Percolation.
I made a movie recently for my advisor. The movie is so pretty, that I thought I’d share it here: may I present to you randomly drawn dots, where two dots are the same color when they touch!
I’ll be a bit more explicit: a dot is drawn at a random location; if it does not overlap any previous dots, it gets a new color. Otherwise, the dot takes the color of the component it touches. Sometimes a new dot connects many components, and in this case, the new component takes on the color of the largest among the old components.
There’s a lot of neat questions to be asked about such a process: for instance, after drawing n dots, how many components should we expect to see? As you can see in the movie, when you draw only a few dots, most of those dots are isolated and have their own color; but after drawing a ridiculously large number of dots, they are all connected and the same color. And inbetween, something more interesting happens.
Here’s an example of “something more interesting” taken from a much larger picture than the above movie:
Posted: July 20th, 2008 under Personal, Questions, Mathematics.
Comments: 4
Comments
Comment from Carlos
Time: November 4, 2008, 10:30 am
What happens when a dot overlaps two dots of different color at the same time? Which color is chosen over the other? When it overlaps three? My question is leading to this: when there are a lot of dots, why don’t we see color transitions (red-blue-green) on much of the dots somewhat haphazardly? The color of the dots seems to tend to green.
Comment from Carlos
Time: November 4, 2008, 10:49 am
If you change the criterion to “color the component upon overlap is the color of the smallest component” what happens? Do you still get convergence to a single color? Is it slower or faster? What if you change it to “color is in the average of the two overlapped components” - do you get a gray color upon convergence (does it?)?
Comment from Carlos
Time: November 4, 2008, 10:50 am
What color does it change to if, in the small chance, the overlap is between two components of the same number?
Comment from Carlos
Time: November 15, 2008, 7:38 pm
Here’s an interesting thing to consider: the way you’ve decided on coloring the components, (information-theoretical) entropy (diversity) regarding color decreases, because you end up with one color. With another scheme, does it increase or decrease?
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