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

### Stand-up Maths

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.
fiblock.info/face/videot/pIOGfJx8imbKc2U.html
www.think-maths.co.uk/ellipsepuzzle
This was my pervious video featuring ellipsoids:
fiblock.info/face/videot/n25kgJupen-6aKg.html
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).
www.ebay.co.uk/itm/363096345270
Bonus content and a deleted scene are available on my Patreon.
www.patreon.com/posts/41274351
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
Fractal
Macey
Sören Kowalick
It all started because of a request I put out on twitter.
standupmaths/status/1301252952930299904

CORRECTIONS:
- 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
www.patreon.com/standupmaths
As always: thanks to Jane Street who support my channel. They're amazing.
www.janestreet.com/
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

## Kommentteja

1. *3blue1brown has entered the ring*

2. So, why is there no formula for it?

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

4. 8:26. You called Ramanujan a "they" 😂.

5. 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. 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. Woohoo! I want to be a Quant!

8. 3 blue 1 brown just peeking up from the bottom of the screen made me laugh

10. 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. why wouldn't integral arc length formula work

1. @Daniel Magee welcome and have a nice day.

2. @Kajal Panda oh ok thank you

3. 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. 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. Imagine calling a crapbook a "laptop"

14. 4:58 Eclipse. Got you.

15. Parker Ellipse Perimeter? 🤔

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

17. 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. 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. 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. 6:53 I would call that a pathagorean average

21. 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. A total ellipse of the chart!

23. RIP Jim Steinman.

24. 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. 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. What if Ramanujan has laptop

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

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

29. Circle vs ellipse...implants vs real ones.

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

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

32. Nice wordplay on ILM. Much appreciated. :-)

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

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

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

38. 15:37 3b1b's pi creature is suspicious 😁

39. a+b π

40. 7:00 : Thats RMS(Root Mean Square) Value

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

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

44. 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. Pythagorean's theorem is helpful here.

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

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

47. Was that 3blue1brown pi?

48. 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. 15:37 lol 3blue1brown

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

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

54. 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. 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. 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. 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. 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. THE PARKER ELIPSE APPROX.

60. 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. Like just because of 'Total Eclipse of the Chart'

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

66. 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. Why not π((a+b)/2)

68. 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. TIL: a circle is a regular ellipse.

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

71. If Ramanujan is still alive we'd understand better

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

74. RIP the Jane Street Hong Kong interns

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

2. 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. The Pi is SUS!!

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

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

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

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

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

82. 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. PI: nice crossover ! 15:38

84. @5:00 The perimeter of an eclipse? 😂

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

1. axis difference

2. simple multiplication

86. 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. 10:52 is all you really need. That is as close to perfect as anything likely to be found.

88. 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. 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. 5:25 he says eclipse!!

91. 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. int(sqrt(diff(sqrt(a²-x²)/a*b,x)^2-1),x=0..a)*4

93. 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. 10:59 mind = blown

95. 5:00 He said "eclipse" two times in less than a minute (he says it again roughly at 5:26)

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

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

100. 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!"