Fermat's Christmas theorem: Visualising the hidden circle in pi/4 = 1-1/3+1/5-1/7+...

NEW (Christmas 2019). Two ways to support Mathologer
(see the Patreon page for details)

Leibniz's formula pi/4 = 1-1/3+1/5-1/7+... is one of the most iconic pi formulas. It is also one of the most surprising when you first encounter it. Why? Well, usually when we see pi we expect a circle close-by. And there is definitely no circle in sight anywhere here, just the odd numbers combining in a magical way into pi. However, if you look hard enough you can discover a huge circle at the core of this formula.

Here is a link to a video in which 3blue1brown about the same hidden circle in Leibniz formula:
And another video by him about a hidden circle in the solution to the Basel problem:

Link to the original Jodocus Hondius engraving of Jodocus Hondius that Google tries to pass of as a portrait of the mathematician Albert Girard

Thank you very much to Marty for all his help with polishing the script of the video and Karl for his idea for the 2019 Easter egg.

Today's t-shirt: google "spreadshirt pi+tree+christmas+math"

Enjoy :)

[mandolinic]

Hey, Mathologer is back after a short interruption of service. Nice to see you again!

[Qwerty]

Merry Christmas!
I think the easiest way to show pi = lim N(r) / r^2 is to say that the squares centered in grid points <= r units away from (0; 0) must cover a circle with radius r - 1 and center (0; 0) but are contained in a circle with radius r + 1 and center (0; 0). Which gives us pi * (r + 1)^2 / r^2 >= N(r) / r^2 >= pi * (r - 1)^2 / r^2 and since both leftmost and rightmost side of this inequality approach pi as r->inf, we get what we wanted

[AmitKureel]

Thanks for recognizing Indian Mathematician Madhava for his work on Calculus and infinite series relating to Pi. More than the credit, I am interested in bringing in awareness among talented Indian youth towards Maths which is currently more influenced by cash-generating Wall-street jobs and related courses.

[MisterMajister]

Let's all just forget about yesterday shall we? Great to see the videos are back and I really hope they work their situation out (IN PRIVATE)!

[MrSigmaSharp]

We dont celebrate Christmas in my country but I celebrate any Mathologer video anytime

[JAlexCarney]

Hooray I was sure that the videos would need to be reuploaded from scratch. Very glad to see the old comments were not lost.

[patrickchristensen]

The disaster has been averted. Glad you’re back!

[harikrishnank.j.]

Madhava is an Indian (Keralite) mathematician. Actually I am watching this video miles away from his native place (Irinjalakkoda, sangamagrama as you said). Nice work sir👏

[VickyGYT]

A big thank you for acknowledging work by an Indian mathematician! First video doing so. You are a gifted teacher.

[Utesfan]

Odd factors of 2020: 1, 5, 101, 505, all good. This tells us there are 16 pairs.

2020 = 16^2 + 42^2 = 38^2 + 24^2. The two permutations and four sign choices yield all cases.

[Mathologer]

Mathologer recently hit 500K subscribers and I would like to thank you all for your interest and your support over the years.


Since I started the channel four years ago, it has pretty much been a one-man labour of love. However, maybe now is a good time to take Mathologer to the next level and hire someone to assist with editing the videos, preparing subtitles, etc. In preparation for this, I recently monetised the videos by switching on the least annoying ads on YouTube. I also just launched a Patreon page a couple of minutes ago:
If you enjoy these videos and you can afford it, please consider taking out one of the Patreon memberships or making a one-time donation via PayPal:
paypal.me/mathologer :)


My plan is to also use this Patreon page as a platform to share more Mathologer materials with you, get into live chats, etc. Let’s see what’s possible and what makes sense here. Merry Christmas :)

[alperyoloyilmaz]

I am really happy to see your videos back. We don't know the value of what we have untill we lose it.

[enricolucarelli]

God save internet! I can not even imagine a life no more without having access to these wonderful, interesting, and enlightening videos like the ones you present every now and then!
Herzlichen Dank, fröhliche Weihnachten, und bitte weiter so.

[erutuon]

My mnemonic for this infinite sum now will have to be that it's related to the number of integer points in one quarter of an infinitely large circle. Pi being the area of the whole unit circle. With the Pythagorean theorem, that connects it to the sums of squares. Beautiful proof. I just rewatched this one. You presented it simply enough for a high school math guy like me to understand.

[davidbarnett]

Glad the issue has been resolved. This is one of the best channels on the internet. I do note the updated 'About' page, removing the GG reference...

[DrMikeE]

OMG! You are a fabulously knowledgeable and effective teacher. I am deeply humbled by your excellence. Thanks so much!


(Dr. Mike Ecker is a rational skeptic and a Ph.D. mathematician - CUNY 1978, Ph.D. Summa Cum Laude - who researches and writes prolifically. He is also a retired PSU mathematics professor and former computer journalist.)

[aaronleperspicace]

Dear Mr Burkard Polster,
I sincerely appreciate the fact that you give credit to the right person for the discovery of a mathematical identity/formula (in this case, the Indian mathematician, Mādhava). Earlier also, I learned through you that the fact that e^ix = cos x + i×sin x was known to Roger Cotes before Euler.
Cheers and merry Christmas!

[chirayu_jain]

Best thing about you, you always tell the real mathematician of the theorem. 😀👍🏻
BTW merry Christmas🥳🥳

[non-inertialobserver]

First the Mathvengers: Eulergame, then Numberphile hit pi million subs, and now a new Mathologer video?! Is it Christmas? Oh wait, it is.

[kutlokebaikanye]

Been following your videos for 4 years and I have never been disappointed. Great stuff mate 👍!