Awhile ago I was thinking about artificial intelligence and creativity. Computers are great at certain things like finding and producing patterns. At the time, it seemed to me that as the development of intelligent machines progresses, the only remaining advantage that humans will have is the ability to create outside of any pattern ("creativity" for the purpose of this thread). This means not only being random (because random events can also form patterns accidentally) but consciously avoiding patterns. On the other hand, maybe even creativity, by this definition, has a pattern. If I am avoiding all known possible patterns, then will my resulting creation not also follow a sort of pattern?

As a more concrete example, consider 2 points on a plot: If we were to place a 3rd point somewhere, where would we place it to maximize the symmetry of the resulting 3-point object? Perhaps we would continue the first 2 points to form a line of 3 points, or perhaps we would use the 3rd point to form an equilateral triangle. These options are shown in the image below, with the original 2 points in black and the (in my opinion) high-symmetry points in red. If you were pushed further, what would you say were the next-highest symmetry positions for the 3rd point? I'd say the orange points but who knows.



The red points seem like the obvious choices to me, but how do we formalize this idea of potential symmetry, PS ("where would adding a point result in the highest symmetry?" )? We would need an algorithm that can pick out regular patterns like lines and grids, as well as amorphous patterns such as rings of n points. The way I've decided to do this is by using NEXAFS (Near Edge X-Ray Absorption Fine Structure), or at least a simulation of it. NEXAFS works by diffracting X-rays off of a cluster of atoms. These X-ray photons which have scattered off of at least one atom each reach some detector where their intensity is measured. If the distance between atoms is roughly the same as the photon wavelength, then the photons will interfere constructively, while they will interfere destructively if the inter-atomic distance is roughly half a wavelength. The scattering is performed with a range of wavelengths, and the resulting wavelength dependence of the detected NEXAFS spectrum represents information pertaining to the structural symmetry of the system.

So, to the point, I use a calculation inspired by NEXAFS to calculate the PS (i.e. the symmetry resulting from adding a point) over a 2D range. This is the result (red is high symmetry, blue is low symmetry):


The initial 2 points are at (0,0) and (0,1). As you can see, the PS is high at the expected 3rd-point locations (equilateral triangle and 3-point line), but the PS plot is far more rich than I could have predicted. I might have guessed that there would be PS peaks at the points forming a square with the first 2 points, but to my surprise that is not the case. At least one aspect (the sharp boundary at radius=3) is probably an artifact of the calculation method. Furthermore, there are many ways in which the calculation could be improved but, since my initial goals were vague and the results are already more fruitful than expected, I don't have much reason to push onward. This calculation of PS will work with any combination of starting points. As another example, here is a PS plot for 3 initial points at (0,0), (0,1), (1,0) (an L shape) (again, red is high symmetry, blue is low symmetry):



So, to address my original question, if I wanted to make a computer program that produces a collection of points that avoids geometrical patterns, I would pick two random points and calculate the PS plot. Then I would pick a 3rd point from the low-symmetry (blue) points and calculate the PS plot for the 3 current points. Then I would pick a 4th point from the low-symmetry points... and so on.

EDIT:
After trying some other combinations of initial points, I found that the scattering algorithm could be improved in various ways. The images above have been subsequently updated.

Hmm. This is interesting, I could see it having some fun applications down the line.

What does the plot look like zoomed out more? I imagine that as the distance from the center of the plot increased, the symmetry should 'smear,' with less severe gradations between red and blue. As the distance from the center increases, smaller and smaller increases in angular rotation produce larger shifts (imagine a rotating right triangle).

Eventually it should reach a point where the new point is so far from the original two that the symmetry is essentially negligible.

Blessings
~ND

First of all, images in the previous post have been updated based on an improved algorithm.

I think that large-scale structure will eventually fade away because, at larger and larger distances, the initial 2 points will essentially blur into each other and we will see the PS pattern resulting from 1 initial point, which is uniform:



Here is the 2-point PS plot zoomed out at different levels. As you can see there is large scale periodic structure which seems to fade away over distance. The bottom-left plot is somewhat different: To find the PS at a point, I integrate over the simulated intensity spectrum, which can encompass as many wavelengths as I care to calculate. Normally I let the wavelength range be from 0 to two times the max distance between the points (2 initial points plus the potential point) used to simulate intensity, but if I fix the wavelength range to something like 0 to 4, I get results like the bottom-right plot.



Here is something new: The PS plot for 3 initial points forming an equilateral triangle (it is slightly saturated to improve contrast). Notice the point at (1,1.5) which completes a diamond shape.



Finally, an animated gif depicting the 3-point PS plot as the 3rd point is shifted from (0.1,0.5) to (1,0.5).