Subjectively deterministic-looking features of scatter plots

[Swamp Thing]

This is a scatter plot of the points where $|\zeta(1/2 + it)|$ has local maxima.

On seeing it, my first thought was that there seems to be a certain amount of deterministic quality, a sort of "texture", in the scatter. There seem to be groups of points that look like skeins, closed curves etc. There are white spaces roughly bounded by closed curves, and "no-go" areas surrounded by dense populations. It is as if some kind of "attractor" phenomenon is at work underlying the randomness.

So as an example of an "attractor" modulating a chaotic scatter, here are two plots of the so-called Hopalong Attractor discovered by Barry Martin and popularized in Scientific American a few decades ago.

The first one is a zoomed-in version of the second one.

But there was always the possibility that the features I perceived in the first plot were all in my head, so I plotted a Gaussian distribution and zoomed in on the middle, just to see if the same apparent groupings appeared:

To my eye, this seems to have a surprising amount of "texture", with groups of points seeming to form somewhat non-random curves, skeins, and even bounded white spaces here and there. This suggests that any deterministic appearance I see in the first plot is entirely subjective, i.e. "all in my head".

But I can't quite make up my mind. After all, it is not unreasonable that the peaks of a quasi-periodic function could show some pattern underneath the randomness. And the first plot (peaks of zeta) has a qualitatively different and more distinct texture.

So... any thoughts?

Also wondering if there have been studies of the psychology of pattern discernment in random or nearly random distributions.

 

[Dale]

On seeing it, my first thought was that there seems to be a certain amount of deterministic quality, a sort of "texture", in the scatter. There seem to be groups of points that look like skeins, closed curves etc. There are white spaces roughly bounded by closed curves, and "no-go" areas surrounded by dense populations. It is as if some kind of "attractor" phenomenon is at work underlying the randomness.

I don't see any of that.

To my eye, this seems to have a surprising amount of "texture", with groups of points seeming to form somewhat non-random curves, skeins, and even bounded white spaces here and there.

I also don't see any of that.

Also wondering if there have been studies of the psychology of pattern discernment in random or nearly random distributions.

I am not sure that this is related, but there are studies that show that it is very difficult for humans to actually produce a random distribution. We tend to produce distributions of points that are much more uniform than is correct for a truly random distribution. If it is random then there will randomly be clumps and voids. A distribution that avoids any clumps and voids would be uniform, not random.

 

[Baluncore]

Also wondering if there have been studies of the psychology of pattern discernment in random or nearly random distributions.

Different people will believe they see different things in a random distribution. That is because the brain tries to extract features to make sense of the pattern. Different people develop different neural networks because they have been subjected to different learning experiences.