How π Emerges From a Forgotten Curve

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⬣ LINKS ⬣
⬡ PATREON: / anotherroof
⬡ CHANNEL: / anotherroof
⬡ WEBSITE: anotherroof.top
⬡ SUBREDDIT: / anotherroof
⬡ TWITCH: / anotherroof
⬣ ABOUT ⬣
Happy Halloween! The Leibniz Formula -- the alternating sum of the reciprocals of odd numbers -- converges on π/4. It's a bizarre result. Even more bizarre is the Witch of Agnesi, a curve with a fascinating history. In this video we marry these together, using the Witch to demonstrate why this alternating sum produces π.
This video was sponsored by Brilliant.
⬣ TIMESTAMPS ⬣
00:00 - Introduction
03:37 - Defining the Witch
07:49 - Two Lemmas
11:56 - Witch Area Part I
13:40 - Etymology of the Witch
18:39 - Witch Area Part II
28:55 - Outro
⬣ INVESTIGATORS ⬣
Nothing for you here. Sorry!
⬣ REFERENCES ⬣
[1] P. Fermat, Œuvres de Fermat, Methodes de Quadrature, Gauthier-Villars et fils (1891).
[2] G. Grandi, "Note al trattato del Galileo del moto naturale accellerato", Opera Di Galileo Galilei (in Italian), vol. III, Florence p. 393 (1728).
[3] M. Agnesi, Instituzioni analitiche ad uso della gioventú (1748).
[4] J. Colson, Analytical institutions, Wingrave & Rivington (1801).
[5] www.smithsonianmag.com/scienc...
[6] D. Struik, A Source Book in Mathematics, Cambridge, Massachusetts: Harvard University Press (1969).
[7] Complete Dictionary of Scientific Biography, Charles Scribner's Sons (Vol. 1.) (2008).
[8] C. Truesdell, Corrections and additions for “Maria Gaetana Agnesi”. Arch. Hist. Exact Sci. 43, 385-386 (1992). doi.org/10.1007/BF00374764
[9] P. Fanfani, Vocabolario dell'uso toscano, p. 334 (1863).
[10] S. Stigler, Studies in the History of Probability and Statistics. XXXIII Cauchy and the witch of Agnesi: An historical note on the Cauchy distribution, Biometrika, Volume 61, Issue 2, August 1974, Pages 375-380, doi.org/10.1093/biomet/61.2.375
[11] T. F. Mulcrone, The Names of the Curve of Agnesi. American Mathematical Monthly, 64, 359 (1957).
[12] www.encyclopedia.com/women/en...
⬣ IMAGE CREDITS ⬣
Cauchy Distribution
upload.wikimedia.org/wikipedi...
Soliton
commons.wikimedia.org/wiki/Fi...
Witches
en.wikipedia.org/wiki/Witch_o...
Implicit Content
www.nicepng.com/downpng/u2w7a...
Maria Agnesi
upload.wikimedia.org/wikipedi...
Portrait of Fermat by Roland Lefevre, public domain
Portrait of Grandi, public domain.
Clifford Truesdell
en.wikipedia.org/wiki/Cliffor...
Pietro Fanfani
upload.wikimedia.org/wikipedi...
Jacopone da Todi
upload.wikimedia.org/wikipedi...
Gabriel Simeoni
upload.wikimedia.org/wikipedi... CREDITS
Cauchy Distribution
upload.wikimedia.org/wikipedi...
Soliton
commons.wikimedia.org/wiki/Fi...
Witches
en.wikipedia.org/wiki/Witch_o...
Implicit Content
www.nicepng.com/downpng/u2w7a...
Maria Agnesi
upload.wikimedia.org/wikipedi...
Portrait of Fermat by Roland Lefevre, public domain
Portrait of Grandi, public domain.
Clifford Truesdell
en.wikipedia.org/wiki/Cliffor...
Pietro Fanfani
upload.wikimedia.org/wikipedi...
Jacopone da Todi
upload.wikimedia.org/wikipedi...
Gabriel Simeoni
upload.wikimedia.org/wikipedi...
⬣ MUSIC CREDITS ⬣
"Night Watch" by Torus.
"Playfully" by V Draganov.
"Ursula Bones" by Roger Gabalda.
"Haunted House" by Danijel Zambo.
"Graveyard Tango" by Oliver Massa.
"Fairytales" by Danijel Zambo.
"Cautious Optimism" by Apex Music.
"The Air we Breathe" by Apex Music.
"Hostage" by Richard Bodgers.

[@AnotherRoof]

To try everything Brilliant has to offer-free-for a full 30 days, visit brilliant.org/AnotherRoof. The first 200 of you will get 20% off Brilliant’s annual premium subscription!

[@playfulmathematician5928]

how did you do that?

[@forivall]

Personally, I'll only sign up for brilliant when they add multivariable calculus, since that's the kind of thing I didn't learn in my compsci degree. Or linear algebra / matrix stuff, since I need a refresher on that.

[@AnotherRoof]

@@forivall Thanks for watching! They have courses on all those things. Advanced Math Level 3 > 3.2: Multivariable Calculus. Advanced Math Level 4 > 4.1: Introduction to Linear Algebra and 4.2: Linear Algebra with Applications. Enjoy learning and don't forget to use my offer code so they know I sent you!

[@forivall]

@@AnotherRoof Aha! Thanks!

[@petevenuti7355]

I lost you about halfway through I'm going to have to watch it again.... But a thought I had while watching this is, is there any interesting relationship between the witch and the hypersphere?

[@pavloslazarou3697]

0:00-1:00 Mathimatician spits bars 1:14-29:37 Mathematician spits facts

[@erikziak1249]

The quality of your videos keeps on increasing. They get better and better. Small details, but they make a difference. It must have taken a lot of work.

[@Redstoner34526]

Sigma notation is a method used to solve for a summation series and is very useful to notate infinite sums such as Riemann sums or power series.

[@Scum42]

I just have to stop watching after that absolutely incredible intro to say this... Since the first video you posted it was clear you were an extremely good educator, and your videos always did an excellent job of explaining the topic at hand. But this intro, in costume+makeup, editing effects, and **written in full verse,** proves you also have talent as an entertainer and general creator. This is one of a few channels that I am extremely proud to say I got in on the ground floor before they had even 10,000 subscribers. At this rate, you'll be at a million in no time.

[@proloycodes]

yeah, same here

[@coarse_snad]

Oh damn. You just made made me realize that this channel is actually small. The production value of the videos does not match the subscriber count :(

[@AnotherRoof]

Thanks for the kind words. Comments like this make my day! And hey @snad_mod we've all got to start somewhere :)

[@STEAMerBear]

You are absolutely correct. Sadly there exists a VAST conspiracy comprised of people who craftily portray (simulate or approximate) the confused and befuddled. These folks tend to neither like nor subscribe when encountering such content despite its being among the best on KZbin.

[@angelowentzler9961]

The collective noun for witches is, naturally, 'coven'.

[@billionai4871]

I just posted that same comment lol! I'm surprised he didn't go for it

[@AnotherRoof]

@@billionai4871 I know of covens, I honestly just thought "gaggle of witches" sounded funny. But note to self: never deliberately say a wrong thing for the sake of a joke 😂

[@DeclanMBrennan]

@@AnotherRoof Gaggle definitely sounds funny but coven sounds more scary. I suppose it depends on the mood you are going for. 🙂

[@billionai4871]

@@AnotherRoof That's totally fair, and it was much funnier indeed

[@jameshart2622]

I tend towards Sir Pterry Pratchett's opinion that the group noun for witch is "argument".

[@neiltarrant7253]

My maths teacher twenty five plus years ago referred to this curve as the 'witch's hat'. Whilst it can look a bit like a witch's hat if sharp enough, I'd guess this is a back-formation - someone looking for an explanation for the name.

[@andrewharrison8436]

Some 50 years ago I was told that it was the path of the witches broom just trying to clear the circular object put in her way.

[@riccardoorlando2262]

In my uni days, the witch's hat was the name of the function family whose graph is a triangle symmetric around x=0, and zero everywhere else, such that it is continuous and its area is 1. This family is a common counterexample to show that the limit of the integral is not the integral of the (pointwise) limit, since as the triangle gets smaller the integral is constant 1 but the functions converge to 0 everywhere except at x=0, and the integral of the limit function is 0.

[@MatthewBouyack]

That intro was absolutely spectacular!

[@betterlifeexe4378]

5:10 A group of witches is called a coven

[@RichRauenzahn]

That implicit content has just the right amount of secs.

[@thabangnhlapo9513]

From 11:00 to 11:48. Just under a minute😂😂

[@billionai4871]

I LOVED the starting bit, well done! worth to note: I would call the family of witch curves a coven Ah, I seem to have commented too early, I just got to the point where you mentioned the origin of the name. Thanks for going into that research hole, and acknowledging such a great mathematician that isn't often mentioned

[@jackevans9574]

Man halloween inspired math videos are right up my alley

[@RubikxsMan]

Is it a dilemma if you have two lemmas?

[@b43xoit]

If they are opposed to each other.

[@TheArtOfBeingANerd]

​@@b43xoitor rather, if they are contradictory statements as in a proof by contradiction

[@firefly618]

An easier way to derive the equation. Let's call x=x₀ and y=y₀ for brevity and let's name the points O=(0,0) I=(0,1) V=(v,y) and also H=(0,y). We can see that OVI is a right triangle because it's inscribed in a semicircle, and we can split it into two right triangles OHV and VHI that are similar to it. This gives the proportion between the two small triangles: v/y=(1-y)/v that we can combine with the small-large triangle proportion: 1/y=x/v. The simultaneous set of equations solves to: y=1/(x²+1).

[@guerom00]

I had no idea about the origin of the name witch of agnesi... I've always called that curve a Lorentzian, i thought it was the "official" name...

[@mikeflowerdew7877]

I had the same thought... It seems that the width is a free parameter in the Lorentzian, but not in the witch. More to the point, the Lorentzian area is normalised to 1, very practical but not so good for this story I guess

[@morgangraley1049]

Amazing video! The explanations were thorough, digestible, and led to such a fantastic explanation for the relationship between the integral of the curve and the alternating series. I’m spooked by how well you explained it all!

[@Rissper.]

The quality on this one is even more impressive than usual! I am most struck by the script, which feels especially well put this time, and I really don't see enough people commenting on that! Love everything you've been putting out since the start of the channel and I really hope your noticeable efforts to create something truly unique here on the KZbin math scene get accordingly rewarded! Thanks for all the amazing videos and happy Halloween

[@SeverinoMonzonZuniga-of7md]

Your intro was great! You made a Mexican Calaverita. I enjoyed your video from start to finish. 👌🏻 Best regards from Mexico.

[@Nikolas_Davis]

Interestingly, _sinus_ for the sin() function is also a mistranslation from Sanskrit via Arabic into Latin. The Sanskrit original, _jīvá_ , means "bow string" or "chord"; it is a rather faithful rendering of the Greek ημίτονο, which means "half string", or "the note produced by a half string" (as the Pythagoreans had established a relation between the tone of sound a string produces when vibrating and its length). This was transliterated in Arabic as _jība_ , and Arabic script lacking vowels, it was interpreted by Europeans as _jaib, jayb_ , meaning "pocket" or "fold". So, the Latin equivalent of _sinus_ was used in the translations, also meaning "fold" or "hollow of surface".

[@Castle3179]

It's called a coven of witches.

[@jayjasespud]

A group of witches aren't necessarily in the same coven.

[@zackbuildit88]

​@@jayjasespudyeah it's kinda like calling a group of thieves "a brotherhood". It's accurate sometimes, but it's not general

[@Castle3179]

@@jayjasespud I didn't think of that...

[@reecec626]

A cauldron of witches x PS does coven rhyme with oven?

[@Castle3179]

@reecec626 Yes, it rhymes with oven.

[@joeeeee8738]

I love when it's explained with detail!!! Great content ❤

[@martinmicheals570]

Thank you very much for this video, and the connection with the surface of a sphere. I still feel like I'm missing out on some of the beauty you seem to be feeling, however, though it does feel a little bit like Gauss' proof regarding the heptadecagon, or Archimedes' On the Sphere and Cylinder. I don't see what the hype is, but I do see there's some 'unjustified' connection.

[@henricobarbosa7634]

Loved the vídeo and the aesthetic! Great as always

[@samcousins3204]

this is a uniquely engaging video. i feel like a child being taught by a kind personal tutor, rather then watching a video - entertaining, educational, and passionate.

[@mikeflowerdew7877]

I loved the implicit content warning! Definitely needed with all the secs that followed 😅 (Apologies, excellent video. I'll see myself out)

[@dumbasgenius7227]

The fact the witch has the same area as the surface area of a sphere makes me think somebody should make a projection from a the earth to the witch. That would be a very fun useless map lol.

[@juanro22]

If chosen wisely, one could send Anctartic or the empty void of the center of the Pacific to the narrow portions extending to the infinite, thus keeping the majority of the interesant bits near the original circle. I wonder how the resulting stretching looks like...

[@BradCowgill]

Loved this video Alex!

[@DeclanMBrennan]

I wonder could there be an area preserving projection from the sphere onto the Witcher? It might make for a really wacky map.

[@DiamondzFinder_]

Absolutely fun and informative video, the rhyming was fun, the mathy learning bit wasn't overcomplicated and of course the citations are very handy, but surprisingly, despite how much more I generally love math, the etymological section was the most interesting to me here.... Well anyway, keep up the great work!

[@FZs1]

Turns out Alex is a great poet, too!

[@dagamerboi]

that start was absolutely incredible!

[@AndyMcEwan19]

Very nice, Alex. Enjoyed that.

[@Draconis_Eltanin]

Killer looks and cool maths with a sprinkle of cool history on top! Keep up the great work :D

[@Spy653]

I always love the absurd amount of rhyming and alliteration you manage to get into the intros

[@HighKingTurgon]

Gosh, the relationship of the areas to the idealized strips of area was SATISFYING

[@dagamerboi]

the end was also pretty cool

[@samuelwaller4924]

Love the multi-minute etymology tangent

[@Schlups]

The point where it got scary for real was when he said he will explain EVERYTHING

[@soninhodev7851]

love the halloween special! many people in the premier were saying calculus was scary... id say its not! ;p

[@davidhowe6905]

Very nice! Regarding the alternating series, it pops up with the Fourier series of the Dirac comb (period T, value zero except at t = 0, +/-T, +/- 2T ….etc., integral over one period = 1 i.e. weird ‘function’ - but turns out to be v. simple Fourier series). Using the result that the integral between -s/2 to +s/2 where s is less than T but greater than zero must also equal 1, by suitable choice of s you get 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ….. = pi/4.

[@Cat-yz1tk]

the rhyming inro was so cool!!

[@falconionio]

Great video!

[@punditgi]

Excellent video! 🎉😊

[@not_David]

27:38 this amount of trig was genuinely the most frightening thing I saw this halloween. This was really cool though, I'm kind of suprised I had never heard of "the witch" or this equivalence (or if I have ive supressed that memory out of fear).

[@AnotherRoof]

And what you saw was the streamlined version -- when I first did this it took a lot longer and didn't even know if it would work >_

[@not_David]

@@AnotherRoof shivers down my spine

[@wojak8453]

There is a cool proof of the value of this sum in David Hilbert's book "Geometry and the Imagination" that directly relates it to the area of a circle approximated by squares arranged in a grid.

[@penguincute3564]

Firstly add some doses of mathrmatical potions, Secondly add some pieces of graph paper, Thirdly add some decimal points, Fourthly add some numbers, Fifthly add some triangles, Sixthly add some symbols of pi, Seventhly add some irrational numbers. If done correctly the pot should contain pi

[@pierrebaillargeon9531]

That was great, but the second part is basically just a rough, non-rigorous integration. So given that we already know the area of the circle and the curve... well, it felt a bit like circular logic. Great intro!

[@tomaspecl1082]

Good job!

[@azireo3921]

Thinking about the "visual" proof at the end, I thought about making the slice in the circle vertical. (if in the vid you took (-phi/2, phi/2), take (90, 90-phi)). Zooming in for small phi, the slice of the circle becomes a triangle, and the slice of the curve a rectangle. And the triangle fills exactly one corner of the rectangle! This works particularly well for this case, but it is harder to visuallize further from zero. Great video, as always, keep going! :)

[@rujon288]

get this man a million subs! the increase in video quality is crazy (:

[@timkw]

Amazing video

[@AWeller92]

̶i̶ ̶t̶h̶i̶n̶k̶ ̶a̶ ̶g̶r̶o̶u̶p̶ ̶o̶f̶ ̶w̶i̶t̶c̶h̶e̶s̶ ̶i̶s̶ ̶c̶a̶l̶l̶e̶d̶ ̶a̶ ̶c̶o̶v̶e̶n̶ sorry, continue

[@publiconions6313]

Always thought it was cool how they changed the rules regarding gloves and historical texts... someone eventually figured out bare fingers were actually safer for the books since gloves negate the gentle grip of fingerprints and make fingers more clumsy

[@enbyarchmage]

The rhymed part of this video already made it deserve more than one like🤣

[@excentrisitet7922]

I think @mathologer would approve! Very nice explanation and proper dive to the past!

[@78Mathius]

This was grand!

[@sunpuppetsofficial]

Best intro of the year

[@IronFairy]

This has everything! Brilliant!

[@rogeraldrich2533]

I find it funny that the pronunciation of "cos" and "sec" is different then when I learned them. Even though they are abbreviated we always pronounced the whole word. Cos was pronounced cosine, sec was secant. Tan could be tan or tangent so the rules were not set in stone. Sin was pronounced sine but you still pronounce it that way, not like "sinning witches."

[@Jaylooker]

Taking the inverse Fourier transform of the exp(-a|t|) gives the general case of the witch at 7:19 after sorting out the constant radius r. The single point with every line passing through that is used to define the witch is a bivector of projective geometry. I think the witch may be a smooth projective curve (variety) and a moduli space since it can parameterized by a value. Assuming this is true and considering the motive over a field of the variety, I wonder what field would be associated to it.

[@agargamer6759]

Top effort!

[@petrosthegoober]

holy effort batman

[@benhsu42]

I just started watching this video and I am amazed at the opening -- I can't tell if its meant to be iambic pentameter or not but its brilliant!

[@serdarakalin2209]

Marvelous!

[@filthyfingers8073]

This should have more views

[@zarsoft]

Excellent poetry!

[@rafakruczek5622]

5:30 and later, I don't WHY English speakers read letter "y" as 'why" But it is even worse when it is mixed with reading "zero" as "NOT" (however it is written with silent "u"/"g"/h"s) :)

[@namantenguriya]

Quality Content.

[@roger7341]

Brilliant!

[@regulate.artificer_g23.mdctlsk]

5:08 I believe the collective noun for witches would be "coven"? Correct me if I'm wrong.

[@danielrybuk1905]

Genius intro!!!

[@clockworkkirlia7475]

Absolutely beautiful maths here. Thank you. The way you light a path and tease out the answers gently and informatively after letting the viewer find what clues they can is something I really appreciate. And, as that sentence structure may indicate, I'm somewhat sleep-deprived, so that I found the video easy to consciously follow despite a very loose grasp of calculus should say a lot! Also, some results from my own experimentation: If you say "Agnesi" four times in the mirror, Pythagoras will jump out of your bath and brain you with an icosceles. EDIT: Also, love the poetic performance! I hope you rewarded yourself with a toffee apple.

[@Macieks300]

22:50-23:20 You can also just use the inscribed angle theorem and get that result immediately.

[@AnotherRoof]

Well spotted!

[@andrewbuchanan5342]

Hard to translate an Italian play on words into English. The nearest I can come up with is "wintch". Winch = "A stationary machine having a drum around which is wound a rope or chain attached to a load being moved." But I am ~300 years too late.

[@TheArtOfBeingANerd]

I did a weid little parametric thing when solving for the witch's formula. First, im an overachiever and did the general case of radius r. And then i used parameter m for the slope of the ray. To find the x and y in terms of m, i solved the system {y=mx, y=2} for x and {yrmx, x²+(y-r)²=r²} for y, getting equations that i then cancelled out m and got a final answer of y= 8r³/(x²+4r²). Which substituting r=½ does give the same formula derived in the video

[@pedroth3]

Really nice proof !

[@reallyWyrd]

This is the content I Am Here For. (I will be checking your math later when I'm sober, so it better check out.)

[@LB-qr7nv]

I your videos are so creative. The only thing I don't like are the time-lapses during the calculations. They make it a bit hard to follow even if I pause many times.

[@robinbruce7838]

Incredible ❤

[@PeterBaumgart1a]

Great. But now I'm exhausted, need a brief nap.

[@harriehausenman8623]

*Sir Mix-a-Lot* salutes you! 🎶

[@noahwiliams7214]

Omg! This was a fantastic video. I’m struck by the fact that the ratio of the area of a circle inscribed in a square to the area of that square is also 𝛑/4. This fact allows you to calculate 𝛑 using random numbers. Maybe there’s another video in that. Keep up the good work. the ratio of the area of a square

[@whannabi]

Why can we comment on something that isn't out yet

[@AWeller92]

why not?

[@wyboo2019]

i immediately recognized 1/(1+x^2) as being a pretty famous function and didnt know it could be derived this way! 1/(1+x^2) is the derivative of the arctangent function and also a great example for radius of convergence of power series; 1/(1+x^2) has pole discontinuities at x = i and x = -i, so if you try to take a taylor series centered at x=0 it will only converge in the range -1 to 1, but if you only look at it as a real-valued function this radius of convergence thing is kinda weird

[@QuantenMagier]

It's a >Coven< of Witches!

[@tilted_skirm]

14:15 John Colson died in 1760 yet managed to publish a translation in 1801? Spooky!😢

[@BobJones-rs1sd]

Just to clarify this, since others may be confused -- Colson did the translation while he was alive (obviously), but only in manuscript. It was pretty much complete, but he never got around to publishing before he died in 1760. Colson was a rather well-known mathematician in his lifetime, so people knew he had undertaken this, but the manuscript required editing. Eventually, John Hellins took up the task of doing the editing and getting it prepared for publication, which finally happened in 1801.

[@reallyWyrd]

Hey, you're pretty good at this I think.

[@b43xoit]

Would the correct modern nautical term be "sheet"? The line that is used to pull in the sail (via the boom)?

[@Jroonk]

A group of witches is called a coven.

[@jamesgill8389]

All the good stuff's at the John Rylands.

[@natashakwa]

Brilliant Math poem!

[@curtiswfranks]

Idk, it takes some witchcraft to survive with, let alone tutor, twenty younger siblings. Props to her!

[@t1mmmmo]

Perhaps a suitable re-mistranslation given the etymology would be "the winch"?

[@snex000]

A group of witches would be a coven.

[@abdirahmansamow7451]

the derivation of the equation looks like de cart's method of deriving gradient of tangents

[@scalesconfrey5739]

I'm surprised at your confusion around 5:09... Obviously, it's a cackle of witches! (I joke, it's actually a coven)

[@jespervanbommel]

Can you please make another video about pi? Especially about this product: 2 * (2/a(1)) * (2/a(2)) * (2/a(3)) * ... = pi/2 Where a(1) = sqrt(2) And a(n+1) = sqrt(2 + a(n))

[@zarsoft]

Behold the math wizard in action...

[@WielkiKaleson]

Witch? Fancy. We call it a Lorenzian.

[@cashkurtz5780]

I think it's starting.

[@dacianbonta2840]

Archimedes would be proud