My pictures were obviously wrong: I had a sign error when doing the computation. They now match those posted by others. One interesting thing is to plot a greyscale image of when it gets within epsilon of 1. Here's within 1e-6: https://imgur.com/a/fGvfnWU On Fri, Sep 4, 2020 at 4:10 PM Mike Stay <metaweta@gmail.com> wrote:
Here's an attempt at looking at that area. These two pictures are 0.01 wide and tall with left x coordinate at -0.72 and 1.18 and y coordinate centered at 0 https://imgur.com/a/ZGuBi2d The colors show how many iterations are required to get to within 1e-6 of 1.
It's hard to see at -0.72, but at 1.18, there's a brighter white region near the solution, indicating it took longer to reach an oscillating pattern.
Here's the program if you want to mess around with it: https://www.khanacademy.org/computer-programming/foias/5924517186453504
On Fri, Sep 4, 2020 at 2:31 PM Dan Asimov <dasimov@earthlink.net> wrote:
Now I'm wondering what all the complex solutions z are of
----- f(1) = z,
f(n+1) = (1 + 1/f(n))^n,
lim_{n —> oo} f(n) = oo -----
—Dan
Kerry Mitchell wrote: ----- If you allow for f(1) < 0 and if I did my analysis correctly, then I see a point similar to George's 1.187452351 at f(1) ~ -0.71774114214285.
Taking this problem as a fractal iteration:
z(n+1) = (1+1/z(n))^n
with z(1) = pixel, then there is a fractal contour connecting the points 1.187452351+0i and -0.71774114214285+0i. -----
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-- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com
-- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com