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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:
And another video by him about a hidden circle in the solution to the Basel problem:
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
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"
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