[math-fun] patterns in numbers : vertical view
Hello everyone, I made one experiment with n*n*phi, phi being the golden ratio. I use a precision of let's say 1000 digits and n = 1 to 16000000. If we look at the square grid made of the numbers (without the decimal point), there are no clear patterns in the lines of digits BUT there is a definite pattern in vertical. for each k (from 1 to 1000) I constructed 1 image made out of the first 16 000 000 digits , making one image of 4000 x 4000. This is the result : http://plouffe.fr/phi_time_n_n%20vertical_base_10/index.html These digits are in base 10, so I used a colourization with blue tones : light blue = 0 , darker blue = 9. The thing is : there is a pattern for every k. Very definite pattern. I used that color because in gray scale it does not comes out clearly enough, the colourization. if we take another constant like pi, e, gamma or even a large rational, the same patterns appear, not surprising. Now, the good question is : ok there is a pattern , then what, what is the difference in those patterns. If n*n*n is used instead of n*n, the same but it does scramble after a few digits. Any explanation for such patterns ? best regards, Simon Plouffe
On 07/01/2017 01:03, Simon Plouffe wrote:
I made one experiment with n*n*phi, phi being the golden ratio.
I use a precision of let's say 1000 digits and n = 1 to 16000000.
If we look at the square grid made of the numbers (without the decimal point), there are no clear patterns in the lines of digits BUT there is a definite pattern in vertical.
for each k (from 1 to 1000) I constructed 1 image made out of the first 16 000 000 digits , making one image of 4000 x 4000. ... These digits are in base 10, so I used a colourization with blue tones : light blue = 0 , darker blue = 9.
The thing is : there is a pattern for every k. Very definite pattern.
I'm not sure I understand. Are n and k the same thing? Does each image show the decimal digits of n^2 phi for some n? (That doesn't seem like it can be right, because you say that if you use n^3 instead of n^2 then "it does scramble after a few digits" but some numbers are both squares and cubes. But maybe the patterns are only visible for smallish n or something?) Another thing I don't think I understand: it looks to me as if the patterns are two-dimensional rather than being "in the lines of digits" or "in vertical". And another, this time in the results rather than in the description: It seems that we get big blocks in which one kind of pattern persists quite stably, and quite abrupt transitions between one block and another. I wouldn't have predicted that sort of phase transition. ... Hmm, it looks to me as if these transitions, when they are visible, occur at fairly consistent positions in the decimal expansion. Maybe they correspond to "interesting" numbers of decimal places or something. -- g
Hello, each image of rank k is : the k'th position of n*n*phi n is from 1 to 16000000 disregarding the decimal point. A pattern appears for every k, The construction of the page uses When n*n*n is used only the first few k appears to have a pattern, the image is quite scrambled already when k = 12. The same thing would appear if we construct an image with the squares starting at position 1 of n*n , we have, 1, 4, 9 , 1, 2, 3, 4, 6, 8, 1, 1, 1, 1, 1, 2, ... : seq A002993. Best regards, Simon Plouffe Le 2017-01-07 à 15:26, Gareth McCaughan a écrit :
On 07/01/2017 01:03, Simon Plouffe wrote:
I made one experiment with n*n*phi, phi being the golden ratio.
I use a precision of let's say 1000 digits and n = 1 to 16000000.
If we look at the square grid made of the numbers (without the decimal point), there are no clear patterns in the lines of digits BUT there is a definite pattern in vertical.
for each k (from 1 to 1000) I constructed 1 image made out of the first 16 000 000 digits , making one image of 4000 x 4000. ... These digits are in base 10, so I used a colourization with blue tones : light blue = 0 , darker blue = 9.
The thing is : there is a pattern for every k. Very definite pattern.
I'm not sure I understand. Are n and k the same thing? Does each image show the decimal digits of n^2 phi for some n? (That doesn't seem like it can be right, because you say that if you use n^3 instead of n^2 then "it does scramble after a few digits" but some numbers are both squares and cubes. But maybe the patterns are only visible for smallish n or something?)
Another thing I don't think I understand: it looks to me as if the patterns are two-dimensional rather than being "in the lines of digits" or "in vertical".
And another, this time in the results rather than in the description: It seems that we get big blocks in which one kind of pattern persists quite stably, and quite abrupt transitions between one block and another. I wouldn't have predicted that sort of phase transition. ... Hmm, it looks to me as if these transitions, when they are visible, occur at fairly consistent positions in the decimal expansion. Maybe they correspond to "interesting" numbers of decimal places or something.
On 07/01/2017 16:21, Simon Plouffe wrote:
Hello,
each image of rank k is : the k'th position of n*n*phi n is from 1 to 16000000 disregarding the decimal point.
Oh! So, just to be absolutely clear: the pixel at (p,q) in image k contains the k'th decimal digit of (4000q+p)^2 phi. If that's right then I take it he "breakpoints" in the pattern happen where the number of digits before the decimal point changes. It would surely make more sense to number the digits relative to the decimal point rather than the first nonzero digit, in which case I bet the sudden changes in pattern would go away. So what we have then is a plot of, in effect, frac(Kn^2 phi) where, for a given plot, K is fixed; and we are seeing kinda-periodic dependence on n. The plots have K = 10^k, but I bet you would see something similar with K not a power of 10. And they plot "first digit after the decimal point of Kn^2 phi" rather than frac(Kn^2 phi) but obviously those plots look very similar. So the question (or at least *a* question) is: why should there be this kinda-periodic dependence of frac(Kn^2 phi) on n? It seems we have something a little like K[(n+t)^2-n^2]phi ~= integer for "many" n and fixed t, though I think this is too crude a description of what we see. That's the same as K[2tn+t^2]phi ~= integer. Perhaps what we actually have is K[(n+t)^2-n^2]phi ~= integer+const in which case the constant would come from Kt^2 phi, and the relevant fact would be that 2Knt phi ~= integer for "many" n, which would e.g. happen if 2Kt phi ~= integer. Of course phi is famously not very well approximated by rational numbers, but *any* number has rational approximations; the best ones for phi are ratios of consecutive Fibonacci numbers; do we have 2K.period a multiple of a "respectably-sized" Fibonacci number in each case? (If you play the same game with cubes instead of squares then the differences will be quadratic instead of linear and so you won't get periodicity.) -- g
participants (2)
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Gareth McCaughan -
Simon Plouffe