Why is there no equation for the perimeter of an ellipse‽

1,2 milj. näkymät165

    Applications for paid internships at Jane Street are open! www.janestreet.com/join-jane-street/internships/
    A Podcast of Unnecessary Detail is out now: festivalofthespokennerd.com/podcast/ Come for the Unnecessary Detail, stay for the A Podcast Of.
    These are my approximation equations:
    perimeter ≈ π[53a/3 + 717b/35 - √(269a^2 + 667ab + 371b^2)]
    perimeter ≈ π(6a/5 + 3b/4)
    If you can do better, submit it to Matt Parker's Maths Puzzles.
    This was my pervious video featuring ellipsoids:
    You can buy the ellipse from this video on eBay. I've written on my two new equations and signed it. All money goes to charity (the fantastic Water Aid).
    Bonus content and a deleted scene are available on my Patreon.
    Huge thanks to all who sent in a recording of them singing "A total ellipse of the chart." Sorry I could not include everyone. These are the people in the video:
    Helen Arney
    Steve Hardwick
    Victoria Saigle
    Andrew McLaren
    Sören Kowalick
    It all started because of a request I put out on twitter.

    - So far the only times (I've noticed that) I say "eclipse" instead of "ellipse" are 5:01 and 05:26 which was just after talking about my wife who is a solar physicist. So I think we split the blame 50/50.
    - It seems everyone but me recognised the Root Mean Square. I think I only associate that with current for some reason! Thanks all.
    - Let me know if you spot any other mistakes!

    Thanks to my Patreons who meant I could spend about a week trying to find approximations for the length of ellipses. "Are you still working on that?" Lucie would - rightfully - ask over the weekend. "I'm going the extra mile for my patreon people!" I would reply. Here is a random subset of those fine folks:
    Benjamin Richter
    Louie Ruck
    Matthew Holland
    Morgan Butt
    Rathe Hollingum
    Jeremy Buchanan
    Sjoerd Wennekes
    Barry Pitcairn
    James Dexter
    Adrian Cowan
    As always: thanks to Jane Street who support my channel. They're amazing.
    Filming and editing by Matt Parker
    Additional camera work by Lucie Green
    VFX by Industrial Matt and Parker
    Music by Howard Carter
    Design by Simon Wright and Adam Robinson
    MATT PARKER: Stand-up Mathematician
    Website: standupmaths.com/
    US book: www.penguinrandomhouse.com/books/610964/humble-pi-by-matt-parker/
    UK book: mathsgear.co.uk/collections/books/products/humble-pi-signed-paperback

    Julkaistu 8 kuukautta sitten


    1. Tom King

      *3blue1brown has entered the ring*

    2. DarkrarLetsPlay

      So, why is there no formula for it?

    3. carsten kurtz

      Why not just the integral sqrt(r^2 + dr/d) from 0 to 2pi

    4. florencefortyseven

      8:26. You called Ramanujan a "they" 😂.

    5. Carbon Roller Caco

      So where h=(a-b)²/(a+b)², the exact perimeter of an ellipse is π(a+b)Σ(n=0 to ∞; (h^n)/(4^n))--WAIT WHAT THE DEUCE IS THAT FIFTH TERM WHY ISN'T IT h⁴/1024 _ARGH_ *looks up why* I quit. I QUIT.

    6. Bananaforscale

      If a goes to 0 then peremiter should be 2b right? So that means the part of the formula with a and b should have 1/pi in it to be able to cancel out with that first pi.

    7. UpAndOut

      Woohoo! I want to be a Quant!

    8. yami dragonborn

      3 blue 1 brown just peeking up from the bottom of the screen made me laugh

    9. 101nka

      Could you please share your code on github

    10. masterjrm

      My Lazy but very good approximation is P= (a+b) + 3*sqrt(a^2+b^2) For a >> b an even more laze formula is P = 4*a+b They have very low percent error for even very high ratios No pi required

    11. Daniel Magee

      why wouldn't integral arc length formula work

      1. Kajal Panda

        @Daniel Magee welcome and have a nice day.

      2. Daniel Magee

        @Kajal Panda oh ok thank you

      3. Kajal Panda

        You can just see that in Wikipedia, and the arc length integral will give you an unintrigrable function. It is also called Leonard Euler elliptic integral. It remains as it is until you put e = 0 for circle and you can get only circle's perimeter formula , I hope you know that the unintrigrable in Reinman's integral is pretty much found in some functions other than elliptic one.

    12. Abhijeet Sarker

      Ihave found a formula which is the perimeter of an ellipse! there is the formula #2πx+(a-x)4θ/sin⁡θ #where x=a^2+b^2-√(a^2 )+b^2 (a-b)/2a, and θ=tan^-1⁡(b/a)

    13. Ygerna

      Imagine calling a crapbook a "laptop"

    14. Zoltan Torok

      4:58 Eclipse. Got you.

    15. Total_DK

      Parker Ellipse Perimeter? 🤔

    16. Bill Volk

      Stupid question: if we can't get the exact value of the perimeter of an elipse, how can we know how inaccurate a formula is for calculating it?

      1. Daniel Magee

        infinite series that uses a very high number of terms like he mentions in the end of the video

    17. Evgeny Varganov

      I was expecting some explanation as to why there's no neat formula, but really it was like a bunch of approximate formulae. Dislike

    18. H. H.C

      can't we have an approximation at every relevamt ellipse (1 .. 5 ..) m-axis increase? Then just use a switch/if statement to construct a composed approximation?

    19. jamcdonald120

      16:00 thats not the point, you could combine the h infinite series with the pi infinite series into 1 infinite series, lets call it PI_h

    20. jamcdonald120

      6:53 I would call that a pathagorean average

    21. Nighthawk

      The eccentricity of an ellipse is always less than 1 (1 being a parabola) and greater than 0 (0 being circle), all things considered the formulas of approximation are great.

    22. Blackhoggaming

      A total ellipse of the chart!

    23. IoEstasCedonta

      RIP Jim Steinman.

    24. Chris Devine

      Hi. I'm from the future. The equation for the circumference of an ellipse was just discovered. It's: C = (a+b) x "cake". (You don't have a symbol for cake yet).

    25. Michael Crosby

      A "Parker Approximation of the Perimeter of an Ellipse" - a formula that not great at giving the perimeter of an ellipse, but a good effort regardless.

    26. Zeta Destroyer

      What if Ramanujan has laptop

    27. Gamespot Live

      Lmao who uses Esentricity when you can subtract a-b to figure it out.

    28. woodchuk1

      The third definition is the quadratic mean, or root mean square.

    29. crystalsnowlion

      Circle vs ellipse...implants vs real ones.

    30. crystalsnowlion

      The only reason I like circles is cuz they're shaped like girls'...😅

    31. TubeRex

      My lazy approximation is: 4x+4-sqrt(x) simple, right

    32. Hagen P

      Nice wordplay on ILM. Much appreciated. :-)

    33. shadrana1

      Prove the formula when the error tends to zero as the term tends to infinity?

    34. Davy Jones

      Don't really feel like the video goes into any detail at all about WHY the equation doesn't exist, just tangents about other approximations. I know there is an integral equation for finding the length of a curve and we know what that equation is, although I assume that the droll answer is that, for an ellipse, it turns into an equation that is impossible to integrate.

    35. Nikola Djuric

      QUADRATIC MEAN, Just to inform you, QM(a,b)=_/((a²+b²)/2). It's well known H.M.

      1. Nikola Djuric

        Geometric mean is lim when n tends 0. Mean of order +infinity is max of numbers, mean of order -infinity is min of those numbers (lim as n tends it)

    36. Mark Fergerson

      Hmmmmmmm. Can an ellipse be considered to be the projection of a circle tilted from the axis of projection, the limits being a circle and a line segment? If so, there should be a relation between the angle of tilt and the eccentricity (and perimeter etc.). of the resulting ellipse.

    37. Karim Abdel

      Did you get my comment? If not, I can rewrite my little approximation of the perimeter 😉.

    38. Prime Phoenix 6174

      15:37 3b1b's pi creature is suspicious 😁

    39. Brad Kemble

      a+b π

    40. Rohit Mungase

      7:00 : Thats RMS(Root Mean Square) Value

    41. Juel Herbranson

      Am I the only one weirding out on that question mark???

    42. famitory

      finding the perimeter of an elipse is easy, just wrap a piece of string around it and then see how long the string is

    43. Geo Profundo

      e^(i * pi)=-1. That's an equation, QED.

    44. Bill Gosper

      Matt, here's an Easter Egg for you. The arclength of one period of sin x = the circumference of an ellipse with semiaxes 1 and √2. [See gosper dot org slash rollingellipse dot png.] In[112]:= #1 == #2 == FunctionExpand@#1== N@# &[ArcLength[Circle[{0, 0}, {√2, 1}]], ArcLength[Sin@x, {x, 0, 2 π]}]] Out[112]= 4 EllipticE[-1] == 4 √2 EllipticE[1/2] == 4 √2 π^(3/2)/Gamma[1/4]^2 + Gamma[1/4]^2/√(2π) == 7.64039557805542 𝚪(¼) is the rightful value of the symbol 𝛕. And someone should write Beckmann II: A History of 𝛕.

    45. Joshua Tyler

      Pythagorean's theorem is helpful here.

      1. Joshua Tyler

        @AMAZINGkaboom You have to use sqrt((a*cos(x))^2 + (b*sin(x))^2). Pythagorean’s theorem with sinusoids.

      2. AMAZINGkaboom

        No, this is because the pythagorean theorm would only calculate the length of the chord at the top of the line of b and a, not the "radius" of the ellipse

    46. Terrance Parsons

      Could you have used a less shitty pen to draw the ellipse?

    47. Jassiel Cano

      Was that 3blue1brown pi?

    48. Sarthak Kalpasi

      like we have integral for area under a curve and derivative for slope of that curve at a point, we should try to find a function which tells us the length of curve between two point.

    49. Baba

      15:37 lol 3blue1brown

    50. Arthur Vermillion

      My approximation "treat it like a funny lookin' circle" If A=πab, then you can use the area to find a circle of the same area, with r² = ab, and then find the circumference of that circle using (√ab) for r, giving a final formula of C = 2π (√ab) Edit: never mind, this thing is stupid and I am stupid

    51. alistair ferguson

      "oh and focal points have a real representation light, mirrors blah blah blah" love how uninterested he is in the real world applications 😂

    52. Charles Mouse

      So... Pi is the irrational number defining a circle. But every other ellipse has it's own equally irrational 'Pi', an infinite number of 'Pi's' in fact. Oh, good! What it the relationship between an irrational number and infinity? Is there any defined relationship between one 'Pi' and another?

    53. Robert Goodman

      Were you trolling comet astronomers when you asked at 13:00 who deals with ellipses of 75:1 or greater?

    54. Robert Goodman

      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.

    55. Harry Nicholas

      is there a video explaining why we use squaring in so many formula, like e = mc2 and PiR2? it seems weird these are all nice round numbers, why isn't ir e - mc almost squared but not a round number?

    56. Slit Fidget Spinner Dab Bodmod

      So interestingly, for the simple approximation if instead of pi you use 23/7 it's less accurate for all values below 13.1(ish) but more accurate for all values above - and beyond 18 the error is never above 1.5% compared to the original's >5% - which makes it both simpler to compute with no irrational (and transcendental) numbers and far better in the long run. You could also use 22/7 as an approximation for pi below 13.1 for decently low error the entire range using only simple rational constants.

    57. harys_john

      So why is there no equation? Is it just because 2 times the integral from -a to a of sqrt(1+(dy/dx)^2)dx where y=b*sqrt(1-x^2/a^2) isn't integrable?

    58. krischan67

      Regarding ellipses 75 times wider than high, we can screw the formula and simply take twice the long symmetry axis as the perimeter, can't we?

    59. Pro Odermonicon


    60. Afonso Luiz Dilda Bucco

      I am happy now because not just my problems are impossible to solve. But there are impossibilities about the deterministic mathematics. So now we can cry together.

    61. Eduardo Fonseca

      Like just because of 'Total Eclipse of the Chart'

    62. FlyingGuitarist

      Can we define a new irrational number like pi but with ellipses and have a relation with pi? Like pi is defined as ratio of circumference of a circle to its radius , the same for a ellipse?

    63. snowiePL

      How do you verify these if there's no exact equation? That's an interesting thing you didn't mention. I've just come up with my version 2*π*((a^p + b^p)/2)^(1/p), p = 184/123*(a/b)^(7/900)

    64. Tom Cook

      Could you not plot the ellipse on a grid (i.e.render it on a screen with a 1 pixel stroke) then count the pixels/apply some basic pythagoras to work out the distances between pixels where the pixels step then multiply the result by the scale factor between screen space and the ellipsis size? Wonder how accurate that would be? As someone not great at maths, but pretty handy with code, this is the approach I would take.

    65. Matthew Hopkins

      You know why? Because no one cares. That's why.

    66. Jonathan O'Brien

      Makes sense, pi is not a number, it is defined as a ratio relative to a circle's r. IE. When a = b. So replacing r with the more generalised a & b using h gives you the more generalised formula at the end of the video for the circumfrance of an ellipse, including the generalised circle. Just when a = b it simplifies to 2 π r, which, as he stated, is still an approximation depending on how accurate you want to define the ratio π.

    67. zelldot

      Why not π((a+b)/2)

    68. Petch85

      I live this video.... But I also hate it. I cannot let this go... And I keep finding problems. How about the perimeter of a superellipse? How du I space my points with equal distance on a ellipse? How du I space my points with equal distance on a superellipse? What about the areas? This makes me go insane.

    69. Sasuke Sarutobi

      TIL: a circle is a regular ellipse.

    70. ubk4242

      How is what he gives at 14:15 not an equation or a well defined function for the perimeter of an ellipse?

    71. Steve Syngkli

      If Ramanujan is still alive we'd understand better

    72. rob hemp

      I can do better: walk around the ellipse with a measuring wheel. May take awhile with some of those pesky comets, changing their orbits and stuff as mass ejection occurs near parent stars.

    73. Riki ponting

      The agonizing blanket postnatally satisfy because trail obviously remove lest a neighborly norwegian. maniacal, worthless catamaran

    74. Prax Zimmerman

      RIP the Jane Street Hong Kong interns

    75. Spacecore-2020

      I really want some input on this because I was doing calc II homework and we learned about how to calculate the lengths of polar functions. So if you had an integral from 0 to 2pi and used the formula you'd be able to calculate the perimeter of an ellipse right?

      1. Spacecore-2020

        Oh. Yeah thats true lol i knew it couldnt be that simple

      2. Thilo Reichelt

        You are right, but exactly that integral does not have a solution in elementary functions. You end up doing an infinite series. Look up "elliptic integral".

    76. Aditya Sharma

      The Pi is SUS!!

    77. Richard Barclay

      Why is 2(pi)r used as standard when I was taught the circumference of a circle it was always (pi)d, sure its the same thing but its tidier to my eye.

      1. Aleš Hace

        We always used 2(pi)r in high school and then always (pi)d at uni.

    78. judy churley

      Do you keep saying 'eclipse' (ie., at 5:00), or am I hearing thing? Or is an 'eclipse' a Parker Ellipse?

    79. Aravind muthu

      I applied regular integration and ended up with a sin^-1 in the formula that doesn't vanish

    80. Quanlai Li

      is there also no equation for the surface area of an ellipsoid?

    81. Nowill

      What happens if you replace ramanujen's 3 with another pi?

    82. Indranil Biswas

      Haha the climax was good! Pi is the real culprit! We use it so often that we sometimes ignore it's an infinite series really

    83. PLATONU

      PI: nice crossover ! 15:38

    84. FatPinarelloRider

      @5:00 The perimeter of an eclipse? 😂

    85. Jätski.fi

      just scale a circle to a different world scale, its like a transform(er)

      1. Jätski.fi

        axis difference

      2. Jätski.fi

        simple multiplication

    86. Rajesh Thipse

      Just a quick suggestion - If there is a predictive way to find the error introduced, why can't we pre-empt that error? E.g. when a/b = 2 and I get an error of 5% extra, then I should divide the result by some f(a,b) which is 1.05 when a/b = 2. I am sure, there must be something beyond what I think :(

    87. ssybesma

      10:52 is all you really need. That is as close to perfect as anything likely to be found.

    88. Luis Rosano

      Why denominators are : 2^2 ; 2^6 ; 2^8 and 2^14 on the infinit serie of 14:16 ? Maybe the next one is 2^22?????

    89. Wojciech Czaderna

      What about placing PI in place of 3 in the lazy Matt Parker formula? Will it make things better, worsen or it's just doeasnt make sense? ;)

    90. Kinshuk Singhania

      5:25 he says eclipse!!

    91. Stephen Schuster

      Here's my approximation, its assumed that b = 1 2*integrate from -a to a(sqrt( 4x^2 / (a^4 - x^2*a^2) + 1 ))dx = circumference

    92. Philipp Petsch


    93. Fada Te

      Maybe we should solve a conics problem ? Or apply a sort of projection to a circle of radius=major ellipse radius (compute the angle of projection from the minor ellipse radius) ?

    94. Han Quan Phoon

      10:59 mind = blown

    95. João Matheus

      5:00 He said "eclipse" two times in less than a minute (he says it again roughly at 5:26)

    96. Jon Name Change

      I thought this video was going to show us the mathematical proof of why you can't derive an equation for the perimeter of an ellipse?!

    97. AlainNaigeon

      I'm disappointed, because the "why" of the title hasn't been explained. We should see again the proof of the formula for the circle, and understand why it cannot be extended or mimicked for the ellipse.

      1. AlainNaigeon

        For instance, I'd like to try something like b=r-epsilon and a=r+epsilon ; this IS an ellipse and the product a*b obviously goes to r^2 when epsilon goes to 0, but there is ANOTHER function going to this limit, which is NOT this product, and we have to understand WHY it cannot be the product, rather then finding alternate approximations. IMHO this would be a much deeper subject.

    98. Neil Peter

      Can't you just integrate to find the arc length?

      1. shadowatom

        Yes, but I think he was meaning a nice equation that just uses a and b. Not to mention integrating the arc length gives a nasty integral to work out.

    99. Zag Zagzag

      ofc i can do better , just use the best ones within the sections

    100. cbmira01yt

      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!"