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

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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 the relevant chapter in Hilbert and Cohn-Vossen’s book Geometry and the Imagination (Google books). I am pretty sure that the idea and proof for the circle proof of the Leibniz formula that I mathologerise in this video first appeared in this book and is due to the authors: https://books.google.com.au/books?id=7WY5AAAAQBAJ&lpg=PA44&pg=PA37#v=onepage&q&f=false

Here is a link to a video in which 3blue1brown about the same hidden circle in Leibniz formula:
https://youtu.be/NaL_Cb42WyY
And another video by him about a hidden circle in the solution to the Basel problem:
https://youtu.be/d-o3eB9sfls

There is also a neat generalisation to what we talked about in this video to the solution of the Basel problem – in terms of the lattice points in a 4-dimensional sphere and the 4-square counterpart of the 4(good-bad) theorem. If you are interested in some details have a look at the last proof in this write-up by Robin Chapman: https://empslocal.ex.ac.uk/people/staff/rjchapma/etc/zeta2.pdf

Links to two Numberphile videos about the one-sentence proof by Don Zagier featuring Matthias Kreck: https://youtu.be/SyJlRUBoVp0 (intro), https://youtu.be/yGsIw8LHXM8 (the math)

Link to the original Jodocus Hondius engraving of Jodocus Hondius that Google tries to pass of as a portrait of the mathematician Albert Girard
https://www.swaen.com/zoomV5e.php?id=8653&referer=antique-map-of.php

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 🙂

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