> When we look at this it can sometimes seem as if there is a pattern emerging from the behaviour. However, if you feel like you can see such a pattern it is entirely coincidental, since the nodes are completely unaware of each other.

bothered me. It seems to me that the fact that patterns can emerge from non-coördinated behaviour is one of the interesting facts about mathematics. See, for a dodgy example—because there is some coördination—firefly flash-locking; and, for a genuine example that I literally just happened to be reading about this morning, the fascinating Nobel-Prize work discussed at https://johncarlosbaez.wordpress.com/2016/10/07/kosterlitz-t... . (I encourage you to go read the latter if you haven't already; as you will expect if you have read Baez's work, it's exposition at a level just about anyone, including non-mathematicians, can probably understand of scientific Nobel-Prize work, and those don't come along very often.) I guess, though, that to make any rigorous statement that this can occur, and isn't just an illusion, one has to make a rigorous definition of 'pattern', and the implicit definition of something like "externally imposed structure" has just as much claim to being 'correct' as anything I could cook up.

For the "memory" example, each particle is moving Uniform(-k, k) at each timestep (where k is some just some fixed distance). So the the distribution of the position of a particle at timestep t is the sum of t identically independent distributions, which will converge to a normal distribution, specifically with Uniform(a, b) having variance (b - a)^2 / 12, you converge to:

N(0, t * k^2 / 3)

So it turns out these particles end up behaving kind of like brownian motion! (Note I didn't take into account hitting walls. I am not sure how to model that?)

For the "velocity" example, empirically you can see that if you just let it run forever the points will just end up bunching at the walls. This makes sense since if there's a positive velocity in either direction it's hard to flip your velocity back into the other direction to get off the wall. By symmetry you expect half to be bunched on one wall and half on the other.

A fun question to ask then is if the velocities will accumulate so much in one direction such that it will always stay stuck on one wall after you run it a long enough time. I believe the answer is no! It will always switch directions again due to the fact that the probability that a random walk on a line returning to the origin has probability 1 as you run it forever (google recurrent random walk in 1 or 2 dimensions). This means the velocity (which is doing the random walk) will have to cross 0 and flip signs at some point. So although you will mostly find these particles hugging the walls, they will keep switching sides forever!

I think there are a lot of other really cool properties in these fancy visualizations waiting to be discovered.

Very nice work. My favorite is this: http://inconvergent.net/generative/trees/

In the human brain, signals are stochastic, so they embed a large portion of randomness with each bit of information. Amazingly, adding noise to signals helps learning. Intelligence is at the edge of order and chaos - the lesson being that chaos is an essential part of it.

If you're interested in these you might also be interested in the book "Computers, Pattern, Chaos, and Beauty":

http://www.worldcat.org/title/computers-pattern-chaos-and-be...

He also has a very interesting Twitter account where he posts his experiments: https://twitter.com/inconvergent