I think that the shapes you perceive are moiré patterns, depending on the
resolution of your plot, the resolution of your screen and the region you
plot.
To understand it, start with a plot of [image:
\operatorname{frac}\left(x^2\right)],as done here
<https://sagecell.sagemath.org/?z=eJwryMkv0chJzE1KSVSosEorSkzWqIgz0tTRMNBRMDTR1FEoACqIL8jPzCsptrUwMNAEAJOQD3M=&lang=sage&interacts=eJyLjgUAARUAuQ==>,
which is a “cut” of your function on any straight line including [image:
(0, 0)]. Even at “high” resolution, the moiré appears at about [image: x=7].
This effect is better illustrated by an implicit plot : at default
resolution, you see sharp (and irregular) patterns, which disappear at
higher resolution (as done here
<https://sagecell.sagemath.org/?z=eJzLzC3IyUzOLIkvyMkv0chJzE1KSVQo1VEos0orSkzWKI0z0i6LM9LUNdAz1VHQMNBRMDTRRGKAdMUX5GfmlRTbWhgYaAIA9woXeA==&lang=sage&interacts=eJyLjgUAARUAuQ==>
).
The same is true for a plot3d (here
<https://sagecell.sagemath.org/?z=eJwryMkvMU7RyEnMTUpJVCjVUSizSitKTNYojVMw0i6LM9LUUdAw0FEwNEFmFAA1xRfkZ-aVFNtaGBhoAgDRxRO1&lang=sage&interacts=eJyLjgUAARUAuQ==>)
and for your complex_plot (see here
<https://sagecell.sagemath.org/?z=eJxLzs8tyEmtiC_IyS_RyEnMTUpJVKiySitKTNZITCrWqNKMM9LUUdAw0FEwNEFmgNTHF-Rn5pUU21oYGGgCAGHYFfM=&lang=sage&interacts=eJyLjgUAARUAuQ==>)
: in all these cases, strong moiré patterns observed at default resolution
disappear (oe ar severely modified) at high resolution.
HTH,
Le mardi 25 février 2025 à 14:00:42 UTC+1, Georgi Guninski a écrit :
> I am looking for both theoretical and technical answer for the following
> plots.
>
> Define f(x,y)=frac(x^2+y^2)
>
> When I complex plot x+I*y to f(x,y) as colormap, the result depends on
> the plotted rectangle.
> In the rectangle $0 < x,y <14$ I see many circles of different sizes.
> In the rectangle $8 < x < 12, 4 <y <6$ I don't see any circles, only lines.
>
> When I do implicit plot of $f(x,y)=0.5$ I see similar results in the
> big and the small rectangles.
>
> Question: Is there fractal-like structure of any of the plots or is
> it just coincidence?
>
> Attaching sage session
>
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