alot of it to this machine.
Who knew there was no single equation. This is a fascinating examination of the perimeter of an ellipse. I am in awe of your wife's performance, well done. Thank you for your insights into this interesting puzzle.
For the physical interpretation of h: it’s a measure of flatness. It should lie within [0, 1] where 0 is a perfect circle (least “flat”) and 1 is a line (either horizontally or vertically, perfectly “flat”).
You just learned that? :D He's well up there with some of the other greats. There's even a "documentary" (more of a dramatization but regardless) of his life called "The man who knew infinity." Wouldn't say its a classic but its not terrible either.
Great video. I didn't know there was no exact formula. When I was at engineering school, a student in my class needed to calculate the perimeter of an ellipse for a software he was coding. I thought about it and came with a (wrong) solution, considering an ellipse is the intersection of a plane and a cylinder (of radius b. The angle between the plane and the cylinder depending on a). Then, "unwrapping" this cylinder (as it was made of paper) to put it flat and measuring the previous intersection as it was (actually, it is not) the hypotenuses of a pair of right-angle triangles, this leads to P=2*sqrt[(pi^2-4)*b^2+4*a^2]. I have just checked this formula against an online calculator that uses Ramanujan's second approximation and found a divergence around 3%.
I'm never not astounded at the genius of Ramanjan wow he was able to do with his just his head what a laptop was only able to do 2 times more accurate... we're talkin margin of errors in the hundredths of a percent as well jeez this guy was a beast edit: just saw his 2nd equation LMAO wtf how was that guy human
It's the difference between solving analytically and solving numerically. Not to say that Ramanujan wasn't brilliant but the two methods just have completely different outcomes, as shown by the error comparisons here.
@@sachinnandakumar1008 by numerically he means computationally making a close approximation through iterative processes, whereas analytically he means solve for a somewhat exact solution by 'traditional' mathematical methods, like algebra and calculus (not that numerical methods don't use those, of course, but that's slightly different).
"I know just enough math to be dangerous" Lol I love this. These videos are so much fun to watch (even if my friends think I'm crazy for watching maths videos in my free time)
your friends are crazy for not watching maths videos in their spare time. or, maybe they've just never tried before, cause as 3b1b discussed many times before, often people just don't know how much they love maths
For that, first we need to delve into the nature of "π". What is π? It is the ratio of circumference to the diameter in a "Circle"(only). Now, Conics are defined by their "eccentricity"(ε) values, which too is a ratio. Conics are, the Circle (ε = 0), Ellipse (0 < ε < 1), Parabola (ε = 1) & lastly Hyperbola (1 < ε < ∞). In these only the circle & Parabola have fixed ε, each (0 or 1). It implies there is only one circle (that can be scaled up to look big) and one Parabola, while there can be an infinite number of Ellipses or (infinite number of) Hyperbolae each of a different eccentricity (ε). Just as for the definition of π (ratio of circumference to the diameter) that is valid for circle, there can be no such a thing for Ellipse. The ratio of circumference to semi-major or minor axis is a continuous variable. So there can be no π, for an Ellipse. Then why do we involve π, in the definition of circumference of an Ellipse (as some would want us to believe)? We don't need π.
I'm actually incredibly impressed by your lazy approximation, it'd seem like such a simple solution multiplying the two axes by fractional constants would have been found earlier. Great work!
I mean it's just a compromise. Sacrifice some accuracy at first for more accuracy later. But I guess in general mathematicians are more interested in symmetry.
How would "4a - (2pi-4)b" do? I think the derivation on that one should be fairly obvious. One thing it would have been nice to see Matt Parker mention would be how the approximations do as eccentricities get large.
I agree, Parker showed himself from his best mathematical side there. I'm still not sure I'll remember this one the day I need it, but it seems the best candidate for those who want to memorize something.
7:00 That is usually called "root-mean-square" (not usually hyphenated but I find it easier to read and more grammatically sensible with hyphens) and comes up in a lot of places. For example, the "voltage" number for the mains electricity in homes and buildings is the root-mean-square of the instantaneous voltages of waveform across one cycle (or equivalently across n cycles or, if you pretend the waveform is infinite, across the whole waveform). It is also the conceptual origin of least-squares regression. You want to minimize the root-mean-square of the errors. Since square-root is a monotonically increasing function, this is the same as minimizing the mean-square of the errors. In general, it is a computationally friendly and integration-friendly way to indicate something similar to average magnitude.
@@YounesLayachi XD. there isn't a single universe where mathematicians, those too of caliber of Matt, wouldn't know of rms. that is something even a petty high schooler knows. Matt was obviously joking.
I read the title by mistake as perimeter of an eclipse. And I was like “that’s a silly mistake to make” But then noticed 5:00 and I’m like okay, great, I’m not the only one.
And Ramanujan did it without the help of computers or calculators. Even without all these means he just smashes Matt's approximiation formula's. He truly was on another level entirely!
Just did some math with a friend of mine lol. It’s 11pm, but we did some good work in my opinion. There are 2 equations, one simple, one more complicated. One where n = 1.5, and one where n = 1 / log(2, pi/2), or approximately 1.53493, where P = 4b((a/b)^n + 1)^(1/n). Not sure if I did the error accuracy thing right, but if I did, we should have under 0.4% error throughout with the complicated equation, and it only gets better as the ellipse becomes longer. Would love if someone wanted to recheck and let me know if I’m right lol
Interesting. I just saw this interesting video yesterday. After that, decided to try a family of solutions: 2*pi*((a^n)+b^n)/2)^(1/n). Started with n=1 and n=2. Noticed that one underestimates, the other overestimates the right answer. So, tried n=1.5. Noticed that it reduced the error to under 1% over the entire eccentricity range. Then I focused on the value that gives the exact answer as the eccentricity goes to infinite. Found exactly the same n you found. That is, n is the reciprocal of the log base 2 of (pi/2). The error is zero when b=a and when b goes to infinity. And it stays under 0.4% over the entire range.
Yup, smiled also. Einstein should've stopped searching after Newton told us what's what. But there was always a a clever-guts Albert in every schoolroom.
@@Mrbobinge Einstein's formula? What about Epstein's formula? Very successful for a long time. A lot of travelling on a plane. Also, a lot of curved surfaces.
Hell if you're clever enough and have too much time on your hands you could build one mega equation that cancels out the other formulas depending on what number range you're using, mixing in functions to give it properties rather than for any mathematical purpose just to say you have an all in one approximation lol
Also, my orbits in Kerbal Space Program...I'm usually too lazy to use the rocket equation properly, and really, *really* like solid fuel boosters for the first stage of my rockets.
Wow, I knew I was an outcast in school when I was the only one who enjoyed mathematics, but this channel brings it to a whole new level. He managed to not only make me understand ellipses where as I had no clue what it was before this video, but he also showed me how it relates to a circle and how pi is a glorified beauty when it comes to the perimeter. And he did it all in a way that kept me attentive and entertained. This man may have not only earned a subscription today, but also may have re-sparked my love for learning more math.
They're just as neat! We're just flawed that our "basic arithmetic operations" / "number system" struggle to deal with then. For want of a metaphor: we're trying to fit a square peg into a round hole. Neither the hole or the peg in isolation can be considered wrong. It's the pairing that is the issue
@@jameshogge Funny, how the metaphor actually goes deeper than I first thought. When you equate a line segment to an arithmetic operation, the square has a simple exact representation, whereas the circle can only be approximated.
For me i often define ellipses in pretty much the same way, but a=1 and b= cos(ß). Since in my application, an ellipse can often be understood as a circle with radius a, seen from an incidence angle ß. For example a rake angle. Really simple. But indeed it's weird that there is no easy approach to circumference!
I was gobsmacked by Ramanujan's second equation. Never in my academic career have I seen it until know, wow. Would have helped so much in undergrad hahah Absolutely love the videos Matt Parker! brilliant, insightful, and helpful.
14:58 Not gonna lie, you had me in the first half. (Adding the "btw, there's no neat equation for the perimeter of a circle either" near the end was sneaky!)
I have tried numerous ways of modeling complex curves for flat spring designs in SolidWorks CAD and failed miserably at defining them with formulae. I could use ellipses to draw segments, but trying to connect them into one poly-line with parametric segment lengths made the model geometry "blow up." In one particularly frustrating design I ended up just freehanding my desired curve and setting that as the definition for the spring shape. I was able to use the brute-force freehand curve to design bending mandrels which made just what I needed. Sometimes real-life is too complicated for computers. It bugged me that I couldn't tell my production people exactly how much flat spring material they needed to build the spring.
@@jasonspudtomsett9089 When modelling/simulating it is usually the norm to be as simple and ideal as possible. But well, all that matters is if it works lol
There is a well defined equation for the perimeter! Parameterize an ellipse and apply some vector calculus. It isn't workable by hand, but it is literally the perimeter. It is also the circumstance of a circle because of how squareroots of squares of trig functions. Take the line integral and you will get your answer.
Matt - The name of that funny square-root-of-average-of-squares thing: It's commonly called the "root mean square," or just the "RMS." It could also be called the "Pythagorean mean." Basically, it's one of a class of generalized means, defined by choosing some monotonic [over some restricted interval, if necessary] function, f(x), and then "transforming" your numbers with it, averaging them, then inverse-transforming the result: M[f](x₁ , ... , xn) = f⁻¹(∑ᵢf(xᵢ) /n) So if f(x) = x; or even ax + b, where a≠0, it's just the ordinary average (arithmetic mean). Interval of applicability is the whole real line. (Or even the whole complex plane!) If f(x) = ln x, it's the geometric mean. Interval of applicability is the positive reals. If f(x) = 1/x, it's the harmonic mean. Interval of applicability is all reals ≠ 0. (Again, could be all complex numbers ≠ 0.) And for your Pythagorean mean, or RMS, f(x) = x². Interval of applicability is the non-negative reals. The same nomenclature can be used for generalized (transformed) sums. Fred
This is awesome. I knew about the generalized mean using a transformation (though I didn't think about monotonicity), but didn't know RMS could also be called Pythagorean mean. That's so cool!
@@Magnasium038 I can't recall for certain, but I think I might have coined the term, "Pythagorean mean," which would be why you hadn't heard it before. The alternative is that I might have picked it up long ago from some other, perhaps obscure, source, which would also explain your not having seen it. Fred
I have no idea why but this has really hooked me in. I am not a mathnetician. I spent all of sunday and several hours this morning drawing elipses and circles on desmos and playing with different equations.
yes and they are derivated, and integrated from [ds^2 + dx^2 + dy^2], sensasionalism??, maybe an introducction why look for approximations?, on why the elliptical functions leads to unelemental integrals??
Idk if this works but when finding the perimeter of planetary orbits, you can use Kepler's equations (with true anomaly) to produce a speed-time function, and then integrate it from the bounds 0 to T, getting total distance traveled in one orbit. This is what I did for my high-school math project and it worked quite well for the planets.
I got interested in this when making bridges with geometrical shapes in a 3D program. Making a fence out of many overlapping shapes, (half-ellipses, but that's irellevant,) I wanted to know how to space them evenly on a bridge surface which was also half an ellipse. Unable to find a good lazy method, I was thankful that particular program approximated the ellipse with a relatively small number of straight segments no matter how large the ellipse was. Thus, I could easily space the fence-bits evenly on each straight section and do the turns by eye. If I do this again on a program which makes smoother ellipses, (which is most of them,) I'll certainly want to try the Parker lazy method in this video, especially because the ratio of such a bridge-ellipse can easily be 10 or more. (Y'know, I'm slightly sad because this post will spoil the number of comments. It was 5,555 before I posted this.)
It was funny the first time. Less so the second time. Excruciating by the third. In fact, i'd estimate it crossed the excruciating line at about 2.718281828...
When 90% of your maths knowledge comes from KSP and you understand (formally) how orbits work. This plus your previous video were a treat. I must admit I really do love derivations, being a maybe inpatient person, who never cared for the maths itself, just how to use it (probably my ultimate downfall) I think its wonderful now to see how the things I know work and why
hypehuman The constant depends more directly on the eccentricity than it does on b/a. To be precise, the constant equal 2π for e = 0 and 4 for e = 1. The dependence on e is given by 4·E(e), where E is the complete elliptic integral of the second kind, in this case as a function of e.
7:11 Missed a trick there: should have been “a total eclipse of the chart” both in reference to getting in the way of the graphics, and to fluffing “ellipse” twice prior to that. :)
Ashutosh Patel Ramanujan had to use his brain to crunch numbers, we can use machines for all the heavy stuff. That was the „mistake“ and the joke, to be born in a time without computers
I found these by integrating a bezier curve: a * [ sqrt(4 + (4 * b/a)² ) + 2 ] --Max 5.682% error a * [ sqrt(2pi + (4 * b/a)² ) + (3+pi)/4 ] -- Max 3.237% error a * [ sqrt(4.905 + (4 * b/a)² ) + pi/2 ] -- Max 3.200% error Edit: Found an even better one For a = 1 and 0
It is also used a lot when trying to measure the output of a system that outputs a sine wave, a good example would be the electrical grid where the AC voltage figure is given as Vrms. Similarly, the most reliable measurements for output of audio systems are usually given as the RMS of the Sound Power Level. In both cases, this is a better approximation of the thing that actually matters than the peak value in terms of audio RMS is closer to perceived loudness as human perception is a continuous function itself, similarly, Vrms of an AC supply more closely aligns with the voltage of a direct current supply a lamp which has a given brightness on 100V DC would require 100Vrms from an AC supply to match that.
@@DarkMage2k Ah ok mostly mentioned it as this seems to be something most don't realize, seeing someone plot an AC waveform from +240V to -240V rather than the more accurate +340V to -340V is quite common the actual max voltages are significantly above the nominal voltage.
6:43 did some thinking on this one, it actually makes a ton of sense!! The key thing is to split the square root so that the numerator and denominator are rooted separately. The numerator is the Pythagorean theorem applied to the major and minor axes, so the value you get is the hypotenuse for the right triangle formed by the axes. Then, that gets divided by square root of 2… where’ve we seen that before? Sin(45) and cos(45)! Dividing by root 2 basically gives us the x and y components of the hypotenuse, ultimately averaging the axes in a very unique way. I’m impressed by the cleverness of this approximation, if I could choose which one was the exact formula for perimeter it’d be this one!
My lazy approximation would be 4a :) The more eccentric the ellipse, the more accurate it gets.
on average it's better than any, but it's practically useless
@@MaoDevHow aliens would describe me in one sentence after studying the human species.
4a is the lower limit for the circumference perimeter of an ellipse.
C or 1, the circle circumference is the upper limit.
@@theglitch312 hahaha!
6:42 It is a *quadratic mean* also (very well) known as *RMS* (Root Mean Square) by Electrical & Electronics Engineers.
The quadratic mean is popular closer to the highest value (Max) or greater than the centered arithmetic mean. The geometric mean, lesser than the arithmetic mean, is near the lowest value (Min), and the harmonic mean is even closer.
The error graph of those means drives us to conclude that the larger axis *_b_* has more influence on the perimeter of an ellipse than the minor axis *_a_* , mainly as eccentricity increases.
We also can realize that such means are the main trunk line in the search for the perimeter of an ellipse:
- The first Ramanujan approximation and the first Parker approximation are some kinds of playing around with weighted arithmetic, quadratic and geometric means... yes, they can all be weighted by multiplier coefficients;
- The second Ramanujan approximation, excellent by the way, is a combination of weighted arithmetic mean and the use of *_h_* has some relation to a weighted quadratic mean;
- The second Parker lazy approximation is a weighted arithmetic mean, relatively good compared to the quadratic one.
15:36 3 Blue 1 Brown's pi is sort of like the Clippy of mathematics:
"It looks like you're trying to find the perimeter of an ellipse!"
now I want someone to make a 3B1B digital assistant
If Clippy were anywhere near that useful, I'd have never turned him off!
Proud to be your 666th upvote :)
@@hoebare Beast Mode! So to speak...
@@hoebare devil
"I know just enough mathematics to be dangerous" - I feel this should be a tshirt.
I'd buy one
This week I worked out that 25 grams of antimatter has the potential energy of a Megaton of TNT. So I feel like I fit into that category.
It's a way of life, that's for certain.
...with Einstein's silhouette and Matt Parker showing his square to Einstein...
I need that
Matt, engineers frequently use the "root mean square" to describe expressions like SQRT((a^2 + b^2)/2).
I think statisticians use it to calculate things like variance, too!
Iirc cuberoot( (a^3 + b^3) / 2) helps get the skew (of a sample of size n=2). I wonder what the skew of a "radius" would be like
I see "root mean square" in a lot of audio plugins, as a way of detecting peaks in the audio (or as an alternative? I donno. It's usually a choice between "peak" and "RMS")
Very useful in machine learning - most models (mostly neural nets) are trained by taking the derivative of the "mean squared error" and following the gradient in the direction that lowers the error. Mean squared error is nice because it's differentiable - well, I guess the absolute value of the error is differentiable when the error is nonzero, but I think you'd be likely to overshoot using gradient descent on absolute value of the error.
kinda surprised he didn't know that considering he studied mechanical engineering in college.
@@josephbrandenburg4373 "RMS" in an electrical context is often a way of getting some sort of "average" because arithmetic mean in a sinusoid (AC signal) doesn't work and it ends up being useful in some areas. considering a lot of audio equipment is analog (and in odd waveforms) it would make sense to use RMS as sort of an average loudness
That moment of realization for 2*pi*r where he says "wait a minute!" is so well timed with the realization for the viewer.
BibiBosh rounded to 100? Approximately 100th? Was it 100. ? ( #BadRounding)
Classic parker
Amazing
It was telegraphed by the title of the video. We could all see it coming.
And the 3 blue 1 brown character popup
If you actually want the answer to "why don't we have a formula", it is simply that the perimeter of an ellipse is the line integral of its parametrisation: an ellipse is the set {(a cos(t), b sin(t)): 0
The real question here is: How do you define which functions are "usual". That's subjective.
@@qborki no it isn't. It's pretty much well defined.
@@qborki Well, I am going to make an assumption here, because I do not know this with absolute certainty, but from what I do know, its math we are talking about. I am pretty sure there is an exact definition of the "usual" function. Its probably just the one you wont understand unless you have a certain level of math knowledge.
The linear integral, which gives you the length the ellipse is unsolvable... This does not mean that there isn't a formula for the perimeter...
@@tomasstana5423 I think he ment elementary functions? Idk, as far as I'm aware of, there are no "usual functions"
There aren't enough comments about how wonderful that 3Blue1Brown π cameo was.
Yes! :D
I think he may be using 3b1bs open source animation software
In which second is that?
@@a.georgopoulou (: at 15:36
@@YambamYambam2 but there is no brown i don't get itt
I am almost 60 years old. I love mathematics and I never, never imagen if somebody could make me laugh watching a math video. Well you did. Mathematics are so amazing, fun and funny too. Thank you so much for this 20 mins. Cheers!
Same here, born 1951
you sound like my grandpa lol!
Loved the little 3Blue1Brown reference.
For those who missed it, see 15:38
Yeah, what a cutie-pi :3
Lol
@@6872elpado what u mean
Saw the reference, came to the comments section looking for this comment. Now back to the rest of the video :)
6:45 thats called the 'root mean squared' value. Read the words in opposite order and you will know why. Very useful in kinetic theory of gases as well as calculations of alternating current.
Or 'quadratic mean.' It's interesting to note that we always have QM>=AM>=GM (quadratic, arithmetic, geometric).
@@alephnull4044 >=HM harmonic mean: 2/(1/a + 1/b) >= min(a,b)
:P
I was surprised that be didn't know that
@@fares8005 Yeah. So HM would be even worse of an approximation than GM.
@@anuragjuyal7614 I guess since RMS is more common in physics and engineering. And not so much in pure maths.
"Who's having an ellipse that is 75 times as long as it is wide?"
An Oort Cloud comet has entered the chat.
@@danieljensen2626 they are much worse.
edit: If i did my math correctly, then something traveling between Uranus and Earth will have that 75 ratio.
But also i feel like at this point just calling it 4a is pretty accurate
And then left and won't be back for a few centuries.
Physicists would approximate this as a line.
An ellipse has totally entered the chart.
There's a comet called Ikeya-Seki. It has an eccentricity of 0.999915. If I calculated correctly, that's 77 times more long than wide. But I think most comets are not that bad. For Hale-Bopp it's 11 something.
13:06 Well, because an object in free fall isn't really tracing out a parabola but instead a highly eccentric elliptic orbit around the earth's gravitational centre, you might in fact need such high eccentricity
I never thaugh about that. It's only a parabola if the force feild is an infinite plane, but on a sherical one, it's an extroardinaraly eccentricical elipse. My whole life is a lie.
@@jackys_handleFor most human-scale projectile motion, the difference is so insignificant that it doesn't make a difference. Local gravitational anomalies, like a mountain or heavy mineral deposit nearby, are going to be more significant, than accounting for the difference between an ellipse and a parabola as the shape of its trajectory.
I wonder if we flatten out an ellipse, since those simple calculations usually tends to treat earths surface as flat, will we actually find a parabola?
@@jackys_handle it's a difference between an eccentricity of 1.0 (parabola) and .9999 (very long ellipse)
Its a parabola if its eccentricity is >= 1 (or is it just greater than? I forget) but an ellipse otherwise. IOW if it is a closed orbit its an ellipse, if it's orbit is open its a parabola.
I just realized that my math teachers frightened me in knowing formulas of perimeter, area and volume of nearly anything, omitting to tell that one was missing.
They shielded you from a dark truth you were not yet ready to accept, that would have shattered your nascent mind
" if ramanujan made 1 major mistake with their mathematical career, it was having it in the past" -matt parker, everybody
Unappreciated joke
I think the mistake I made with my career as engineer on a starship is not having my career hundreds of years in the future.
This is why I love Matt.
Did Ramanujan prefer "their" as a pronoun, or did you just disrespectfully choose the pronoun that was more comfortable for you? Oh, my... I shouldn't have assumed "you" to be the correct term either.... nevermind...
"The future is now old man"
The way he connects the whole thing together by stating reminding us that pi is an infinite series at the end is phenomenal
Yeah, I loved that bit. :)
Makes me wonder if we could get a nicer equation is we took away pi and put a and b into the pi series....
Pi is an infinite series if you live in world of integers. Integers are infinite series if you live in a world of Pis.
@@notabene7381 tf
Considering the quality and amount of output, with very little formal training, and dying way too young, Ramanujan must be the greatest mathematician of all time.
I am NO mathematician, but programming, while accidentally seeing this.
The information density of your beautyful feature is high AND entertaining, while i can learn in ease.
I was browsing 20 unnecessary Sites to veryfy a typo in a book of Physics and found this comprehensive while deep and refreshing channel of yours.
THANKS a LOT for occupying my screen, talking with purpose. I secretly like Maths in awe and i see you love it too. Being rewarded.
8:35 "His mistake was doing math in the past."
Honest mistake, we'll try to do better next time.
One of the few mathematicians in the western canon that you can say that about. I feel that your joke is underappreciated.
Unfortunately, Ramanujan's mistake was deadly.
@@jansamohyl7983 being born leads to death... so we all made the mistake
@@jaredjones6570 I mean, I haven’t made that mistake yet, and I’d be kind of freaked out if you have.
Actually there were no gendered pronouns used in the video. It's hard to miss. Everything is "they".
15:30 Matt: *slaps Pi”
“This bad boy can fit an infinite series of fractions in it’
Good meme
This is the best comment.
Robert Slackware Why? Can‘t you e.g. do sth like 110100100010000...?
@Robert Slackware
π is in the open interval from 0 to 3.5, so it is not infinite.
@Robert Slackware LOL! rock on, man...
Whenever Mathematicians are scratching their heads on a problem, a wild Ramanujan appears
Wild?
@@thebiggestcauldron he is wild (commentor)
@jocaguz18 Yes.
And ... then an even wilder Ramanujan appears. This formula C = π(a+b) ((12 + h)/8 - √((2 - h)/8)) fits much better than Ramanujan's (which is C = π(a+b) (3 - √(4 - h)), when expressed in terms of h). We're onto his game!
@@dgarrard100 Gotta catch both of 'em!
Another approach is to use the integral formula for the curve length. This integral can't be presented as a well-defined function, so you have to use a Simpson rule, for instance.
With the Simpson rule, you can also estimate an error.
That was my solution, the antiderivative ends up being pretty complicated.
@@JosephEaorle but it would be exact, so the claim that there is no exact equation is false; there is no simple, exact equation; but there is an exact equation.
Yeahhhhhh maybe, but with the Simpson rule you'd get dragged down by having to write it over and over on a chalkboard.
For further Reference on the subject one should consider the Extensively studied field of Elliptic Integrals [ en.wikipedia.org/wiki/Elliptic_integral ] and for Numerical Calculation of the Integrals one could use Adaptive Gaussian Quadrature schemes like Patterson methods [ en.wikipedia.org/wiki/Gaussian_quadrature ] which provides Much Better results than Simpson Rule, or for a simply Naive but much Better than Simpson calculation one could take Romberg Integration schemes.
Title: Why is there no equation for the perimeter of an ellipse?
Trick answer: There is, but it involves an infinite series.
Plot Twist Just like the equation for the perimeter of a circle.
This is where I ended up in my reasoning as well, which I guess was the point of the video. My intuition was telling me that pi was to circles what some other unknown constant would be to ellipses, and then my intuition also wondered if each ellipse might have its own unique "pi"-like constant.
Best plot twist on UA-cam's history
@@guillermogarciamanjarrez8934 this
@@geshtu1760 So, given a/b [which is consistent with his setting b=1, and by the way it makes more sense to use b/a -- and set a=1 -- because b can go to zero, unless you prefer that a can go to infinity] -- okay, given a/b, the perimeter equals 2*pilike(a/b)*avg(a,b)? Or perhaps 2*pilike(a/b)*a? Then the complications of figuring out the formula for pilike(a/b) are exactly the complications that he walks thru in the video. So, yes.
Ok. But is there an equation that "hides" the infinite series for an ellipse? If not, then I have a suggestion for a sequel.
"Ignore what happens a lot further that way. It's not relevant."
*disapproves in Big O Notation*
Theta(n!) is so fast it even beats Theta(2n)!, if are range is 0 to 3 hehe
@@macicoinc9363 What is theta? Are you using it to mean Big O?
Oversimplified, Big O means "grows not as fast as", little o means "grows faster than" and theta means "grows roughly the same as"
@@t0mstone581 Thanks!
T0mstone wooo computational mathematics is so fun...
"And who's having an ellipse which is 75 times as wide as it is high?"
As it turns out, there is the Hale-Bopp comet which, according to Wikipedia:
Semi major axis = 186 AU
eccentricity = 0.995086
Semi major / Semi minor = 203.5
Incidentally, Haley's Comet is pretty eccentric, but still below 75:
Semi major axis = 17.834 AU
eccentricity = 0.96714
Semi major / Semi minor = 30.4
Glad you said this. When he made that comment, I shouted "COMETS" at the screen.
TY so much for this as I was wondering about comets eccentricity's.
How did you calculate the Semi major / Semi minor ?
@@laurgao -- Using the eccentricity.
Where did you get your major/minor from?
I was under the understanding that a/b=(1-e^2)^-0.5 , which gives me 10.0 and 3.93.
The interesting fact I noticed about the "bouncing" approximation is that for certain values of ratio they give a 0% error
A broken clock is correct twice a day
Also, the sine function perfectly approximates the value of 0 infinitely many times, but that doesn’t make it a good approximation of 0
I would venture to guess that those certain values would be irrational?
@@BeauDiddley87 and transcendental, going on a limb here
@@fi4re Sadly that just works for analog clocks lol. Digital ones have a more nihilistic approach.
I actually discovered *4(a + b) - ln(4a + 1)* at ~10AM on 08/04/2021 as my own Approximation! It only ever reaches 1.6813% (-When b = 1) error and eventually approaches -0.0297% error- 0.000% error.
I found a more general Approximation of *4(a + b) - ln(4a/b + 1)b.* It always maxes at only 1.6813% error.
@@Inspirator_AG112 That's very clean! Well done.
I think if you put 'h' inside the ln term, may be possible to find a better one.
No pi in the equation? That makes it even more awesome!
It actually approches perfection. (Correction 7 months later.)
8:33 "I know just enough mathematics to be dangerous" this surely enters my top five best statements ever to be stated
I expected at least a mention of an integration approach
Yeah, I was waiting for it too...a bit disappointed that he didn't mention it
The origin of the elliptic integral.
I thought I was the only one disappointed after watching it. No mention whatsoever of the elliptic integral.
I was expecting this too, before the infinite series (like, where does that come from?)
Me too
If I had a nickel for every time Matt Parker called an ellipse an "eclipse", I'd have two nickels. Which isn't a lot, but it's weird that it happened twice.
they rehearsed that song too often before recording ;)
I thought he did this more than twice, but I was not counting.
Anyone count lipses? Lips'? Lips's? Yeah, yeah. Anyone count lips's?
Definitely more than twice - he did it twice just between 5:00 and 5:30. Using Keppler's approximation and the duration of this video (21 min), I'd say, he could've confused ellipses with eclipses as many as 84 times.
I blame Bonnie Tyler.
There's actually some deeper math hiding beneath the surface here. The elliptic integral (which is a non-elementary integral that calculates the circumference exactly) is related to elliptic functions and elliptic curves (which were used to prove Fermat's last theorem).
I was going to comment Matt was wrong. You don't need an infinite series, just integrals.
@@revcrussell Right, an integral who's solution can only be written as an infinite series... You can also write an integral equation for Pi, but that doesn't really get you anywhere.
@@danieljensen2626 Even further: what are integrals in general, but succinctly notated limits of infinite series.
@@anteroinen4239 Good point, although some of the ones we like to use converge to algebraic or even rational numbers.
Wrong
The term you're looking for at 6:46 is "root mean square" or rms, and is used a lot in AC electricity voltage computations.
Huh, I always called it the quadratic mean
Also molecular velocity
Yaap...it’s also used to equate kinetic energy of gas.
It’s a incredible way of getting rid of negative value when finding a average.
I was looking to see if someone made this very common. Thank you.
Encountered it in molecular kinetics, average speed of particles in a gas
When I was doing my GCSEs, I was doing Graphic Design, and I was building my design, a diorama using concentric elliptical curves of clear plastic with designs drawn on them to create an interesting parallax image. I ran into an issue though, I didn't know how long I needed to cut my plastic sheets. I knew how I would work it out if they were half-circles, but not if they were half-ellipse. So I asked my teacher how to work out the circumference of an ellipse, and tbh, he was stumped - so together we looked it up, and we discovered that it was a lot harder to do than we first thought it would be
Excellent, Excellent reporting! Wow! Ramanujen's brilliance was in finding something that freaking simple to do such a fantastic job. That kind of accuracy is good enough to land a probe on a comet. I enjoyed your improved lazy approximation, and I REALLY enjoyed the nice vocalist who sang Total Elipse of the Chart.
Doesn't he say "eclipse" numerous times when referring to an "ellipse"? Maybe I'm just going crazy :-)
He does, I caught that too:))
Well everyone, atleast most of us do it.
5:00 one example I found
I wonder if it was on purpose 🧐
It’s weird I saw this comment and I found a few
"So what are the traits of an ellipse?"
"Oh well there's the major and minor axes, two focal points, an eccentricity and h."
"What's h?"
*leaves*
@1:50
*Insert h meme here
Incredibly incorrect and flippant answer here, but I think it's some inverse of the hypotenuse between the ends of a and b.
Considering the weight of the problem, probably Plancks constant
if put a=kb then h = (k-1)² / (k+1)² for (k>=1)
People in 100 years: if Matt Parker made one major mistake, it was having his mathematical career in the past.
And with his mathematical insight, I've got something he didn't have, I've got a quantum computer.
................................................
so even though I only know juuust enough mathematics to be hazardous I can outsource alot of it to this machine.
That's a Parker Square of a career timing
ONE major mistake?
@@motazfawzi2504 I love the idea of this, and hope things like this persist like memes online for centuries LOL
you nailed it.
Who knew there was no single equation. This is a fascinating examination of the perimeter of an ellipse. I am in awe of your wife's performance, well done. Thank you for your insights into this interesting puzzle.
Take a shot every time Matt calls an ellipse an eclipse :p
makes me wanna do a parker square...
Only twice though, so you won't get many shots.
@@SumNutOnU2b Well, it's a Parker drinking game. It works somewhat okay, but not great.
An eclipse is a parker ellipse
@@wolframstahl1263 brilliant!
For the physical interpretation of h: it’s a measure of flatness. It should lie within [0, 1] where 0 is a perfect circle (least “flat”) and 1 is a line (either horizontally or vertically, perfectly “flat”).
oww, it's that h from the standard equation of 2 degree in 2 variables??
anyways, thanks for it
What we learned today: Ramanujan was hot stuff
You just learned that? :D He's well up there with some of the other greats. There's even a "documentary" (more of a dramatization but regardless) of his life called "The man who knew infinity." Wouldn't say its a classic but its not terrible either.
Speak for yourself there! So brilliant and original that the Brits had to teach him to speak math like they do just so they could understand him
@@enginerdy You mean speak maths? :D
You made my day bro
I swear he must have had a mathematical IQ of like 200 or more!
Great video. I didn't know there was no exact formula. When I was at engineering school, a student in my class needed to calculate the perimeter of an ellipse for a software he was coding. I thought about it and came with a (wrong) solution, considering an ellipse is the intersection of a plane and a cylinder (of radius b. The angle between the plane and the cylinder depending on a). Then, "unwrapping" this cylinder (as it was made of paper) to put it flat and measuring the previous intersection as it was (actually, it is not) the hypotenuses of a pair of right-angle triangles, this leads to P=2*sqrt[(pi^2-4)*b^2+4*a^2]. I have just checked this formula against an online calculator that uses Ramanujan's second approximation and found a divergence around 3%.
I'm never not astounded at the genius of Ramanjan wow he was able to do with his just his head what a laptop was only able to do 2 times more accurate... we're talkin margin of errors in the hundredths of a percent as well jeez this guy was a beast
edit: just saw his 2nd equation LMAO wtf how was that guy human
It's the difference between solving analytically and solving numerically. Not to say that Ramanujan wasn't brilliant but the two methods just have completely different outcomes, as shown by the error comparisons here.
he was a human
you are not
@@sachinnandakumar1008 by numerically he means computationally making a close approximation through iterative processes, whereas analytically he means solve for a somewhat exact solution by 'traditional' mathematical methods, like algebra and calculus (not that numerical methods don't use those, of course, but that's slightly different).
I mean, he was known for pioneering achievements in sequence and series. Pretty much expected.
Creativity unbound by the labor and limitations of programming.
"I know just enough math to be dangerous" Lol
I love this. These videos are so much fun to watch (even if my friends think I'm crazy for watching maths videos in my free time)
your friends are crazy for not watching maths videos in their spare time. or, maybe they've just never tried before, cause as 3b1b discussed many times before, often people just don't know how much they love maths
I spat my meds out upon hearing that..... note to self: dont watch Parker when taking your meds
I love Matt's identity as 'StandupMaths' -- literally making Maths enjoyable to the wider public by making it into comedy. Pure genius.
@@Shrooblord doesnt it come from him doing that math/science comedy show with Steve Mould?
"He knows maths. Enough to be dangerous. Matt Parker in Parker Eclipse."
Parker Duck! Let's get dangerous!
*maths 🙈
5:00
@@witerabid I'm using a mix of British and American English, whatever I feel like :D but I'll change it just for you.
@@YuureiInu 😅 I was just preempting the Brits. I usually say "math" too. 😉
For that, first we need to delve into the nature of "π". What is π? It is the ratio of circumference to the diameter in a "Circle"(only). Now, Conics are defined by their "eccentricity"(ε) values, which too is a ratio. Conics are, the Circle (ε = 0), Ellipse (0 < ε < 1), Parabola (ε = 1) & lastly Hyperbola (1 < ε < ∞). In these only the circle & Parabola have fixed ε, each (0 or 1). It implies there is only one circle (that can be scaled up to look big) and one Parabola, while there can be an infinite number of Ellipses or (infinite number of) Hyperbolae each of a different eccentricity (ε).
Just as for the definition of π (ratio of circumference to the diameter) that is valid for circle, there can be no such a thing for Ellipse. The ratio of circumference to semi-major or minor axis is a continuous variable. So there can be no π, for an Ellipse. Then why do we involve π, in the definition of circumference of an Ellipse (as some would want us to believe)? We don't need π.
Thank you for this explanation.
because you touch yourself at night
based!
So why is π involved in the area formula of an ellipse then?
0:26 Matt - “It’s a more generalised version”
and like all good mathematicians
“And my goodness, is it lovely!”
3:31 Also, like all good mathematicians, he completely disregarded the actual usefulness of the focal points "light, mirrors, bla bla bla"
@@luisramos123 I'd have it no other way!
I'm actually incredibly impressed by your lazy approximation, it'd seem like such a simple solution multiplying the two axes by fractional constants would have been found earlier. Great work!
I mean it's just a compromise. Sacrifice some accuracy at first for more accuracy later. But I guess in general mathematicians are more interested in symmetry.
And it is easy to remember as well, once you write 3, 4, 5, 6 in an appropriate circle thing and « fill in the gaps » with a, b, and fraction bars!
The fact that it gives the circumference of a circle as 1.95pi radians is bad starting point, but it *is* very #ParkerMaths
How would "4a - (2pi-4)b" do? I think the derivation on that one should be fairly obvious. One thing it would have been nice to see Matt Parker mention would be how the approximations do as eccentricities get large.
I agree, Parker showed himself from his best mathematical side there. I'm still not sure I'll remember this one the day I need it, but it seems the best candidate for those who want to memorize something.
That's a Parker Approximation right there. #ParkerSquare
We don't need to keep making these jokes any more, because I've generalised it:
"This is a Parker N"
Parker approximations... that's two layers of haphazardness!
This is a Parker Joke
@@robinw77 that was a parker reply
I paused the video just to look up for this XD
7:00
That is usually called "root-mean-square" (not usually hyphenated but I find it easier to read and more grammatically sensible with hyphens) and comes up in a lot of places. For example, the "voltage" number for the mains electricity in homes and buildings is the root-mean-square of the instantaneous voltages of waveform across one cycle (or equivalently across n cycles or, if you pretend the waveform is infinite, across the whole waveform).
It is also the conceptual origin of least-squares regression. You want to minimize the root-mean-square of the errors. Since square-root is a monotonically increasing function, this is the same as minimizing the mean-square of the errors.
In general, it is a computationally friendly and integration-friendly way to indicate something similar to average magnitude.
Many engineering programs even have an RMS function, even if in most of them it is trivial to define one yourself.
Thanks, I hate it
when i first saw him being oblivious of the rms, i assumed he is joking. there is no way he doesnt know that an rms is well known average
@@Mayank-mf7xr it has nothing to do with maths, so, I'm not sure what you're expecting
@@YounesLayachi XD. there isn't a single universe where mathematicians, those too of caliber of Matt, wouldn't know of rms. that is something even a petty high schooler knows. Matt was obviously joking.
I read the title by mistake as perimeter of an eclipse. And I was like “that’s a silly mistake to make”
But then noticed 5:00 and I’m like okay, great, I’m not the only one.
Wdym?
@@Bozzigmupp He says "eclipse" instead of "ellipse" at those times.
Looking at Matt's monstrosity of an equation next to Ramanujan's elegant simplicity makes me feel like there should be a sensor bar over it!
😂😂😂
Lmfao same here 😂😂
What does the Wii have to do with this?
And Ramanujan did it without the help of computers or calculators. Even without all these means he just smashes Matt's approximiation formula's. He truly was on another level entirely!
Just did some math with a friend of mine lol. It’s 11pm, but we did some good work in my opinion. There are 2 equations, one simple, one more complicated. One where n = 1.5, and one where n = 1 / log(2, pi/2), or approximately 1.53493, where P = 4b((a/b)^n + 1)^(1/n). Not sure if I did the error accuracy thing right, but if I did, we should have under 0.4% error throughout with the complicated equation, and it only gets better as the ellipse becomes longer. Would love if someone wanted to recheck and let me know if I’m right lol
Interesting. I just saw this interesting video yesterday. After that, decided to try a family of solutions: 2*pi*((a^n)+b^n)/2)^(1/n). Started with n=1 and n=2. Noticed that one underestimates, the other overestimates the right answer. So, tried n=1.5. Noticed that it reduced the error to under 1% over the entire eccentricity range.
Then I focused on the value that gives the exact answer as the eccentricity goes to infinite. Found exactly the same n you found. That is, n is the reciprocal of the log base 2 of (pi/2). The error is zero when b=a and when b goes to infinity. And it stays under 0.4% over the entire range.
“But what about orbits?” That’s when you know you married a right partner.
Sorta helps his wife is a physicist involved in satellite science. :P
@spim randsley Dammit, if only Earth had a moon as marker - save all that chalky maths stuff.
What about the perimeter of a testee?
@@pluto8404 Test these.
@spim randsley Bread + moon cheese squared. That's gotta be the solution.
I laughed so hard when Matt swept the infinite expansion under the π.
pi = 3, why bother with those stupid fractions
lmao me too
for anyone else who sees this, it happens at 15:16
Best Matt Parker moment ever!
However, PI is incomplete without its LE.
Best part of this: "I stopped searching for a function when I found that Kepler had developed an approximation."
Yup, smiled also. Einstein should've stopped searching after Newton told us what's what. But there was always a a clever-guts Albert in every schoolroom.
Nothing serious, I hope?
@@Mrbobinge Einstein's formula? What about Epstein's formula? Very successful for a long time. A lot of travelling on a plane. Also, a lot of curved surfaces.
I think you can make a pretty accurate one with conditionals. 1-2 range use formula A, 2-4 use formula B, 4-8 use formula C, 8-infinite use formula D.
Hell if you're clever enough and have too much time on your hands you could build one mega equation that cancels out the other formulas depending on what number range you're using, mixing in functions to give it properties rather than for any mathematical purpose just to say you have an all in one approximation lol
Parker: "And who's having an ellipse which is seventy-five times as wide as it is high?"
Halley: "Hold my slide rule."
Halley's comet isn't that eccentric though....
I thought this too, but Halley's comet has an eccentricity of 0.967, which means that its orbit is only 3.93 times wider than it is high.
Also, my orbits in Kerbal Space Program...I'm usually too lazy to use the rocket equation properly, and really, *really* like solid fuel boosters for the first stage of my rockets.
C= Tau•R
Wonder if some of the complexity drops if we adopt Tau instead of Pi?
@@Trevor21230 same
2:33 “super extreme” is an understatement. It’s literally an ellipse where the ratio of a to b is infinite
That can be achieved by setting b to zero. Essentially it's a straight line of infinite length.
@@DavidSmith-vr1nb Or 2 straight lines if b is not zero
My bad, I was wrong. It's actually a parabola.
@@Owen_loves_Butters lol
@@DavidSmith-vr1nb Not of infinite lenght. If b=0, then the line is of lenght 2a.
The perimeter is 4a, btw.
@@juanausensi499 the point was that the ratio is infinite, not the length
Edit: my bad, misread the comment you replied to ....
"When are you going to get a job!"
...
"In the future... I'm not gonna make the same mistake as Ramanujan..."
One of the approximations is the RMS value of a & b. The root of mean of squares one.
Wow, I knew I was an outcast in school when I was the only one who enjoyed mathematics, but this channel brings it to a whole new level. He managed to not only make me understand ellipses where as I had no clue what it was before this video, but he also showed me how it relates to a circle and how pi is a glorified beauty when it comes to the perimeter.
And he did it all in a way that kept me attentive and entertained.
This man may have not only earned a subscription today, but also may have re-sparked my love for learning more math.
Same here. I too was a loner in school. I was too different from others. I am glad those days are behind me.
That man is a national treasure.
Are you telling me nonelementary antiderivatives aren't neat equations?
Integrals for the win!!!
Yeah...neat
They're just as neat! We're just flawed that our "basic arithmetic operations" / "number system" struggle to deal with then.
For want of a metaphor: we're trying to fit a square peg into a round hole. Neither the hole or the peg in isolation can be considered wrong. It's the pairing that is the issue
@@jameshogge Funny, how the metaphor actually goes deeper than I first thought. When you equate a line segment to an arithmetic operation, the square has a simple exact representation, whereas the circle can only be approximated.
@@niklaskoskinen123 Wow, that's... kind of deep. At least, deeper than the peg will go into the hole.
14:00 I've been suffering for 14 minutes wondering how you were getting a % error if no equation existed, but ahh the good ole infinite series
went comment surfing at 10:52. to try to ease my suffering.
For me i often define ellipses in pretty much the same way, but a=1 and b= cos(ß). Since in my application, an ellipse can often be understood as a circle with radius a, seen from an incidence angle ß. For example a rake angle. Really simple. But indeed it's weird that there is no easy approach to circumference!
Your vision is usefull for area of an ellipse but didn't help for the circumference.
why is eszett here
I was gobsmacked by Ramanujan's second equation. Never in my academic career have I seen it until know, wow. Would have helped so much in undergrad hahah
Absolutely love the videos Matt Parker! brilliant, insightful, and helpful.
14:58 Not gonna lie, you had me in the first half. (Adding the "btw, there's no neat equation for the perimeter of a circle either" near the end was sneaky!)
My approximation: "4a".
Work great if a is huge compared to b. The error goes to 0 then
I wonder at wich point it becomes better than the best approximation we have
I have an approximation that works perfectly if a=0
Genius. Now try to sell it to NASA.
Oooh, you jusye made me realize how ridiculous it is to measure the approximation relatively to excentricity
(( 2rPi-4r)a/r)+4r where a is always smaller than r, wrong ?
The quality in this video is amazing! Thank you.
He said "Ratio", "Major", and "Minor" in the same sentence and it wasn't about music
Music ⊆ Maths ?
@@TheYahmez yeah, i always laught inside me when someone says "i love music but hate math" :D
Everything is just applied maths
@@ali709aliali And math is applied philosophy
@@gileee No, it's the other way around.
I like these simple geometric videos. They remind me why I fell in love with maths all those years ago.
Engineers be like "Ehh, it's close enough. Who cares....."
I can confirm this.
The correct observation; “It’s over engineered so it’ll work if we just let it ride.”
I have tried numerous ways of modeling complex curves for flat spring designs in SolidWorks CAD and failed miserably at defining them with formulae. I could use ellipses to draw segments, but trying to connect them into one poly-line with parametric segment lengths made the model geometry "blow up." In one particularly frustrating design I ended up just freehanding my desired curve and setting that as the definition for the spring shape. I was able to use the brute-force freehand curve to design bending mandrels which made just what I needed. Sometimes real-life is too complicated for computers. It bugged me that I couldn't tell my production people exactly how much flat spring material they needed to build the spring.
@@jasonspudtomsett9089 When modelling/simulating it is usually the norm to be as simple and ideal as possible. But well, all that matters is if it works lol
Wouldn't it be so much easier if Pi was 3? How accurate do we need this result? An order of magnitude? Great, Pi = 3.
There is a well defined equation for the perimeter! Parameterize an ellipse and apply some vector calculus. It isn't workable by hand, but it is literally the perimeter. It is also the circumstance of a circle because of how squareroots of squares of trig functions. Take the line integral and you will get your answer.
I was expecting to find an integral that would give the path length and was surprised when none were mentioned.
Yeah but to my knowledge there’s no analytical solution
en.wikipedia.org/wiki/Ellipse#Metric_properties
The ellipse circumference in general is not an elementary function.
@@badbeardbill9956 Correct. And pi is irrational number, so does it mean there's no number of length of circle?
@@leonidfro8302pi is a number a transcendental number. Means it is not countable.
I hope there's a second part to this where you talk about elliptic integrals. Please, I want to know more!
6:54 Root mean square? I mean that would be the fourth most common mean after arithmetic, geometric and harmonic mean.
Yeah, the quadratic mean. I remember studying the hierarchy of which mean is greater when the values used differ from one another.
What about trimmed mean?
@@peterflom6878 That's more for messy real world data, whereas the others actually turn up in many exact formulas.
ooh whats harmonic mean that sounds fun! my first guess would be 1/(1/a + 1/b)
@@joeyhardin5903 almost. I guess you meant 2/(1/a + 1/b).
Matt - The name of that funny square-root-of-average-of-squares thing: It's commonly called the "root mean square," or just the "RMS."
It could also be called the "Pythagorean mean."
Basically, it's one of a class of generalized means, defined by choosing some monotonic [over some restricted interval, if necessary] function, f(x), and then "transforming" your numbers with it, averaging them, then inverse-transforming the result:
M[f](x₁ , ... , xn) = f⁻¹(∑ᵢf(xᵢ) /n)
So if f(x) = x; or even ax + b, where a≠0, it's just the ordinary average (arithmetic mean). Interval of applicability is the whole real line. (Or even the whole complex plane!)
If f(x) = ln x, it's the geometric mean. Interval of applicability is the positive reals.
If f(x) = 1/x, it's the harmonic mean. Interval of applicability is all reals ≠ 0. (Again, could be all complex numbers ≠ 0.)
And for your Pythagorean mean, or RMS, f(x) = x². Interval of applicability is the non-negative reals.
The same nomenclature can be used for generalized (transformed) sums.
Fred
This is awesome. I knew about the generalized mean using a transformation (though I didn't think about monotonicity), but didn't know RMS could also be called Pythagorean mean. That's so cool!
Awsome information! But, instead of "root mean square", it was told to me as just "quadratic mean"
@@guigazalu I'd agree that that term works.
Fred
@@Magnasium038 I can't recall for certain, but I think I might have coined the term, "Pythagorean mean," which would be why you hadn't heard it before.
The alternative is that I might have picked it up long ago from some other, perhaps obscure, source, which would also explain your not having seen it.
Fred
I've known this as the Quadratic Mean. Through the QM-AM-GM-HM inequality.
I have no idea why but this has really hooked me in. I am not a mathnetician. I spent all of sunday and several hours this morning drawing elipses and circles on desmos and playing with different equations.
See also the elliptic integrals, invented for just this purpose.
I believe you because of your beard.
yes and they are derivated, and integrated from [ds^2 + dx^2 + dy^2], sensasionalism??, maybe an introducction why look for approximations?, on why the elliptical functions leads to unelemental integrals??
the whole deductive scheme of the problem....
discovered..
@@jonnydonny9270 invented
"I only know juuuuust enough mathematics to be dangerous" - Matt Parker
Hilariously well-timed to my scrolling.
"Who has an ellipse 75 times long than it is high?"
Laughs in comet inbound from the Oort cloud.
12:58
so you know how to steal from the comments section
Idk if this works but when finding the perimeter of planetary orbits, you can use Kepler's equations (with true anomaly) to produce a speed-time function, and then integrate it from the bounds 0 to T, getting total distance traveled in one orbit. This is what I did for my high-school math project and it worked quite well for the planets.
I got interested in this when making bridges with geometrical shapes in a 3D program. Making a fence out of many overlapping shapes, (half-ellipses, but that's irellevant,) I wanted to know how to space them evenly on a bridge surface which was also half an ellipse. Unable to find a good lazy method, I was thankful that particular program approximated the ellipse with a relatively small number of straight segments no matter how large the ellipse was. Thus, I could easily space the fence-bits evenly on each straight section and do the turns by eye. If I do this again on a program which makes smoother ellipses, (which is most of them,) I'll certainly want to try the Parker lazy method in this video, especially because the ratio of such a bridge-ellipse can easily be 10 or more.
(Y'know, I'm slightly sad because this post will spoil the number of comments. It was 5,555 before I posted this.)
This “a total ellipse of the chart” gag might be the weirdest one Matt has ever done
But, weirdly lovely.
It feels jarringly out-of-place, yet also perfectly at home in this video.
As soon as that popped up I hit the like button. I couldn't help myself.
It was funny the first time. Less so the second time. Excruciating by the third. In fact, i'd estimate it crossed the excruciating line at about 2.718281828...
When 90% of your maths knowledge comes from KSP and you understand (formally) how orbits work. This plus your previous video were a treat. I must admit I really do love derivations, being a maybe inpatient person, who never cared for the maths itself, just how to use it (probably my ultimate downfall) I think its wonderful now to see how the things I know work and why
Thanks, Matt for being so MATTematically precise in your videos.
Now I want a graph showing the "pi-ish value" for every ellipse.
That's a great idea! And I made it! See here: www.geogebra.org/m/mdfbg46y
@@hypehuman looks like it wants to converge on something, which is very interesting...
Edit: oh, i'm an idiot - it converges on 4. Of course.
@@alexjago51 yeah I had that same train of thought :) It's 4 at b/a=0, and I expect it will approach 4 again as b/a approaches infinity.
hypehuman The constant depends more directly on the eccentricity than it does on b/a. To be precise, the constant equal 2π for e = 0 and 4 for e = 1. The dependence on e is given by 4·E(e), where E is the complete elliptic integral of the second kind, in this case as a function of e.
@@hypehuman thanks
I found an approximation for when "h" is near 1 (really flat ellipse). Try "s=sqrt(2*(a^2+b^2))*(acosh(1/h)+6*sqrt(2))/3"
6:42 "I don't know, what is it?" It's good old root-mean-squared (RMS) isn't it?
My thought exactly :)
Was searching for a comment on this. Does math not have Root-mean-squared anywhere ??
Electrical engineers know this damn well
@@atharvachoudhary6974
Yeah. In statistics we sometimes use it.
It shows up a lot in physics but I've never seen it in pure maths.
Finally someone addresses the question that has kept me up at night for years.
7:11 Missed a trick there: should have been “a total eclipse of the chart” both in reference to getting in the way of the graphics, and to fluffing “ellipse” twice prior to that. :)
I think that's the joke. he mis-said it, got in the way of the graph, and had the music bits, all together as a running joke.
With all those lines and numbers everywhere, I'm surprised that Matt hadn't lost the plot.
Lol
He didn't because he 'totally grips all the parts'. 😁
At 5:00: "The perimeter of an eclipse."
Only Bonnie Tyler knows that function.
And 5:28, eclipse again lol
Bonnie Tyler also shared that knowledge w/ Nicki French.
Its as he becomes a little hypoxic. you can see he looks a little weaker and slurrs, pauses a little bit too.
@@adriano-moraes _5: *25_
So interesting. A small point: I would have liked a quick reminder of the formula for 'h'.
15:36 Grant? Is that you? Show yourself......
"parker eclipse approximation"
i was not expecting that.
that was brilliant.
silly ramanujan he shouldn't have had his career in the past what a silly mistake
Rookie mistake
Can anyone explain this line to me in layman term , I am teenager tho--
Imagine being born in the 19th century instead of the 21st lmao IDIOT
@@curiash if ramanujan had their career nowadays, they would've had access to modern computers, but in the past they didn't.
Ashutosh Patel Ramanujan had to use his brain to crunch numbers, we can use machines for all the heavy stuff. That was the „mistake“ and the joke, to be born in a time without computers
wtf, how can a math video be so captivating that I randomly and willingly put 20 minutes to watch it fully
I’m probably the worst person at math but I can’t help to enjoy every single minute of your videos. If only you were my math teacher
I found these by integrating a bezier curve:
a * [ sqrt(4 + (4 * b/a)² ) + 2 ] --Max 5.682% error
a * [ sqrt(2pi + (4 * b/a)² ) + (3+pi)/4 ] -- Max 3.237% error
a * [ sqrt(4.905 + (4 * b/a)² ) + pi/2 ] -- Max 3.200% error
Edit: Found an even better one
For a = 1 and 0
Looks like python code 😉
6:43
It's called root mean square or rms in short. Used in thermodynamics and kinematics a lot. Especially thermodynamics and kinetic theory
It is also used a lot when trying to measure the output of a system that outputs a sine wave, a good example would be the electrical grid where the AC voltage figure is given as Vrms. Similarly, the most reliable measurements for output of audio systems are usually given as the RMS of the Sound Power Level. In both cases, this is a better approximation of the thing that actually matters than the peak value in terms of audio RMS is closer to perceived loudness as human perception is a continuous function itself, similarly, Vrms of an AC supply more closely aligns with the voltage of a direct current supply a lamp which has a given brightness on 100V DC would require 100Vrms from an AC supply to match that.
@@seraphina985 ah I hate that part of physics so I didn't include it lol
@@DarkMage2k Ah ok mostly mentioned it as this seems to be something most don't realize, seeing someone plot an AC waveform from +240V to -240V rather than the more accurate +340V to -340V is quite common the actual max voltages are significantly above the nominal voltage.
I love the sneaky 3B1B pi that pops in to say hi
6:43 did some thinking on this one, it actually makes a ton of sense!! The key thing is to split the square root so that the numerator and denominator are rooted separately. The numerator is the Pythagorean theorem applied to the major and minor axes, so the value you get is the hypotenuse for the right triangle formed by the axes. Then, that gets divided by square root of 2… where’ve we seen that before? Sin(45) and cos(45)! Dividing by root 2 basically gives us the x and y components of the hypotenuse, ultimately averaging the axes in a very unique way. I’m impressed by the cleverness of this approximation, if I could choose which one was the exact formula for perimeter it’d be this one!
It's called the root mean squared