Hello everybody, here is something peculiar about Viète formula for Pi. Vieta or Viète formula here : https://proofwiki.org/wiki/Vieta%27s_Formula_for_Pi Viète was the most famous person from Vendée (France). as you may know the formula converges to Pi (or Pi/2 or 2/Pi depending on which side you put the square roots in the expression), Here is the peculiar thing : there is a definite pattern in the successive terms of the binary expansion of those numbers when they converge to Pi. Here is a sample code for Pari-gp for doing this at high precision : \p 4000 p0 = 2000 p1 = sqrt(2^p0) p2 = 2.0 v = p1/p2 for (n=1, 8000 , p1=sqrt(2.0+p1); v=v*p1/2) print(binary(v)) By doing so, it produces a stream of binary digits at each step of the formula. from these lines of bits, I can do 1 image , colourized for a better view of the contrast. And here is the result : http://plouffe.fr/dessins/vi%c3%a8te.png that image is 20000 x 5000 , which is 20000 binary digits of the first 5000 terms of Viète formula. we see the convergence of the formula, if we look from a distance : we see nothing at all BUT if we take a closer look at the patterns in the binary expansion we see this : http://plouffe.fr/dessins/formule%20de%20vi%c3%a8te.PNG Many things can be said about this : - There is a definite pattern in the binary expansion of those numbers that we do not see individually but when they are grouped we see something. - This pattern appears also with the obvious generalization of Viète formula replacing the '2' by 1, 2, 3, 4, 5, ... the pattern is the same. so, numbers like 4*Pi*sqrt(3)/2, Pi/2, 4/5*log(phi)*sqrt(5) , where phi is the golden ratio and sqrt(3)/2*log(2+sqrt(3)). etc. shows similar patterns. - If we do the same method for series like the Newton-Leibniz for Pi/4 : nothing at all appears, the same with sum(1/(n*2^n), n=1..infinity) = log(2), nothing to see either. - Viète process appears so far as being the only one able to produce a pattern, I have tried many , this is the only example so far. - What is surprising is the way the pattern persist, in diagonal as well as vertical. NOTE : to produce your own image : 1- run the small program with Pari-GP. 2- Clean the output to have only ones and zeros for each line 3- Find a way to transform this text file into a pure Unix file with LF at the end of each line OR a pure binary file of let 's say 10000000 million bits. 4- rename the file by adding .raw at the end. 5- Open Photoshop and open the file, the program will detect that it is something in binary and propose to open it in raw format :, just specify the width and depth of the file properly, 6- convert the file to 'indexed colors', 7- change it to a black body color scheme, this will color the ones and the zeros in blue and black or light brown and white depending of your choise, and voilà the image. Any comment on this ? best regards, Simon Plouffe